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@@ -0,0 +1,2224 @@
+MNRAS 000, 1–14 (2022)
+Preprint 1 February 2023
+Compiled using MNRAS LATEX style file v3.0
+Tree-based solvers for adaptive mesh refinement code FLASH - IV: An
+X-ray radiation scheme to couple discrete and diffuse X-ray emission
+sources to the thermochemistry of the interstellar medium
+Brandt A. L. Gaches,1,2★ Stefanie Walch,1,3 Richard Wünsch4 and Jonathan Mackey5
+1I. Physikalisches Institut, Universität zu Köln, Zülpicher Straße 77, 50937, Köln, Germany
+2Department of Space, Earth and Environment, Chalmers University of Technology, Gothenburg SE-412 96, Sweden
+3Center for Data and Simulation Science (CDS), University of Cologne, www.cds.uni-koeln.de, Germany
+4Astronomical Institute, Czech Academy of Sciences, Bo˘ciní II 1401, 141 00 Prague, Czech Republic
+5Centre for AstroParticle Physics and Astrophysics, DIAS Dunsink Observatory, Dunsink Lane, Dublin 15, Ireland
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+X-ray radiation, in particular radiation between 0.1 keV and 10 keV, is evident from both point-like sources, such as compact
+objects and T-Tauri young stellar objects, and extended emission from hot, cooling gas, such as in supernova remnants. The
+X-ray radiation is absorbed by nearby gas, providing a source of both heating and ionization. While protoplanetary chemistry
+models now often include X-ray emission from the central young stellar object, simulations of star-forming regions have yet to
+include X-ray emission coupled to the chemo-dynamical evolution of the gas. We present an extension of the TreeRay reverse
+raytrace algorithm implemented in the Flash magneto-hydrodynamic code which enables the inclusion of X-ray radiation from
+0.1 keV < 𝐸𝛾 < 100 keV, dubbed XrayTheSpot. XrayTheSpot allows for the use of an arbitrary number of bins, minimum and
+maximum energies, and both temperature-independent and temperature-dependent user-defined cross sections, along with the
+ability to include both point and extended diffuse emission and is coupled to the thermochemical evolution. We demonstrate the
+method with several multi-bin benchmarks testing the radiation transfer solution and coupling to the thermochemistry. Finally,
+we show two example star formation science cases for this module: X-ray emission from protostellar accretion irradiating an
+accretion disk and simulations of molecular clouds with active chemistry, radiation pressure, protostellar radiation feedback from
+infrared to X-ray radiation.
+Key words: astrochemistry -– radiative transfer -– methods:numerical — ISM:clouds -– X-rays: general — X-rays: ISM.
+1 INTRODUCTION
+Molecular gas is subjected to radiation across the electromagnetic
+spectrum. Hard radiation, such as X-ray and gamma-ray radiation,
+can penetrate deep into molecular gas and drive the thermochemistry
+of dense gas (Spitzer & Tomasko 1968; Maloney et al. 1996; Yan
+1997; Wolfire et al. 2022). X-rays provide an important source of ion-
+ization in dense gas, driving the ion-neutral chemistry and providing
+heating through photo-electrons (Lepp & Shull 1983; Maloney et al.
+1996; Dalgarno et al. 1999). Using the typical molecular gas photoab-
+sorption cross sections (Maloney et al. 1996), the 𝜏 = 1 surface for 1
+keV photons is approximately 4 × 1021 cm−2 (compared to ≈ 10−18
+cm−2 for UV radiation). However, the photoabsorption cross sections
+scale roughly as 𝐸−2.5 (Mackey et al. 2019), so harder radiation pen-
+etrates much further into the cloud. Therefore, in regions near bright
+X-ray sources, the cloud structure can become dominated through-
+out by the X-ray radiation. Regions in which the thermochemistry
+is regulated primarily through X-ray radiation are often denoted as
+X-ray Dominated Regions (XDRs) (coined by Maloney et al. 1996),
+in analogue to photo-dissociation regions (PDRs).
+★ E-mail: [email protected] (BALG)
+X-ray radiation drives ionization primarily through secondary, in-
+duced processes. While the primary ionization cross sections are
+low, the resulting ejected fast electrons can produce a cascade of
+secondary ionizations and pumped far ultraviolet (FUV) radiation
+through the excitation and subsequent de-exictation of H and H2
+(Prasad & Tarafdar 1983; Dalgarno et al. 1999; Meijerink & Spaans
+2005). These fast electrons can also provide heating through pho-
+toelectric heating of dust grains. In this way, X-ray radiation acts
+in a very similar manner as cosmic rays, and untangling the two
+contributions can be difficult (see e.g. Meijerink et al. 2006).
+Molecular clouds are immersed in a bath of X-ray radiation, with
+contributions from both external and internal sources. Externally,
+molecular gas can be irradiated through supernovae and their rem-
+nants (e.g. Yamane et al. 2018; Brose et al. 2022), X-ray binaries
+(e.g. White et al. 1988; Remillard & McClintock 2006; Reig 2011;
+Mineo et al. 2012; Lutovinov et al. 2013; Giacobbo et al. 2018),
+nearby activate galactic nuclei (AGN) (e.g. Sunyaev et al. 1993; Sun-
+yaev & Churazov 1998; Harada et al. 2013; Churazov et al. 2017;
+Mingozzi et al. 2018; Cruz-González et al. 2020). Internally, young
+stellar objects, including embedded accreting protostars and more
+evolved T-Tauri stars (Calvet & Gullbring 1998; Feigelson & Mont-
+merle 1999; Feigelson et al. 2007), and high-mass stars just reaching
+© 2022 The Authors
+arXiv:2301.13237v1  [astro-ph.IM]  30 Jan 2023
+
+2
+Gaches et al.
+the main sequence can become X-ray bright (e.g. Cassinelli et al.
+1994), whether through accretion or magnetic powered radiation or
+coronal emission. Finally, gas heated through feedback processes,
+such as winds and supernovae, can become warm enough to emit
+X-ray radiation while they cool (Raymond & Smith 1977). Observa-
+tional X-ray surveys of molecular gas and star-forming regions show
+substantial amounts of diffuse emission and a sizable number of point
+sources (Sunyaev et al. 1993; Feigelson et al. 2013; Townsley et al.
+2014, 2019).
+Despite their potential importance, their inclusion into simulations
+of molecular clouds has been sparse. There has been substantial focus
+on thermochemical models of protoplanetary disks (e.g. Glassgold
+et al. 1997; Igea & Glassgold 1999; Ercolano et al. 2008a, 2009;
+Owen et al. 2011; Meijerink et al. 2012; Cleeves et al. 2017; Picogna
+et al. 2019; Waggoner & Cleeves 2019) and models of molecular
+gas near external sources or compact objects (e.g. Krolik & Kallman
+1983; Lepp & McCray 1983; Draine & Woods 1991; García-Burillo
+et al. 2010; Hocuk & Spaans 2010; Meijerink et al. 2011; Odaka
+et al. 2011; Orlando et al. 2011; Mackey et al. 2019). These methods
+typically utilize Monte Carlo methods (Ercolano et al. 2008a; Odaka
+et al. 2011; Molaro et al. 2016; Walls et al. 2016; Cleeves et al. 2017),
+or ray-trace schemes and focus primarily either on the inclusion of
+point sources or external radiation fields (e.g. Wise & Abel 2011;
+Mackey et al. 2019; Khabibullin et al. 2020)
+In this paper, we will present an X-ray extension of the reverse
+ray tracing scheme TreeRay (Wünsch et al. 2021), which allows
+for the inclusion of an arbitrary number of point sources and diffuse
+radiation. The module is a TreeRay extension of the diffuse X-ray
+module presented in Mackey et al. (2019), which enabled diffuse X-
+ray irradiation at the domain boundary. Our implementation enables
+up to 100 energy bins at arbitrary locations between 0.1 and 100
+keV and temperature-dependent photoabsorption cross sections. In
+Section 2 we give an overview of the X-ray TreeRay algorithm,
+called XRayTheSpot, and the coupling of it to the X-ray-driven
+chemistry. In Section 3 we show the performance of the module with
+different radiation transfer tests and a benchmark against the Cloudy
+code. In Sections 4 and 5 we demonstrate the use of this module for
+protostellar emission irradiating a surrounding disk and in a star
+formation simulation, respectively. Finally, in Section 6 we discuss
+the future extensions and scientific applications of XRayTheSpot.
+2 METHODS
+Our new XRayTheSpot module is able to treat radiation from
+0.1 keV to 100 keV. We describe below in detail the adopted pho-
+toabsorption cross sections and the module’s implementation within
+TreeRay.
+2.1 XRay Cross Sections
+X-ray radiation is attenuated as it propagates through gas via a com-
+bination of photoionization, at lower energies, and the Compton pro-
+cess, at high energies. The previous module, described in Mackey
+et al. (2019), used the low-energy approximation for the X-ray cross
+section, 𝜎𝑥, from Panoglou et al. (2012):
+𝜎𝑥 = 2.27 × 10−22𝐸−2.485
+𝛾
+cm2
+(1)
+per H-nucleus, where 𝐸𝛾 is the photon energy. However, this cross
+section is valid only for cold, neutral gas and for solar metallicity.
+We include now, as input during run time, temperature-dependent
+cross sections, which can be re-computed for problems with different
+metallicities. For photoionization, we use the analytic fits from Verner
+& Yakovlev (1995), 𝜎pi, where
+𝜎pi = 𝜎0𝐹(𝐸𝛾/𝐸0),
+(2)
+where
+𝐹(𝑦) =
+�
+(𝑦 − 1)2 + 𝑦2
+𝑤
+�
+𝑦−𝑄 �
+1 +
+√︁
+𝑦/𝑦𝑎
+�−𝑃
+,
+(3)
+𝑦 = 𝐸𝛾/𝐸0, 𝑄 = 5.5 + 𝑙 − 0.5P, 𝑙 = 0, 1, 2 is the subshell orbital
+quantum number, and 𝜎0, 𝐸0, 𝑦𝑤, 𝑦𝑎 and P are fit parameters from
+the associated public ViZieR catalog1. However, to utilize these cross
+sections, the ionization level populations must be known. We assume
+collisional ionization equilibrium and use the ChiantiPy package
+(Dere 2013), using version 9 of the Chianti atomic database (Dere
+et al. 1997, 2019) to compute the ionization fraction as a function of
+temperature.
+Figure 1 shows the equilibrium ionization fractions as a function
+of temperature for the 15 different elements (see Table 1) we include
+in the cross sections. These computations show that, particularly for
+𝑇 > 105 K, there are multiple ionization states for metals which
+contribute to the photoionization cross section.
+We also include the cross section for the Compton effect, which
+becomes particularly important at higher energies. We use the total
+Klein-Nishina (KN) cross section (Klein & Nishina 1929; Longair
+2011), 𝜎KN,
+𝜎KN = 𝜋𝑟2
+𝑒𝑥−1
+��
+1 − 2(𝑥 + 1)
+𝑥2
+�
+ln(2𝑥 + 1) + 1
+2 + 4
+𝑥 −
+1
+2(2𝑥 + 1)2
+�
+,
+(4)
+where 𝑟𝑒 is the classical electron radius and 𝑥 = 𝐸𝛾/(𝑚𝑒𝑐2). While
+most applications will be in the limit of Thomson scattering, we
+include the full Compton cross sections to enable more flexibility
+in the choice of energy bins. In the cross-section plots, we show the
+total Compton cross-section weighted by the number of free electrons
+contributed by each species. The total cross section, 𝜎𝑥, is thus
+𝜎𝑥(𝐸) =
+𝑁elem
+∑︁
+𝑖
+𝑥𝑖𝜎pi,i(𝐸) + 𝜎KN,i(𝐸),
+(5)
+where 𝑥𝑖 is the abundance of element 𝑖 with respect to hydrogen and
+the sum is carried out including the cross sections for the 𝑁elem ele-
+ments. Table 1 shows the elements we include in the photoabsorption
+cross section and their fiducial abundances relative to hydrogen.
+Figure 2 shows the photoionization cross sections and free-electron
+contributions to the Compton cross section as a function of energy for
+gas with𝑇 = 105 K. As shown, for some elements, the Compton cross
+section becomes more important than photoionization, in particular
+for hydrogen and helium above 1 keV, and for carbon and oxygen
+above 30 keV. For hydrogen, the Compton effect is dominant due to
+the negligible neutral fraction at T = 105 K. Figure 3 shows the total
+cross section as a function of energy for 𝑇 = 105 K and each of the
+total elemental contributions. Here, the X-ray photoabsorption cross
+section is dominated by helium (< 0.3 keV), then carbon (0.3 − 0.6
+keV) and oxygen (0.8−4 keV). At energies above 4 keV, the hydrogen
+and helium Compton cross sections dominate with a contribution
+from the iron photoionization cross section around 10 keV. However,
+many of the metals contribute equally to the total cross section around
+1 keV.
+Finally, Figure 4 shows 𝜎𝑥 as a function of energy and temperature
+from 𝑇 = 104 to 107 K. At low temperatures, we recover the analytic
+1 https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+AS/109/125
+MNRAS 000, 1–14 (2022)
+
+XRayTheSpot: X-raying Molecular Gas
+3
+Table 1. Elements included in our photoabsorption cross section calculation
+and their fiducial abundances, 𝐴𝑋, reported as 𝐴𝑋 = log(𝑁𝑥/𝑁𝐻) + 12
+(Asplund et al. 2009).
+Element
+Abundance (𝐴𝑋)
+H
+12
+He
+10.986
+C
+8.443
+O
+8.783
+N
+7.913
+Ne
+8.103
+Na
+6.353
+Mg
+7.593
+Al
+6.523
+Si
+7.573
+S
+7.193
+Ar
+6.553
+Ca
+6.383
+Fe
+7.503
+Ni
+6.283
+cross section previously used, although around 10 keV there is an in-
+crease in the cross section due to iron. However, the rather significant
+temperature dependence highlights the necessity of including a tem-
+perature dependent cross section: at high temperatures, the higher
+thermal ionization state leads to a reduction of nearly two orders of
+magnitude in the cross section, thereby making the gas significantly
+more optically thin to the X-ray radiation, producing less heating and
+enabling more X-rays to escape. Below 104 K, the cross section does
+not noticeably change since hydrogen is not significantly collisional
+ionized. Therefore, for the results of this paper, for colder gas, we
+use the 𝑇 = 104 photoabsorption cross section. The module though
+allows for the user to define their own temperature dependent cross
+sections across any temperature range.
+For a given set of energy bins, {(𝐸𝑙,𝑖, 𝐸𝑟,𝑖)}, where 𝑖 = 1, 𝑁bin
+for 𝑁bin bins, we define:
+𝐸𝑐,𝑖 = 1
+2
+�𝐸𝑙,𝑖 + 𝐸𝑟,𝑖
+� ,
+(6)
+where 𝐸𝑙,𝑖 is the left bound of the 𝑖th bin, 𝐸𝑟,𝑖 is the right bound,
+and 𝐸𝑐,𝑖 is the midpoint of the bin. We derive bin-averaged cross
+sections, such that
+exp
+�
+− ⟨𝜎𝑋,𝑖⟩
+𝜎𝑐
+�
+=
+1
+𝐸𝑙,𝑖 − 𝐸𝑟,𝑖
+∫ 𝐸𝑟,𝑖
+𝐸𝑙,𝑖
+exp
+�
+−𝜎𝑥(𝐸𝛾)
+𝜎𝑐
+�
+𝑑𝐸,
+(7)
+where ⟨𝜎𝑥,𝑖⟩ is the bin-averaged cross section for bin, 𝑖, and 𝜎𝑐 =
+𝜎𝑥(𝐸𝑐,𝑖). Our fiducial tests use 𝑁bin = 8 between 1 – 10 keV using
+logarithmically spaced bins.
+All of these cross section data, and the initialization and storage
+of the bins and bin-averaged cross sections are kept in a new Flash
+module, XrayCommon. Flash is a highly module public magneto-
+hydrodynamic code (Fryxell et al. 2000) written in Fortran and
+highly-scalable with MPI. The scripts necessary to compute the X-
+ray cross sections are publicly available on GitHub2. This module
+enables the coupling of X-ray physics to multiple other modules.
+Plasma models and the necessary X-ray data are also stored within
+this module, as a unified location.
+2.2 TreeRay
+TreeRay is a novel reverse ray tracing scheme, described fully in
+2 � https://github.com/AstroBrandt/XRayCrossSections
+Wünsch et al. (2021), implemented in Flash. Simply, TreeRay
+enables an efficient method to compute the contributions of radiation
+from every cell, for every cell. It does so through the combination of
+a reverse ray-trace algorithm with a tree (Wünsch et al. 2018), which
+also currently is used in the gravity solver. Below we describe briefly
+the different aspects of the TreeRay algorithm and XRayTheSpot
+extension and refer the reader to Wünsch et al. (2021) for more
+details.
+2.2.1 Building the Tree
+The foundation of the TreeRay algorithm is an octtree which stores
+all necessary variables for the various TreeRay modules. At min-
+imum, the tree stores the mass and center of mass coordinates for
+the respective cell, or leaf, or higher nodes. For XRayTheSpot, two
+further quantities are stored onto the tree: the bin-integrated X-ray
+luminosity in each energy bin and the gas temperature. While the
+bin-integrated X-ray luminosity is purely additive, the temperature is
+stored as a mass-weighted average of each set of eight sub-nodes (or
+leaves).
+2.2.2 Ray Structure
+Before the tree walk is executed for a given cell, rays are generated by
+casting 𝑁pix = 12𝑁2
+side rays from each cell using directions defined
+by the HealPix (Górski et al. 2005) algorithm. HealPix tessellates
+the unit sphere into areas representing equal solid angles with a unit
+vector pointing to the center of each of these surface areas from the
+sphere’s center. TreeRay allows for 𝑁side = 1, 2, 4, 8, ..., with higher
+values representing higher angular resolution. The rays are split into
+𝑁𝑟 evaluation points, set by the grid resolution, Δ𝑥, the allowed
+length of the ray, 𝐿ray, which is set to three-dimensional diagonal of
+the computational domain, and a free parameter, 𝜂𝑅.
+Along each ray, the radial coordinate point of the ith evaluation
+point is
+𝑟𝑖 = Δ𝑥𝑖2
+2𝜂2
+𝑅
+,
+(8)
+leading to segments with increasing lengths. This behavior coincides
+well with the geometric acceptance criterion described below for
+deciding whether or not to accept a tree node. The total number of
+evaluation points is
+𝑁𝑅 = 𝜂𝑅 × floor
+�√︂
+2𝐿ray
+Δ𝑋
+�
++ 1.
+(9)
+2.2.3 Tree Walk
+The mapping of the cells/nodes onto the rays requires two factors: a
+multipole acceptance criterion (MAC) and a weighting function to
+map from the tree onto the different radial evaluation points. When
+the MAC is met, a node is accepted and used. The simplest MAC is
+the Barnes-Hut (BH) geometric MAC (Barnes & Hut 1986), where
+a node of size ℎ𝑛, at a distance 𝑑, from the cell is opened if
+ℎ𝑛/𝑑 < 𝜃lim
+(10)
+where 𝜃lim is a user-defined opening angle with a sensible choice
+being 𝜃lim =
+√︃
+4𝜋/𝑁pix3. We also utilize the ‘Src MAC’ (Wünsch
+3 The resulting 𝜃lim for 𝑁side = 1, 2, 4, 8 is 1.0, 0.5, 0.25, 0.125, respec-
+tively. For the results of this paper, we adopted these recommended values of
+MNRAS 000, 1–14 (2022)
+
+4
+Gaches et al.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ionization Fraction
+I
+II
+I
+II
+III
+IV
+V
+VI
+I
+II
+III
+IV
+V
+VI
+VII
+VIII
+H
+He
+C
+O
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ionization Fraction
+I
+II
+III
+IV
+V
+VI
+VII
+I
+II
+III
+IV
+V
+VI
+VII
+VIII
+IX
+X
+I
+II
+III
+IV
+V
+VI
+VII
+VIII IX X
+XI
+XII
+XIII
+I
+II
+III
+IV
+V
+VI VII
+VIII
+IX
+X
+XI
+XII
+N
+Ne
+Si
+Mg
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ionization Fraction
+I
+II
+III
+IV
+V
+VI
+VII
+VIII IX X XIXIIXIIIXIV
+XV
+I
+II
+III
+IV
+V
+VI
+VII
+VIII
+IX
+X XI
+XIIXIII
+XIV
+XV
+I
+II
+III
+IV
+V
+VI
+VII
+VIII
+IX
+X
+XI
+I
+II
+III
+IV
+V
+VI
+VIIVIII
+IX
+X
+XI
+XII
+S
+Fe
+Na
+Al
+104
+105
+106
+107
+Temperature (K)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Ionization Fraction
+I
+II
+III
+IV
+V VIVII
+VIII
+IX
+X XI XIIXIIIXIVXV
+I
+II
+III
+IV
+V
+VI
+VIIVIIIIX
+X
+XI
+XIIXIIIXIVXV
+I
+II
+III
+IV
+V
+VI
+VII VIII IX
+X
+XI
+XIIXIII
+XIV
+XV
+Ar
+Ca
+Ni
+Figure 1. Ionization fraction for different elements as a function of temperature. Annotated in the text are the peaks of different ionization levels for each element.
+MNRAS 000, 1–14 (2022)
+
+XRayTheSpot: X-raying Molecular Gas
+5
+10
+15
+10
+12
+10
+9
+10
+6
+10
+3
+ (Mbarn/H-nucleus)
+Photoionization
+Compton
+H
+He
+C
+O
+10
+10
+10
+8
+10
+6
+10
+4
+ (Mbarn/H-nucleus)
+N
+Ne
+Si
+Mg
+10
+11
+10
+9
+10
+7
+10
+5
+ (Mbarn/H-nucleus)
+S
+Fe
+Na
+Al
+102
+103
+104
+105
+Energy (eV)
+10
+10
+10
+8
+10
+6
+ (Mbarn/H-nucleus)
+Ar
+Ca
+Ni
+Figure 2. Photoionization (solid) and Compton process (dashed) cross sec-
+tions for each element as a function of energy, assuming thermal ionization
+equilibrium at 𝑇 = 105 K. Each elemental contribution is weighted by the
+assumed abundance with respect to hydrogen.
+et al. 2021), where a node with sources is opened if
+ℎ𝑛/𝑑 < 𝜃src,
+(11)
+where 𝜃src is a user-defined parameter.
+Quantities on the tree are mapped onto the radial evaluation points
+of a ray through the use of kernels. We utilize both a piece-wise
+third-order polynomial, ����𝑝(𝛿), and a kernel derived to ensure it
+meets the requirements of the radiation transfer equation, 𝑊 𝑓 (𝛿),
+where 𝛿 = (𝑟𝑖 − 𝑑)/ℎ𝑛 and 𝑑 is the distance from the node center
+of mass and the ray evaluation point (see Wünsch et al. 2021) and
+ℎ𝑛 is the node’s linear size. The node quantites are weighted by the
+overlap of the volume of the ray segment and the node.
+𝜃lim for the corresponding 𝑁side. See Wünsch et al. (2018) for 𝜃lim resolution
+tests in the context of TreeRay/OpticalDepth
+102
+103
+104
+Energy (eV)
+10
+10
+10
+8
+10
+6
+10
+4
+10
+2
+tot (Mbar/H-nucleus)
+H
+He
+C
+O
+N
+Ne
+Si
+Mg
+S
+Fe
+Na
+Al
+Ar
+Ca
+Ni
+Figure 3. Total photo-absorption cross section (black) with each element
+contribution highlighted (colors) as a function of energy for gas at temper-
+ature, 𝑇 = 105 K. Each elemental contribution is weighted by the assumed
+abundance with respect to hydrogen.
+102
+103
+104
+105
+Energy (eV)
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+Cross section (Mbarn)
+103
+104
+Energy (eV)
+10
+6
+10
+5
+10
+4
+Cross section (Mbarn)
+4.0
+4.5
+5.0
+5.5
+6.0
+6.5
+7.0
+Temperature
+Figure 4. Total photoabsorption cross section as a function of energy and
+temperature (color). The black dashed line shows the previously used analytic
+cross section from Panoglou et al. (2012). Inset: Zoom-in to 1 – 10 keV.
+Following the tree walk, the rays from a cell outward store the
+mass, center of mass, gas temperature and bin-integrated luminosi-
+ties. These provide all the necessary information to solve the equation
+of radiation transfer along each ray.
+2.2.4 Solving the Radiation Transfer Equation
+Along the rays, the one-dimensional radiation transfer equation is
+solved:
+𝑑𝐼𝜈
+𝑑𝑠 = −𝜖𝜈 + 𝛼𝜈𝐼𝜈
+(12)
+where 𝑠 is the distance along the ray, and 𝜖𝜈 and 𝛼𝜈 are the emission
+and absorption coefficients. The band-integrated flux, 𝐽(𝐸), irradiat-
+MNRAS 000, 1–14 (2022)
+
+6
+Gaches et al.
+ing a cell 𝑖 due to the band-integrated luminosity, 𝐿𝑋, emitting from
+node 𝑗 can be simply written
+𝐽 𝑗𝑖(𝐸) = 𝐿𝑋, 𝑗 (𝐸) 𝑒−𝜏𝑥
+4𝜋𝑟2
+𝑖 𝑗
+(13)
+where 𝑟𝑖 𝑗 is the distance between the centers of cell 𝑖 to node 𝑗 and
+𝜏𝑥 =
+∫
+𝑛H(𝑠)𝜎𝑥(𝐸,𝑇(𝑠))𝑑𝑠 ≈
+∑︁
+𝑘
+𝜌𝑘
+𝜇𝑚𝐻
+⟨𝜎𝑋 (𝐸,𝑇𝑘)⟩ 𝛿𝑠
+(14)
+is the X-ray opacity between cell 𝑖 and node 𝑗 and 𝜌𝑘 is the density
+at evaluation point 𝑘 along the ray, 𝑛H is the hydrogen nuclei density,
+and 𝛿𝑠 = 𝑟𝑘 − 𝑟𝑘−1. We store the solution as an energy density,
+𝜀𝑥 = 𝐽/𝑐, where 𝑐 is the speed of light, onto the grid to be used in
+chemistry, described below. The total energy density is the sum over
+the HealPix rays:
+𝜀𝑖 =
+𝑁pix
+∑︁
+𝑘
+𝜀𝑘𝑖(𝐸).
+(15)
+For the solution, we also store the cell mass and temperature onto
+the tree and map these to the rays using 𝑊𝑝. In our algorithm, no
+assumption is made with respect to what produces the X-ray energy
+density, enabling both point sources (with their radiation spread over
+their host cells) and diffuse emission produced via cooling of hot gas
+in the cell.
+2.3 Pre-existing Chemistry
+We briefly describe here the previous treatment of X-ray radiation
+within the chemical network (see Mackey et al. (2019) for more
+details). The chemical network consists of 17 species, of which 9 are
+solved numerically and the rest are followed through conservation
+equations. We solve the non-equilibrum species H+, H2, C+, CO,
+HCO+, CHx, OHx, He+ and M+. CHx is a proxy species for simple
+hydrocarbons, e.g. CH, CH2, etc, and simple ions CH+, CH2+, etc.
+Similarly, OHx is a proxy for OH, H2O and ions OH+, H2O+, etc. M
+is a proxy for metals that can become the primary source of electrons
+in shielded regions of molecular clouds, where reaction rates treat
+M as Si. The network is primarily based on the ‘NL99’ network of
+Glover & Clark (2012), which uses the hydrogen chemistry from
+Glover & Mac Low (2007a,b) with the CO chemistry of Nelson &
+Langer (1999) including updated reaction rates from Gong et al.
+(2017). For this work, all photodissociation rates have been updated
+using the KIDA astrochemistry database (Wakelam et al. 2012).
+X-ray radiation is coupled to the thermochemistry through the
+following primary processes (see also Mackey et al. 2019):
+• Dust heating, following the analytic prescription in Yan (1997).
+• Primary ionization of a species by X-rays. Note though that is is
+relatively unimportant for our considered species, and plays a minor
+role in the heating and ionization for hydrogen species and helium.
+• Secondary ionization through collisional ionization by fast elec-
+trons produced following primary ionizations (e.g. Dalgarno et al.
+1999; Meijerink & Spaans 2005).
+• Induced FUV radiation generated by H2, which is collisionally
+excited by fast electrons and the subsequent ionizations and dissoci-
+ations (Prasad & Tarafdar 1983; Gredel et al. 1987; Maloney et al.
+1996; Meijerink & Spaans 2005).
+• Coulomb heating of the gas via energy exchange between the
+produced fast electrons and other charged particles (Dalgarno et al.
+1999).
+These processes have all been generalized for the arbitrary number
+of energy bins and the temperature-dependent cross sections. The
+input X-rays are computed by the XRayTheSpot module. Since we
+use band-integrated radiative variables, the heating parameter for a
+particular cell 𝑖 due to the impinging X-ray radiation is
+𝐻𝑥,𝑖 =
+𝑁bin
+∑︁
+𝑛
+𝑗𝑖(𝐸𝑛) ⟨𝜎𝑥(𝐸𝑛,𝑇𝑖)⟩ .
+(16)
+3 TESTS AND BENCHMARKING
+Here we show various numerical tests of the radiation transfer and a
+benchmark of the thermochemsitry against Cloudy. For our bench-
+marks, we fiducially use 8 bins, logarithmically spaced between 1 -
+10 keV, following Meijerink & Spaans (2005).
+3.1 Point Source Test
+Our first test is a single central point source with a constant luminosity
+distribution, 𝐿𝑥,𝑛 = 1 L⊙ for all 𝑁bin bins, embedded in a volume
+with a uniform density of 𝑛(𝐻) = 2 × 103 cm−3 and a spatially
+constant temperature 𝑇 = 10 Kelvin in a (30 pc)3 volume. We use
+a constant luminosity to better compare the solutions of different
+energy bins. We then compute the radial profiles of the energy density
+and compare against the analytic solution:
+𝐽(𝐸, 𝑟) = 𝐿𝑥(𝐸) 𝑒−𝜎𝑥 (𝐸)𝜌𝑟
+4𝜋𝑟2
+(17)
+where the energy density, 𝜖 = 𝐽(𝐸)/𝑐.
+Figure 5 shows the performance of XRayTheSpot for a single
+bright point source as a function of radius. The results in figure 5
+used 𝜂𝑅 = 4, 𝑁side = 8 and 𝑁block = 8, where 𝑁block is the number of
+blocks of cells per spatial dimension, and one block consists of a cube
+of 83 cells. The radial range was chosen that for the lowest energy bin,
+the emission transitions from optically thin to strongly optically thick,
+with the maximum radius corresponding to 𝜏(𝐸 = 1.17eV) ≈ 10.
+The left panel shows the comparison between the ray trace solution
+and the analytic solution. These solutions agree well with each other
+with the lines largely overlapping. The right panel shows the relative
+error, defined as
+𝛿𝑐 = |𝑐𝜀 − 𝐽(𝐸, 𝑟)|
+𝐽(𝐸, 𝑟)
+(18)
+where 𝜀 is the solution from XRayTheSpot. The relative error is
+rather insensitive to the optical depth but more sensitive to how
+strongly the radiation field is coupled to the gas (e.g. the magnitude
+of the photoabsorption cross section). The 10% error shown for the
+most optically thick bin at low energies is due to the mapping of
+the density structure onto the rays using the kernel. For X-ray optical
+depths greater than 𝜏𝑥 ≈ 10, the error starts to increase towards unity,
+but at these energy densities, the X-rays have a negligible impact on
+the thermochemistry. Therefore, these relative differences will have
+no discernible impact on the thermochemical evolution of the gas.
+In order to highlight the differences of the new module with the
+previous cross section implementation presented in Mackey et al.
+(2019), we perform a second calculation imposing a strong temper-
+ature gradient such that the radial temperature profile is
+𝑇(𝑟) = 5 × 105 [1 − tanh(𝑟 − 8 pc)] + 100 K.
+(19)
+This temperature is artificial and chosen such that the X-ray radiation
+transitions from optically thin to optically thick due to the change in
+MNRAS 000, 1–14 (2022)
+
+XRayTheSpot: X-raying Molecular Gas
+7
+𝑁block
+𝑁side
+𝜂𝑅
+Initialization (s/proc)
+Evolution (s/proc)
+4
+2
+2
+2.3
+1.3
+4
+4
+2
+5.5
+3.6
+4
+4
+4
+6.1
+4.6
+4
+8
+2
+22.1
+14.0
+8
+4
+2
+17.4
+29.3
+8
+8
+4
+94.0
+167.2
+Table 2. Timing for the pont source test for the different runs in Figure 7.
+Each row gives the model parameters of 𝑁block, 𝑁side and 𝜂𝑅 and the time
+in seconds per processor for the initialization of the tree and a ray trace step.
+cross section. Figure 6 shows the result of this comparison and as
+expected, the emission for the lower energy bins is up to an order of
+magnitude greater than the low-temperature solution and maintains
+an 𝑟−2 trend until a much greater radius.
+Figure 7 shows the performance of XRayTheSpot for a range of
+parameters, exploring both low- and high- spatial and ray resolutions.
+We find that grid resolution is the primary source of deviations at
+small radius, while the ray resolution increases the accuracy at larger
+radii. At large distances from the source, the solution tends to slightly
+under predict for low ray and angular resolution due to overestimation
+of the column density. At small radii, the solution over-predicts the
+resulting flux. For optically thin radiation bins, the solution almost
+exactly matches the analytic. Therefore, the deviations come about
+due to mapping the mass from the cells and tree nodes onto the rays
+using the kernel.
+For science uses, the number of blocks, 𝑁block, is determined by
+the necessary resolution to resolve crucial gas dynamics (e.g. the
+Jeans length for gravity simulations). Increasing both 𝑁side and 𝜂𝑅,
+while producing more accurate radiation transfer solutions, leads to
+substantially higher computational costs. Table 2 shows the compu-
+tational time per processor for the initialization and per ray trace step
+for the models in Figure 7. Between the lowest and highest accuracy
+tests, (𝑁block, 𝑁side, 𝜂𝑅) = (4, 2, 2) and (8, 8, 4), respectively, the
+increase in cost of the initialization and ray trace was a factor of ≈
+40 and ≈ 130, respectively. The time for the raytrace is dominated
+(≥ 95%) by the tree walk. We find using 𝑁side = 4 is the best balance
+of time and accuracy.
+3.2 Shadow Test
+Our next test is a shadow test to verify the solution of the solver when
+sources are placed near dense regions. Here, we have a 𝐿𝑋 = 10 L⊙
+source with a spectrum, 𝑑𝐿/𝑑𝐸 ∝ 𝐸−2, placed near a dense core with
+a hydrogen-nuclei number density of 𝑛H = 103 cm−3. We consider
+radiation between 1 - 10 keV, moving from optically thick bands to
+optically thin. Figures 8 and 9 show the results of this test, for both
+low- and high- ray resolution which use (𝑁block = 8, 𝑁side = 4,
+𝜂𝑅 = 2) and (𝑁block = 8, 𝑁side = 8, 𝜂𝑅 = 4), respectively. For both
+cases, the test reveals the expected results that the low-energy X-rays
+are absorbed by the dense core and this creates a wide-angle shadow,
+while higher energy X-rays are barely attenuated, producing smaller
+to no shadows. The high-ray resolution test also shows the expected
+drop in ray-tracing artifacts.
+Figure 10 shows a one-dimensional cut along the z-axis from the
+source through the dense blob of gas for the two ray resolutions
+compared. The figure shows that higher ray resolution leads to a
+smoother attenuation of the flux for the optically thick, lower energy
+bins while there is very little change for higher energy bins which
+are substantially less attenuated. This is most pronounced for the
+𝐸𝑐 = 1.33 keV bin
+3.3 Benchmark against Cloudy
+The final benchmark tests the X-ray radiation coupling to the ther-
+mochemistry. We place a point source with a physical size of 1016
+cm and total X-ray luminosity, 𝐿XR = 1036 erg s−1, and a luminosity
+spectrum 𝑑𝐿/𝑑𝐸 ∝ 𝐸−2 between 1 - 10 keV in a (1.3 pc)3 volume
+filled with a gas number density of 𝑛H = 103 cm−3, 𝑁side = 4,
+and 𝜂𝑅 = 2. and compare with a one-dimensional model using the
+Cloudy code. The volume and resolution were chosen such the inner
+XDR is resolved by ≈ 20 cells while the optically thick regime is
+also traced. In particular, we use 7 maximum adaptive-mesh reso-
+lution levels refining on the density and temperature, such that the
+maximal resolution is 1.3 × 10−3 pc. The one-dimensional Cloudy
+model used a “sphere” geometry, an input power-law spectrum be-
+tween 1 - 10 keV for the X-ray radiation source, a cosmic microwave
+background and a cosmic-ray ionization rate of 3 × 10−17 s−1. Fur-
+ther, we turn off grain physics, induced radiative processes, radiation
+pressure, radiation scattering, outward line radiation transfer and
+molecule freeze-out, since the Flash simulations do not have these
+processes. Finally, we set refractory metal abundances to zero, with
+the exception of silicon which Flash uses as the proxy for metals
+for the chemistry (as described above). The Cloudy script used is
+shown in Appendix A.
+Using uniform-spaced grids, even with substantial AMR levels
+and, it is in practice difficult to fully capture sharp thermochemical
+transition regions, such as that shown below as captured by Cloudy.
+Further, the source encompasses several cells at the highest reso-
+lution, rather than an infinitely small point source. Capturing such
+ionization and dissociation fronts entirely is numerically intensive
+and generally requires the use of one-dimensional models tailored to
+do so (as with Cloudy).
+Figure 11 shows the result of this benchmark, using . Near the
+source, the temperature and chemistry solutions well match the
+Cloudy solution. The Flash and Cloudy solutions qualitatively
+reproduce the chemical structure, although due to the larger cell-size
+of the Flash grids, the sharp HI transition seen at 𝑁(𝐻) ≈ 5 × 1020
+cm−2 is not fully captured and is instead smoothed over a few cells.
+The temperature solutions agree within a factor of a few. However,
+Cloudy solves the line cooling and level excitations in a much more
+robust manner than the included Flash thermochemistry, including
+a full non-equilibrium solution with many more electronic and ion-
+ization states. Such inclusions though are not numerically feasible
+for in-situ thermochemistry in three-dimensional MHD simulations.
+Further, Cloudy solves the full radiation transfer solution from radio
+through X-ray radiation with substantially more bins. Given the con-
+straints of these physics, the found solution is deemed to be adequate
+and matches the overall trends as determined by Cloudy.
+4 PROTOSTELLAR DISK
+Evolved protostellar objects, in particular Class II objects in which the
+lack of a surrounding gaseous envelope leaves the central protostar
+and disk exposed, are known to be X-ray emitters. These X-rays
+can become important for disk dynamics and planet formation (e.g.
+Ercolano et al. 2008b; Mohanty et al. 2013). For these stars, the
+X-ray emission is thought to come from a combination of accretion
+and magnetospheric emission (Hartmann et al. 2016). As a first test
+science case, we model the X-ray radiation transport from a central
+protostar into a protostellar disk.
+The surface density follows from the often used truncated power-
+law (e.g. Lynden-Bell & Pringle 1974; Andrews et al. 2011; Cleeves
+MNRAS 000, 1–14 (2022)
+
+8
+Gaches et al.
+100
+101
+r (pc)
+10
+20
+10
+18
+10
+16
+10
+14
+ (erg/cm3)
+1.17 keV
+1.56 keV
+2.07 keV
+2.77 keV
+TreeRay
+3.69 keV
+4.92 keV
+6.56 keV
+8.75 keV
+Analytic
+100
+101
+E (keV)
+10
+24
+10
+23
+10
+22
+ (cm
+2)
+1.17 keV
+1.56 keV
+2.07 keV
+2.77 keV
+TreeRay
+3.69 keV
+4.92 keV
+6.56 keV
+8.75 keV
+Analytic
+2
+4
+6
+8
+10
+r (pc)
+10
+4
+10
+3
+10
+2
+10
+1
+100
+Relative Error 
+c
+Figure 5. Radial profile test for a single source in a constant density and temperature medium. Left: Radiation density versus radius for each bin for the TreeRay
+(solid) and analytic solution (dotted). Left inset: Bin-averaged cross sections (black points) and the analytic cross section in Eq. 1. Right: Relative errors for each
+bin as a function of radius.
+100
+101
+r (pc)
+10
+20
+10
+18
+10
+16
+10
+14
+ (erg/cm3)
+1.17 keV
+1.56 keV
+2.07 keV
+2.77 keV
+3.69 keV
+4.92 keV
+6.56 keV
+8.75 keV
+Const T
+Temp Grad
+Const T
+Temp Grad
+103
+104
+105
+106
+Temperature (K)
+Figure 6. X-ray energy density versus radius for single point source. The
+solid line uses the constant temperature at 𝑇 = 10 K (same as Figure 5) while
+the dashed-dotted line uses the temperature profile shown by the red dotted
+line.
+et al. 2016):
+Σ𝑔(𝑅) = Σ𝑐
+� 𝑅
+𝑅𝑐
+�−𝛼
+exp
+�
+−
+� 𝑅
+𝑅𝑐
+�2−𝛼�
+(20)
+between an inner and outer radius, 𝑅in and 𝑅out, respectively, 𝑅𝑐
+is the critical radius where the surface density distribution becomes
+exponential, 𝛼 is the power law index and Σ𝑐 is the characteristic
+surface where the disk transitions to an exponential profile. For the
+initial conditions, we assume the gas is in hyrostatic equilibrium,
+such that the density follows
+𝜌𝑔(𝑅, 𝑧) = Σ𝑔(𝑅)
+√
+2𝜋ℎ
+exp
+�
+−
+� 𝑧2
+2ℎ2
+��
+(21)
+where ℎ = 𝑐𝑠/Ω is the disk scale height, 𝑐𝑠 =
+√︃ 𝛾𝑘𝑏𝑇𝑔
+𝜇𝑚H , 𝑘𝐵 is
+Boltzmann’s constant, 𝛾 = 5/3 is the adiabatic index, 𝑇𝑔 is the gas
+temperature, 𝜇 = 2.33 is the mean mass per particle for molecular gas,
+𝑚𝐻 is the mass of the hydrogen atom, Ω = 3
+4
+√︃
+𝐺𝑀∗
+𝑅3
+is the Keplerian
+rotational frequency and 𝑀∗ is the mass of the central protostellar
+object. For this fiducial test, we set 𝑀∗ = 0.7 M⊙, Σ𝑐 = 64 g cm−2,
+𝑅𝑐 = 100 AU, 𝛼 = 1. The temperature profile is given by
+𝑇(𝑅) = max
+�
+𝑇0
+�
+𝑅
+1𝐴𝑈
+�−0.5
+, 10 K
+�
+,
+(22)
+where we fiducially take 𝑇0 = 50 K. The disk is initialized to be
+rotating in Keplerian motion around the central protostellar object.
+We assume the disk is magnetized with an initial toroidal field such
+that the ratio of the magnetic to thermal pressure, 𝜇𝑀 = 10−5. We
+simulate the domain in a 240 AU box with a maximal resolution of
+1 AU.
+The central protostar is put in by hand, with active accretion. For
+the X-ray emission, we assume an accretion floor of 10−9 M⊙ yr−1,
+similar to rates observed in young stellar objects (e.g. Ingleby et al.
+2013). The simulation is run using the Bouchut-5 MHD solver, grav-
+ity, and XRayTheSpot. The X-ray emission is derived by assuming
+there is an accretion shock, with properties following “hot spot” ac-
+cretion (Hartmann et al. 2016) with accretion columns filling 10% of
+the protostar surface, thermally emitting X-ray emission. The ther-
+mal X-ray emission is computed using a one-temperature Raymond-
+Smith plasma model (Raymond & Smith 1977). The implementation
+of a coronal model is left for a future work.
+Figure 12 shows a slice of the density, gas temperature, X-ray
+emission at 1.17 keV (1st bin) and 6.56 keV (8th bin), the heating
+rate per H nucleus, 𝐻𝑥, and 𝐻𝑥/𝑛, which is often used as a diagnostic
+for the importance of the X-ray heating (Wolfire et al. 2022). We find
+that the lowest energy X-rays are all absorbed near the protostar or
+escape through the outflow. However, the harder X-rays at 6.56 keV
+are able to permeate much of the domain. The 𝐻𝑋 and 𝐻𝑥/𝑛 slices
+MNRAS 000, 1–14 (2022)
+
+XRayTheSpot: X-raying Molecular Gas
+9
+101
+2 × 100
+3 × 100
+4 × 100
+6 × 100
+r (pc)
+10
+20
+10
+18
+10
+16
+10
+14
+ (erg/cm3)
+Nblock = 4, Nside = 4,
+R = 4
+1.17 keV
+1.56 keV
+2.07 keV
+2.77 keV
+TreeRay
+3.69 keV
+4.92 keV
+6.56 keV
+8.75 keV
+Analytic
+2
+4
+6
+8
+10
+r (pc)
+10
+3
+10
+2
+10
+1
+100
+c
+1.17 keV
+1.56 keV
+2.07 keV
+2.77 keV
+TreeRay
+3.69 keV
+4.92 keV
+6.56 keV
+8.75 keV
+Analytic
+101
+2 × 100
+3 × 100
+4 × 100
+6 × 100
+r (pc)
+10
+20
+10
+18
+10
+16
+10
+14
+ (erg/cm3)
+Nblock = 4, Nside = 2, 
+R = 2
+2.5
+5.0
+7.5
+10.0
+r (pc)
+10
+3
+10
+2
+10
+1
+100
+c
+101
+2 × 100
+3 × 100
+4 × 100
+6 × 100
+r (pc)
+10
+21
+10
+19
+10
+17
+10
+15
+10
+13
+ (erg/cm3)
+Nblock = 4, Nside = 4, 
+R = 2
+2.5
+5.0
+7.5
+10.0
+r (pc)
+10
+3
+10
+2
+10
+1
+100
+c
+101
+2 × 100
+3 × 100
+4 × 100
+6 × 100
+r (pc)
+10
+20
+10
+18
+10
+16
+10
+14
+ (erg/cm3)
+Nblock = 4, Nside = 8, 
+R = 2
+2.5
+5.0
+7.5
+10.0
+r (pc)
+10
+3
+10
+2
+10
+1
+100
+c
+101
+2 × 100
+3 × 100 4 × 100
+6 × 100
+r (pc)
+10
+21
+10
+19
+10
+17
+10
+15
+10
+13
+ (erg/cm3)
+Nblock = 8, Nside = 4, 
+R = 2
+2.5
+5.0
+7.5
+10.0
+r (pc)
+10
+3
+10
+2
+10
+1
+100
+c
+101
+2 × 100
+3 × 100 4 × 100
+6 × 100
+r (pc)
+10
+20
+10
+18
+10
+16
+10
+14
+ (erg/cm3)
+Nblock = 8, Nside = 8, 
+R = 4
+2.5
+5.0
+7.5
+10.0
+r (pc)
+10
+3
+10
+2
+10
+1
+100
+c
+Figure 7. . Energy density versus radius for the different model parameters, annotated in the top left of each subfigure. Inset: Relative error, 𝛿𝑐, of the numerical
+solution against the analytic solution as a function of radius from the source.
+MNRAS 000, 1–14 (2022)
+
+10
+Gaches et al.
+15
+10
+5
+0
+5
+10
+15
+y (pc)
+Density
+1.33 keV
+1.78 keV
+15
+10
+5
+0
+5
+10
+15
+y (pc)
+2.37 keV
+3.16 keV
+4.22 keV
+10
+0
+10
+x (pc)
+15
+10
+5
+0
+5
+10
+15
+y (pc)
+5.62 keV
+10
+0
+10
+x (pc)
+7.50 keV
+10
+0
+10
+x (pc)
+10.00 keV
+0
+2
+4
+nH (cm
+3)
+18.0
+17.5
+17.0
+16.5
+16.0
+15.5
+15.0
+14.5
+14.0
+Fi (erg cm
+2 s
+1)
+Figure 8. Shadow test, consisting of a point source illuminating a constant
+density core. Top left corner: Number density distribution for a z-axis slice.
+Others: X-ray flux in the given energy band for a z-axis slice using 𝑁block = 8,
+𝑁side = 4, 𝜂𝑅 = 2.
+15
+10
+5
+0
+5
+10
+15
+y (pc)
+Density
+1.33 keV
+1.78 keV
+15
+10
+5
+0
+5
+10
+15
+y (pc)
+2.37 keV
+3.16 keV
+4.22 keV
+10
+0
+10
+x (pc)
+15
+10
+5
+0
+5
+10
+15
+y (pc)
+5.62 keV
+10
+0
+10
+x (pc)
+7.50 keV
+10
+0
+10
+x (pc)
+10.00 keV
+0
+2
+4
+nH (cm
+3)
+18.0
+17.5
+17.0
+16.5
+16.0
+15.5
+15.0
+14.5
+14.0
+Fi (erg cm
+2 s
+1)
+Figure 9. Same as Figure 8, but with 𝑁block = 8, 𝑁side = 8 and 𝜂𝑅 = 4.
+100
+101
+zsrc (pc)
+0
+10
+18
+10
+17
+10
+16
+10
+15
+10
+14
+e (erg/cm3)
+1.33 keV
+1.78 keV
+2.37 keV
+3.16 keV
+4.22 keV
+5.62 keV
+7.50 keV
+10.00 keV
+Fiducial
+High Res.
+Fiducial
+High Res.
+100
+101
+102
+103
+104
+Hydrogen Density (cm
+3)
+Figure 10. X-ray energy density versus distance along the z-axis from the
+source for the shadow test. The solid line uses the ray resolution in Figure 8 and
+the dashed-dot uses the ray resolution in Figure 9. The dotted red line shows
+the hydrogen nuclei density highlighting the location of the high-density blob.
+clearly show that the disk midplane is left relatively unheated by the
+X-rays, although the X-rays become important in the cavity and outer
+disk regions. In particular, most of the cavity exhibits very warm gas,
+even with only X-ray emission included, due to the rapid absorption
+of soft X-ray emission. The cavity heats to temperatures exceeding
+104 Kelvin, potentially becoming bright in hydrogen recombination
+lines. The inclusion of EUV radiation will heat the diffuse gas further,
+along with further ionizing the surrounding low-density cavity.
+5 MOLECULAR CLOUD
+We present an example application for XRayTheSpot, to demon-
+strate how all the different TreeRay energy bands work together: a
+virialized, magnetized turbulent cloud. We consider a 2 pc region of
+a molecular cloud resolved with 2563 cells. We produce an initial
+turbulent field by stirring the domain with a flat power spectrum
+between the largest wave modes 𝑘 = 1...3 for 10 crossing times at
+a velocity dispersion of 0.72 km s−1, consistent with the observed
+linewidth-size relationship (McKee & Ostriker 2007). During the stir-
+ring, we use periodic boundary conditions and chemistry to achieve
+more accurate initial conditions for the abundances before collapse.
+The choice of stirring for 10 crossing times is to ensure the chemistry
+has reached a more quiescent state, with the kinetic energy spectrum
+generally being reached after two crossing times (Federrath et al.
+2010). We assume the cloud is nearly virialized, such that the virial
+parameter
+𝛼 ≡ 5𝜎2𝑅
+𝐺𝜌𝐿3 = 2
+(23)
+where 𝑅 = 𝐿 is the box length, resulting in 𝜌 = 5 × 10−21 (g cm−3)
+and a total box mass of 𝑀 = 590 M⊙. Before stirring, we initialize
+a magnetic field in the 𝑧-axis with a magnitude such that the plasma
+beta,
+𝛽 ≡
+𝜌𝑐2𝑠
+𝐵2/8𝜋 = 103.
+(24)
+After the turbulence is initialized, gravity and source particles (stars)
+are turned on and the boundary conditions are changed to “diode”
+MNRAS 000, 1–14 (2022)
+
+XRayTheSpot: X-raying Molecular Gas
+11
+1019
+1020
+1021
+Hydrogen Column Density
+101
+102
+103
+104
+Temperature (K)
+Cloudy
+Flash
+1019
+1020
+1021
+Hydrogen Column Density
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+Abundances
+H/Htot
+2H2/Htot
+Figure 11. Flash vs Cloudy benchmark. Left: Temperature versus hydrogen column density from the central point source for Flash (black) and Cloudy
+(blue). Right: Atomic (solid) and molecular (dashed) hydrogen abundances versus total hydrogen column density from the source, where Htot = H+ + H + 2H2.
+t = 100 yr
+10
+20
+10
+19
+10
+18
+10
+17
+10
+16
+10
+15
+10
+14
+102
+103
+104
+10
+10
+10
+9
+10
+8
+10
+7
+10
+6
+10
+23
+10
+22
+10
+21
+10
+20
+10
+19
+10
+18
+10
+17
+10
+29
+10
+28
+10
+27
+10
+26
+10
+25
+10
+24
+10
+23
+10
+10
+10
+9
+10
+8
+10
+7
+10
+6
+Figure 12. Protostellar disk example case usage. Top row: Slice plots at 𝑧 = 0 for the density (left), gas temperature (middle) and 1.17 keV radiation energy
+density. Bottom row: X-ray heating rate, H𝑥 (left), H𝑥/n diagnostic term (middle) and 6.56 keV radiation energy density.
+such that gas can flow out of the domain. During the simulation, the
+cloud is irradiated by an FUV radiation field of 𝜒 = 1.7 in units
+of the Habing field (Habing 1968). The simulation is run using the
+chemistry described above, and all TreeRay modules:
+• OpticalDepth for the external radiation field (Wünsch et al.
+2018). OpticalDepth solves for the column density from a cell to
+the external boundary and attenuates a prescribes external radiation
+flux (𝜒 = 1.7). In this study, it is only used for the FUV radiation,
+while (Mackey et al. 2019) implemented the ability to include an
+impinging X-ray flux.
+• OnTheSpot for the EUV emission (Wünsch et al. 2021). This
+module solves for UV-ionizing radiation from arbitrary sources and
+MNRAS 000, 1–14 (2022)
+
+12
+Gaches et al.
+iterates to convergence. The UV photon flux is coupled to the ther-
+mochemistry to model photochemistry.
+• RadPressure to account for the thermal radiation and radiation
+pressure (Klepitko et al. 2022). This module enables the inclusion
+of thermal radiation from point and diffuse sources and the resulting
+radiation pressure. The thermal radiation is included in the chemistry
+through radiative dust heating.
+• XRayTheSpot, described above.
+Sink particles representing protostars are injected when the den-
+sity exceeds 𝜌thresh ≥ 4.59×10−18 g cm−3. Further criteria are used:
+there are checks to ensure a local gravitational potential and a con-
+verging flow. The protostar evolution follows the Offner et al. (2009)
+model and implemented in Flash (Klepitko et al. 2022). Protostellar
+emission consists of the intrinsic and accretion luminosities, where
+the total accretion luminosity is
+𝐿acc = 𝑓acc
+𝐺𝑀∗ �𝑀∗
+𝑅∗
+,
+(25)
+where 𝑀∗ is the mass of the protostar,
+�𝑀∗ is the accretion rate,
+𝑅∗ is the protostar’s radius and we take 𝑓acc = 0.33. The X-ray
+spectrum was computed by assuming hot-spot accretion, described
+above, which provides the temperature and the density of the accre-
+tion shocks near the protostellar surface (Calvet & Gullbring 1998;
+Hartmann et al. 2016) and a single temperature plasma model (Ray-
+mond & Smith 1977). Due to the low resolution, we set a minimum
+of �𝑀∗ = 10−9 M⊙ yr−1. This is needed since when the protostar
+particles first form, the burst of accretion blows out HII regions, and
+the low resolution inhibits resolving the proper structure around the
+cores. The infrared to EUV spectrum, used for RadPressure and
+OnTheSpot is computed assuming the emission is composed of two
+blackbodies: one for the intrinsic spectrum of the protostar at the
+photosphere, such that
+𝑇∗ =
+�
+𝐿∗
+4𝜋𝜎sb𝑅2∗
+�1/4
+,
+(26)
+which is provided by the protostellar evolution model, and another
+assuming the accretion luminosityisreprocessedprimarily as a black-
+body with temperature 𝑇acc, such that
+𝑇acc =
+�
+𝐿acc
+4𝜋𝜎sb𝑅2∗
+�1/4
+,
+(27)
+where 𝜎sb is the Stefan-Boltzmann constant. Therefore, the total
+infrared luminosity from the protostar is described as
+𝐿∗,IR = 𝑓∗,IR(𝑇∗)𝐿∗ + 𝑓acc,IR(𝑇acc)𝐿acc
+(28)
+and the EUV luminosity as
+𝐿∗,EUV = 𝑓∗,EUV(𝑇∗)𝐿∗ + 𝑓acc,EUV(𝑇acc)𝐿acc
+(29)
+where 𝑓IR(𝑇) and 𝑓UV(𝑇) are the fraction of the blackbody emis-
+sion in each of these bands (𝐸 < 13.6 eV and 13.6 eV ≤ 𝐸 ≤ 100
+eV, respectively). The X-ray emission was computed assuming the
+“hot-spot” model, described above. While there may be some double
+counting of emission by treating the total spectrum in the two differ-
+ent methods, we find this impact is marginal as the X-ray emission
+generally accounts for only a small fraction (≤ 10%) of the total
+protostellar luminosity.
+Figure 13 shows the column density, and density-weighted projec-
+tions of the gas and temperature, radiation temperature, EUV photon
+density and X-ray energy densities after ≈ 1 Myr of evolution with
+gravity. The star formation, as traced by heated knots of gas, is oc-
+curring along a main filament structure. The high temperatures here
+are primarily caused by the EUV photons, which are rapidly ab-
+sorbed in the nearby gas. The X-ray emission is found to be highly
+absorbed along the main filament structure, and instead traces out the
+more diffuse turbulent structure of the molecular cloud. As expected,
+higher energy X-ray bands showcase more extended emission with
+the brightest emission in the 3.7 keV band. In all X-ray bands, the tur-
+bulent structure of the molecular cloud is seen in the density-weighted
+integrated emission. This case study highlights the new capabilities
+of including protostellar radiative feedback from infrared to X-ray in
+star formation simulations.
+6 DISCUSSION/FUTURE WORK
+We have presented the new X-ray radiation transfer module,
+XRayTheSpot using the reverse ray-tracing scheme TreeRay im-
+plemented in Flash (Wünsch et al. 2021). XRayTheSpot enables
+an arbitrary number of point or diffusive sources of X-ray emission,
+and an arbitrary number and position of energy bins. The module
+uses temperature dependent cross sections assuming gas in thermal
+ionization equilibrium. However, the module is flexible enough such
+that the user can provide their own cross section data to be used. The
+module produces the expected behavior for X-ray point sources and
+shadow tests and is able to reasonably reproduce the thermochemistry
+compared to Cloudy, despite the significantly simpler treatment of
+X-ray chemistry and grain-processes in Flash.
+We demonstrated the utility of this module with two example sci-
+ence cases focusing on protostellar X-ray emission. First, we mod-
+elled the emission of an 0.7 M⊙ protostar with an accretion rate of
+10−9 M⊙ yr−1 through a protostellar disk. We find that soft X-rays
+are rapidly absorbed at the disk surfance, with most of the emission
+escaping through the outflow cavity. However, harder X-rays are able
+to permeate the disk due to their significantly lower optical depth.
+The X-ray heating was also strong within the outflow cavity, with no
+X-ray heating towards the midplane of the disk, as expected. Second,
+we perform a low-resolution star formation simulation of a turbulent
+molecular cloud. In this simulation, protostars are self-consistently
+formed and the X-ray emission modelled on the fly. This simulation
+includes the entire range of different TreeRay radiation modules:
+diffuse FUV (OpticalDepth Wünsch et al. (2018)), EUV (OnTheS-
+pot Wünsch et al. (2021)), thermal radiation and radiation pressure
+(RadPressure Klepitko et al. (2022)) and X-ray emission from 1
+keV to 10 keV. Since the X-ray emission in the simulation comes en-
+tirely from accretion onto the protostars, the X-ray emission is highly
+variable. Due to the lower resolution and the inclusion of ionizing
+radiation, the accretion occurs in bursts followed by the expansion of
+HII regions, which cut off accretion. With higher resolution, accre-
+tion may still be able to occur through disks, instabilities and more
+porous density structures. In future work, we will perform higher
+resolution simulations to model star formation including chemistry
+and radiation feedback across the electromagnetic spectrum.
+In this work, we focus primarily on point sources. However,
+XRayTheSpot makes no differentiation between point sources ver-
+sus extended more diffusion emission. Future studies will include
+diffuse X-ray emission from cooling hot gas and shocked gas. Our
+module currently includes the computation of X-ray emission from
+accretion onto protostars, and future work will include X-ray models
+for more types of point sources such as X-ray binaries. The module
+presented in this work will allow the first-generation of simulations
+of star formation and galaxies with the inclusion of a wide range of
+X-ray sources.
+MNRAS 000, 1–14 (2022)
+
+XRayTheSpot: X-raying Molecular Gas
+13
+t = 1.08 Myr
+10
+2
+10
+1
+100
+101
+102
+10
+15
+20
+25
+30
+8
+10
+12
+14
+16
+10
+10
+10
+8
+10
+6
+10
+4
+10
+2
+10
+22
+10
+21
+10
+20
+10
+19
+10
+18
+10
+17
+10
+19
+10
+18
+10
+17
+10
+19
+10
+18
+10
+17
+10
+16
+10
+18
+10
+17
+10
+16
+Figure 13. Panel plots highlighting the features of a 2 pc piece of a molecular cloud after 𝑡 = 1.08 Myr of evolution. For all fields except the column density, the
+panel is showing the density-weighted projection. All projections are along the z-axis. The figure shows a simulated molecular cloud after 1 Myr of gravitational
+evolution including protostar sink particles and radiation feedback from infrared to X-rays. While the EUV radiation is rapidly absorbed (indicated by the black
+background color), the infrared and X-ray emission is able to penetrate much further into the cloud.
+ACKNOWLEDGEMENTS
+BALG and SWG acknowledges support by the ERC starting grant
+No. 679852 ‘RADFEEDBACK’. SWG and BALG thank the German
+Science Foundation (DFG) for funding through SFB956 project C5.
+We also thank the Regional Computing Center Cologne (RRZK)
+for hosting our HPC cluster, Odin, on which the simulations have
+been performed. RW acknowledges the support by project 20-
+19854S of the Czech Science Foundation and by the institutional
+project RVO:67985815. JM acknowledges support from a Royal
+Society-Science Foundation Ireland University Research Fellowship
+(20/RS-URF-R/3712) and an Irish Research Council Starting Lau-
+reate Award (IRCLA\2017\83). The authors thank Andre Klepitko
+for many helpful discussions. Andre Klepitko also implemented the
+protostellar evolution model into the code. The software used in this
+work was in part developed by the DOE NNSA-ASC OASCR Flash
+Centre at the University of Chicago (Fryxell et al. 2000). The fol-
+lowing Python packages were utilized: NumPy (Harris et al. 2020),
+SciPy (Virtanen et al. 2020), Matplotlib (Hunter 2007), yt (Turk
+et al. 2011), ChiantiPy (Dere 2013).
+DATA AVAILABILITY
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+APPENDIX A: CLOUDY BENCHMARK SCRIPT
+We present the Cloudy script which was used for the X-ray bench-
+marking. The Cloudy model consists of a uniform density medium
+with 𝑛H = 103 solved using the “sphere” command. We turn off most
+induced and grain processes and set the abundances for most metals
+to zero to better match the methods used in our Flash simulations.
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–14 (2022)
+
+XRayTheSpot: X-raying Molecular Gas
+15
+1 t i t l e XDR source
+2 ## r a d i a t i o n
+s o u rces
+3 CMB
+4 t a b l e SED " plaw . sed "
+5 l u m i n o s i t y
+35 range
+73.5
+to
+735 Ryd
+6 ##Geometry
+7 r a d i u s
+16.1938
+8 hden
+3.0
+9 sphere
+10 ## Stopping
+and
+i t e r a t e
+11 stop H2 column
+d e n s i t y
+24.0
+12 stop
+t e m p e r a t u r e
+l i n e a r
+3.0
+13 i t e r a t e
+to
+convergence
+14 ##ISM and
+Grain
+physics
+15 cosmic
+ray
+r a t e
+−16.523
+16 abundances ISM
+17 g r a i n s ISM no
+qheat
+0.56
+18 no
+g r a i n x−ray
+t r e a t m e n t
+19 no induced
+p r o c e s s e s
+20 no
+r a d i a t i o n
+p r e s s u r e
+21 no
+s c a t t e r i n g
+o p a c i t y
+22 no
+g r a i n
+p h y s i c s
+23 no
+g r a i n
+molecules
+24 no
+l i n e
+t r a n s f e r
+25 ##Abundances
+26 element
+carbon
+abundance
+−3.853872
+27 element
+helium
+abundance −1
+28 element
+oxygen
+abundance
+−3.494850
+29 element
+s i l i c o n
+abundance −7
+30 element
+n i t r o g e n
+o f f
+31 element
+s u l p h u r
+o f f
+32 element
+neon
+o f f
+33 element
+aluminium
+o f f
+34 element
+phosphor
+o f f
+35 element
+c h l o r i n e
+o f f
+36 element
+argon
+o f f
+37 element
+calcium
+o f f
+38 element
+chromium
+o f f
+39 element
+n i c k e l
+o f f
+40 element
+l i t h i u m
+o f f
+41 element
+b e r y l l i u m
+o f f
+42 element
+f l u o r i n e
+o f f
+43 element
+potassium
+o f f
+44 element
+scandium
+o f f
+45 element
+t i t a n i u m
+o f f
+46 element
+vanadium
+o f f
+47 element
+manganese
+o f f
+48 element
+c o b a l t
+o f f
+49 element
+copper
+o f f
+50 element
+zinc
+o f f
+51 ## output
+52 save
+overview
+l a s t
+" xdr . ovr "
+53 save
+molecules
+l a s t
+" xdr . mol "
+54 save
+abundances
+l a s t
+" xdr . abund "
+55 save
+continuum
+l a s t
+" xdr . cont "
+56 save PDR l a s t
+" xdr . pdr "
+Listing 1: Input file for Cloudy benchmark
+MNRAS 000, 1–14 (2022)
+

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+Algebraic Model Management: A Survey
+Patrick Schultz1, David I. Spivak1, and Ryan Wisnesky2
+1 Massachusetts Institute of Technology
+2 Conexus AI
+Abstract. We survey the field of model management and describe a
+new model management approach based on algebraic specification.
+1
+Introduction
+In this paper we survey the field of model management and describe a new
+model management approach based on techniques from the field of algebraic
+specification, with the hope of establishing an interlingua between the two fields.
+By “model management” we mean “meta-data intensive” database management
+in the sense of Bernstein & Melnik [4], which we define in Section 2. By “a new
+algebraic model management approach” we mean our particular way [19] [20] of
+specifying database schemas and instances using algebraic (equational) theories.
+We first noticed a connection between model management and algebraic spec-
+ification while investigating applications of category theory [3] to data integra-
+tion [5]. These investigations are described in [19] and [20], and we present no
+substantial new results in this paper. We assume readers have basic proficiency
+with category theory [3], algebraic specification [17], and SQL.
+Outline. In Section 2 we describe the traditional approach to model manage-
+ment and in Section 3 we describe our algebraic approach. Also in Section 3 we
+describe the open-source CQL (Categorical Query Language) tool, available for
+download at http://categoricaldata.net, which implements our approach in
+software. We conclude in Section 4 by comparing our approach with the tradi-
+tional approach.
+2
+Model Management
+To quote from Melnik [16]:
+Many challenging problems facing information systems engineering in-
+volve the manipulation of complex metadata artifacts, or models, such
+as database schemas, interface specifications, or object diagrams, and
+mappings between models. The applications that solve metadata ma-
+nipulation problems are complex and hard to build. The goal of generic
+model management is to reduce the amount of programming needed
+to develop such applications by providing a database infrastructure in
+which a set of high-level algebraic operators, such as Match, Merge, and
+Compose, are applied to models and mappings as a whole rather than
+to their individual building blocks.
+arXiv:2301.04846v1  [cs.LO]  12 Jan 2023
+
+In the paragraph above the word “model” is defined to mean a metadata ar-
+tifact such as a schema, which conflicts with the definition of the word “model”
+as a structure satisfying a theory. In this paper, we use the phrase “model man-
+agement” to mean the field identified above, and use the word “model” to mean
+a structure satisfying a theory.
+Today model management is a large sub-field of information management
+with a research literature containing hundreds of published articles [4]. There is
+a consensus in that literature [4] that model management is concerned with at
+least the problems described in the next sections.
+2.1
+Schema mapping
+Given two database schemas S and T, the schema mapping problem [7] is to
+construct a “mapping” F : S → T that captures some user-specified relationship
+between S and T. Different model management systems use different notions
+of schema, including SQL, XML, and RDF [4]. The most common mapping
+formalism studied in the literature is that of “embedded dependencies” (EDs) [5]:
+formulae in a fragment of first-order logic with useful computational properties.
+We will use SQL schemas and EDs in our examples in this section. Consider
+the following SQL schema S, consisting of two tables connected by a foreign key:
+CREATE TABLE N2(ID INT PRIMARY KEY, age INT)
+CREATE TABLE N1(ID INT PRIMARY KEY, name STRING, salary INT,
+f INT FOREIGN KEY REFERENCES N2(ID))
+and the following SQL schema T, consisting of one table:
+CREATE TABLE N(ID INT PRIMARY KEY, age INT,
+name STRING, salary INT).
+These two SQL schemas are displayed graphically in Figure 1.
+An example schema mapping F : S → T expressing that the target table N
+is the join of source tables N1 and N2 along the column f is:
+∀id1, id2, a, n, s. N1(id1, n, s, id2) ∧ N2(id2, a) → N(id1, a, n, s).
+Two instances satisfying the above ED are shown in Figure 1. In general, many
+EDs can map between two SQL schemas.
+2.2
+Query generation
+Given a schema mapping F : S → T, the query generation problem [4] is to
+construct a query which converts databases on S to databases on T in a way
+that satisfies F. The query languages typically studied include SQL, XQuery,
+and various comprehension- and λ-calculi [4].
+A SQL query to implement the example mapping from Section 2.1 is:
+
+String
+◦
+N1•
+name
+�
+salary
+�
+f
+� N2•
+age
+�
+◦
+Int
+F
+−−−→
+String
+◦
+N•
+name
+�
+age
+�
+salary � ◦
+Int
+N1
+ID name salary f
+1 Alice $100 1
+2
+Bob $250 2
+3
+Sue
+$300 3
+N2
+ID age
+1 20
+2 20
+3 30
+�∆F �
+←−−−−−−
+�ΠF �,�ΣF �
+−��−−−−−−−−→
+N
+ID name salary age
+1 Alice $100 20
+2
+Bob $250 20
+3
+Sue
+$300 30
+Fig. 1. Example Data Migrations, with Foreign Keys (see Sections 2.1, 3.2)
+INSERT INTO N
+SELECT N1.ID, N1.age, N2.name, N1.sal
+FROM N1, N2
+WHERE N1.f = N2.ID
+Technically, the INSERT portion of the above SQL code is not a “query”, but
+rather an “update”, and in practice the code generated from a query generation
+task will often store the results of the query. An example of running the above
+SQL is shown as the left-to-right direction of Figure 1. In general, many or no
+SQL queries may implement a set of EDs [5]. EDs can also be directly executed
+by an algorithm called “the chase” [5].
+2.3
+Mapping Inversion
+Given a schema mapping F : S → T, the mapping inversion problem [10] is to
+construct a schema mapping F −1 : T → S that undoes F with respect to query
+generation (i.e. the queries generated from F and F −1 should be inverses).
+The natural candidate ED to invert the schema mapping of Section 2.1 ex-
+presses that N projects onto N1 and N2:
+∀id1, a, n, s. N(id1, a, n, s) → ∃id2. N1(id, n, s, id2) ∧ N2(id2, a)
+and a possible SQL implementation of this ED is:
+INSERT INTO N1
+SELECT ID, name, sal, ID
+FROM N
+INSERT INTO N2
+SELECT ID, age
+FROM N
+
+However, the above ED is not an inverse to the ED of Section 2.1, as is seen by
+taking ∅ = N1 ̸= N2. Indeed, it is rare for an ED, or set of EDs, to be invertible,
+and weaker notions of inverse, such as “quasi-inverse” [10], are common in the
+literature [10]. An example of running the above SQL is shown as the right-to-left
+direction of Figure 1.
+2.4
+Mapping Composition
+Given schema mappings F : S → T and G : T → U, the mapping composition
+problem [8] is to construct a schema mapping G ◦ F : S → U that is equivalent
+with respect to the query generation problem (i.e. running the query generated
+from G ◦ F should have the same effect as running the query generated from G
+on the results of the query generated from F).
+The composition of the ED from Section 2.1 with the ED from Section 2.3 is
+∀id1, id2, n, s, a. N1(id1, n, s, id2) ∧ N2(id2, a) → ∃x. N1′(id, n, s, x) ∧ N2′(x, a)
+where N1’, N2’ are target “copies” of source tables N1, N2. This composed ED is
+not the identity, thereby showing that the ED from Section 2.3 does not invert the
+ED from Section 2.1. In the case of EDs, composed mappings may not exist [8],
+but some restrictions and extensions of EDs are closed under composition [8].
+2.5
+Schema matching
+Given two database schemas S and T, the schema matching problem [5] is to au-
+tomatically find “correspondences” between S and T and to automatically infer
+schema mappings S → T from these correspondences. In general, inference of en-
+tire mappings cannot be fully automated and the focus of the matching problem
+is to reduce the human effort required to construct a schema mapping by e.g.,
+suggesting partial mappings that can be completed by users. There are many
+techniques for schema matching ranging from comparison of column names by
+string similarity to machine learning algorithms; for an overview, see [5]. In the
+example from Section 2.1, two correspondences that are easy to automatically
+find are (N1, N) and (N2, N) and tools such as Clio [14] can create the ED from
+Section 2.1 from these two correspondences.
+2.6
+Further References
+In this paper we will focus on the problems described in the previous sec-
+tions, but many other problems are studied in the model management litera-
+ture [4], and many of these problems are related to algebraic specification. For
+example, schema/instance merge problems [4], which arise often in data inte-
+gration scenarios [2], can be formalized as pushouts in suitable categories of
+schemas/instances [20], and such pushouts are related to model-theoretic con-
+cepts such as model amalgamation [15].
+
+Many software products solve model management problems [4], including
+ETL (Extract, Transform, Load) tools [5], which extract data from separate
+databases, apply user-specified transformations, and then load the result into
+a target system such as a data warehouse; query mediators [5], which answer
+queries about a “virtual” integrated database by combining queries about sep-
+arate source databases; and visual schema mapping tools [14] which allow users
+to create schema mappings by visually connecting related schema elements with
+lines, as shown in Figure 2.
+There have been at least two attempts to provide a “meta semantics” for
+model management operations. In [16] Melnik gives a “state based” meta se-
+mantics to some of the above operations by defining a schema mapping S → T
+to be an arbitrary binary relation between instances on S and instances on T;
+the ED-based semantics described above is an instantiation of this meta seman-
+tics. In [2] and [13] the authors give an “institution theoretic” meta semantics
+to some of the above operations by defining a schema mapping S → T to be a
+morphism in a suitable category of schemas; CQL’s semantics is an instantiation
+of this meta semantics.
+Fig. 2. A schema mapping in Clio [14]
+
+file:students.xsml (managed)
+口
+Source
+Target
+S File:StudentsSource.xsd
+S File:StudentsTarget,xsd
+qa
+e? targetDB
+e gradEnrolls
+e Evaluations
+* gradEnroll [o,*]
+[* eval [0.*]
+e: sid (xsistring)-
+e: name (xsistring)--
+e: grade (xsiint)
+(buuisisn) pp a
+e? File (μsistring)
+..
+@ Enrollment
+e: File (xsistring)-
+e* Student [0.*]
+e* underGrad [o,*]
+e: sid (xsistring)
+e: name (rsistring)
+.
+(ouuisisi) pis a
+e Courses
+e: name (xsistring)
+e: address (xsistring)
+[* course [o,*]
+e enrolls
+e eid (xsint) ..-
+[ enroll [o,*]
+e: addr (xsistring)
+e cid (xsistring)
+e: sid (rsistring)3
+Algebraic Model Management
+Our approach to model management is based on the algebraic approach to
+databases, data migration, and data integration we describe in [19] and [20].
+Those works, and hence this work, extend a particular category-theoretic data
+model that originated in the late 1990s [11] and was later extended in [21] and [23]
+and implemented in CQL (http://categoricaldata.net).
+In the next section we describe our formalism for database schemas and
+instances and introduce CQL. The subsequent sections implement the model
+management operations from Section 2 using our formalism. In this section we
+abbreviate “algebraic theory” as “theory”.
+3.1
+Algebraic Databases
+In our formalism [20], database schemas and instances are defined as theories of
+a certain kind, which we describe in the next sections. For ease of exposition, we
+will sometimes conflate schemas and instances as defined in our formalism with
+their CQL equivalents.
+Type sides We first fix a theory, Ty, called the type side of our formalism. The
+sorts of Ty are called types and the functions of Ty are the functions that can
+appear in schemas and instances.
+CQL allows arbitrary theories to be used as type sides. But we have found
+that in practice, CQL users almost always want to use the theory of an existing
+programming language, say java, for their type side. The ability to “bind” CQL
+to an existing language is particularly important in model management because
+input data may only be accessible through, e.g., a java API. For this reason,
+CQL allows a type side to be defined by specifying, for each sort s, a java class
+Cs and a java function String → Cs that tells CQL how to interpret the strings
+it encounters in CQL programs as objects of Cs.
+An example CQL type side about integers and strings is shown in Figure 3.
+This type side defines a theory with two sorts and infinitely many constants –
+all the java strings and integers – and no equations. The java code for Int says
+that whenever a string x is encountered in an CQL program and a term of sort
+Int is required, that java’s parseInt function should be applied to x to yield
+the desired Int. The keyword literal, used in many places in CQL, indicates
+a literal (user-defined constant) definition.
+Schemas A schema on type side Ty is a theory extending Ty with new sorts
+(called entities), new unary functions from entities to types (called attributes),
+new unary functions from entities to entities (called foreign keys), and new equa-
+tions (called data integrity constraints) of the form ∀v : s. t = t′, where s is an
+entity and t, t′ are terms of the same type, each containing a single free variable v.
+The restrictions in the preceding sentence (e.g., no functions from types to enti-
+ties) are necessary to use our formalism for model management purposes [19] [20].
+Figure 4 shows the CQL schemas corresponding to Figure 1. These schemas
+contain no equations and are both on the type side Ty defined in Figure 3.
+
+typeside Ty = literal {
+java_types
+String = "java.lang.String"
+Int = "java.lang.Integer"
+java_constants
+String = "return input[0]"
+Int = "return java.lang.Integer.parseInt(input[0])"
+}
+Fig. 3. CQL type side Ty
+schema S = literal : Ty {
+entities
+N1
+N2
+foreign_keys
+f : N1 -> N2
+attributes
+name : N1 -> String
+salary : N1 -> Int
+age : N2 -> Int
+}
+schema T = literal : Ty {
+entities
+N
+attributes
+name : N -> String
+salary : N -> Int
+age : N -> Int
+}
+Fig. 4. CQL schemas S and T on type side Ty
+instance I = literal : S {
+generators
+1 2 3 : N1
+equations
+name(1) = Alice
+salary(1) = 100
+age(f(1)) = 20
+name(2) = Bob
+salary(2) = 250
+age(f(2)) = 20
+name(3) = Sue
+salary(3) = 300
+age(f(3)) = 30
+}
+Fig. 5. CQL instance I on schema S
+Fig. 6. Initial algebra for CQL instance I
+
+Delta - 9:36:11 PM 1 (exec: 0s)(gui: 0s)
+Select:
+Tables
+Type Algebra
+DP
+Text
+typeside Ty
+N1 (3)
+N2 (3)
+schema S
+ID 
+name
+salary
+f
+ID 
+age
+schema T
+[1] 
+Alice
+100
+[1.f]
+[1.f]
+20
+mapping F : S -> T
+[2]
+Bob
+250
+[2.f]
+[2.f]
+20
+instance I : S
+[3] 
+Sue
+300
+[3.f]
+[3.f]
+30Instances An instance I on schema S is a theory extending S with new 0-ary
+function (constant) symbols called generators and non-quantified equations. An
+example CQL instance on schema S (Figure 4) is shown in Figure 5.
+The intended meaning of an instance I, written �I�, is the term model (i.e.,
+initial algebra) for I which contains, for each sort s, a carrier set consisting of the
+closed terms of sort s modulo provability in I. A morphism of instances I → J
+is a homomorphism (natural transformation) of algebras �I� → �J�.
+Figure 6 shows the meaning of the instance I from Figure 5 in the CQL tool.
+The CQL tool visually displays term models as sets of tables, one per entity e,
+each with an ID column corresponding to the carrier set for e. The tables in
+Figure 6 are isomorphic to the left tables in Figure 1.
+In the following sections we implement the model management operations
+from Section 2 using the preceding definitions of schema and instance.
+3.2
+Schema mapping
+Given schemas S, T, the schema mapping problem (Section 2.1) is to construct
+a “mapping” F : S → T that captures some relationship between S and T.
+Let S and T be CQL schemas on the same type side Ty. An CQL schema
+mapping F : S → T is defined as a “derived signature morphism” [18] from
+S to T that is the identity on Ty. That is, F : S → T assigns to each entity
+e ∈ S an entity F(e) ∈ T, and to each attribute / foreign key f : s → s′ a
+term F(f), of type F(s′) and with one free variable of type F(s), in a way that
+respects equality: if S ⊢ t = t′, then T ⊢ F(t) = F(t′). We have found that
+many mappings arising in practice cannot be expressed using plain signature
+morphisms and require the more general notion of “derived” signature morphism.
+Whereas a schema mapping in Section 2.1 was an ED (formula in a fragment
+of first-order logic), which induces a single binary satisfaction relation between
+instances, CQL schema mappings are derived signature morphisms and induce
+three relations between instances, which we will describe in the next section.
+An example CQL schema mapping F : S → T is shown in Figure 7, where
+CQL schemas S and T are defined in Figure 4. This mapping is also shown
+graphically in Figure 1.
+3.3
+Query generation
+Given a mapping F : S → T, the query generation problem (Section 2.2) is to
+use F to construct a query which converts databases on S to databases on T.
+In our formalism, the database instances and morphisms on a schema S
+constitute a category, denoted S–Inst, and a schema mapping F : S → T induces
+a functor ΣF : S–Inst → T–Inst defined by substitution. The functor ΣF has a
+right adjoint, ∆F : T–Inst → S–Inst, which corresponds to the “model reduct
+functor” when our formalism is described in institution-theoretic terms [2]. The
+functor ∆F has a right adjoint, ΠF : S–Inst → T–Inst. See [19] for proof
+that ∆F always has left and right adjoints. As adjoints, ∆F , ΠF preserve limits
+and ∆F , ΣF preserve colimits, implying many useful properties; for example,
+ΣF (I + J) ∼= ΣF (I) + ΣF (J) and ΠF (I × J) ∼= ΠF (I) × ΠF (J).
+
+mapping F = literal : S -> T {
+entities
+N1 -> N
+N2 -> N
+foreign_keys
+f -> lambda x:N. x
+attributes
+name -> lambda x:N. name(x)
+salary -> lambda x:N. salary(x)
+age -> lambda x:N. age(x)
+}
+Fig. 7. CQL schema mapping F : S → T
+Note that unlike Section 2.1, where there was a single query associated with a
+schema mapping (ED), in our algebraic approach there are three queries, one for
+each of ∆F , ΣF , ΠF . The conditions under which ∆F ,ΣF , ΠF can be expressed
+in SQL and vice-versa are characterized in [23].
+Although it is possible to give explicit formulae to define ∆F , ΣF , ΠF [19]
+we instead give examples in Figures 1 and 8. Note that in these examples we are
+not showing instances (theories) as defined in Section 3.1; we are showing term
+models. For this reason, we surround ∆F , ΣF , ΠF with denotation brackets �� in
+these examples. In addition, as adjoints ∆, Σ, Π are only defined up to unique
+isomorphism, so we arbitrarily make up names for IDs and in these examples.
+Figures 1 and 8 show an CQL schema mapping F which takes two distinct source
+entities, N1 and N2, to the target entity N. The �∆F � functor projects in the
+opposite direction of F: it projects columns from the single table for N to two
+separate tables for N1 and N2, similar to FROM N AS N1 and FROM N AS N2 in
+SQL. When there is a foreign key from N1 to N2, the �∆F � functor populates it
+so that N can be recovered by joining N1 and N2. The �ΠF � functor takes the
+cartesian product of N1 and N2 when there is no foreign key between N1 and
+N2, and joins N1 and N2 along the foreign key when there is. The �ΣF � functor
+disjointly unions N1 and N2; because N1 and N2 are not union compatible (have
+different columns), �ΣF � creates null values. When there is a foreign key between
+N1 and N2, �ΣF � merges the tuples that are related by the foreign key, resulting
+in a join. As these examples illustrate, ∆F can be thought of as projection,
+ΠF can be thought of as a product followed by a filter (which can result in a
+join), and ΣF can be thought of as a disjoint union (which does not require
+union-compatibility) followed by a merge (which can also result in a join).
+3.4
+Mapping Composition
+Given schema mappings F : S → T and G : T → U, the mapping composition
+problem (Section 2.4) is to construct a schema mapping G ◦ F : S → U that is
+equivalent with respect to query generation.
+In one sense, the mapping composition problem is trivial [19] for our for-
+malism: ∆F ◦G ∼= ∆G ◦ ∆F , ΠF ◦G ∼= ΠF ◦ ΠG, and ΣF ◦G ∼= ΣF ◦ ΣG. But
+
+String
+◦
+N1•
+name
+�
+salary
+�
+N2•
+age
+�
+◦
+Int
+F
+−−−→
+String
+◦
+N•
+name
+�
+age
+�
+salary � ◦
+Int
+N1
+ID name salary
+1 Alice $100
+2
+Bob $250
+3
+Sue
+$300
+N2
+ID age
+4 20
+5 20
+6 30
+�∆F �
+←−−−−−−
+N
+ID name salary age
+1 Alice $100 20
+2
+Bob $250 20
+3
+Sue
+$300 30
+N1
+ID name salary
+1 Alice $100
+2
+Bob $250
+3
+Sue
+$300
+N2
+ID age
+4 20
+5 20
+6 30
+�ΣF �
+−−−−−−→
+N
+ID
+name
+salary
+age
+1
+Alice
+$100
+age(1)
+2
+Bob
+$250
+age(2)
+3
+Sue
+$300
+age(3)
+4 name(4) salary(4)
+20
+5 name(5) salary(5)
+20
+6 name(6) salary(6)
+30
+N1
+ID name salary
+1 Alice $100
+2
+Bob $250
+3
+Sue
+$300
+N2
+ID age
+4 20
+5 20
+6 30
+�ΠF �
+−−−−−−→
+N
+ID name salary age
+1 Alice $100 20
+2
+Bob $250 20
+3
+Sue
+$300 20
+4 Alice $100 20
+5
+Bob $250 20
+6
+Sue
+$300 20
+7 Alice $100 30
+8
+Bob $250 30
+9
+Sue
+$300 30
+Fig. 8. Example Data Migrations (see Section 3.2)
+
+this solution is not wholly satisfactory because in practice a mixture of ∆, Σ, Π
+functors may be needed to accomplish any particular task (similarly, in SQL a
+mixture of joins and unions may be needed to accomplish any particular task).
+The following results are proved in [19] and [23]:
+– Every composition ΣF ◦∆G is isomorphic to ∆F ′ ◦ΣG′ for some F ′, G′. This
+statement is also true if ΣF is replaced with ΠF .
+– Pairs of the form (F, G), denoting ΣF ◦ ∆G, are closed under composition.
+This statement is also true if ΣF is replaced with ΠF . Such pairs can be spec-
+ified in an intuitive “select-from-where” syntax, described in [19] and [20].
+– Triples of the form (F, G, H), denoting ΣF ◦ ΠG ◦ ∆H, are closed under
+composition, provided that F is a discrete op-fibration [3], which is exactly
+the “union compatibility” condition [5] that ΣF performs unions over tables
+whose columns match; Figure 1 is not a discrete op-fibration.
+3.5
+Mapping Inversion
+Given a schema mapping F : S → T, the mapping inversion problem (Sec-
+tion 2.3) is to construct a mapping F −1 : T → S that somehow “undoes” F.
+Our formalism has strong inversion properties but does not have inverses per
+se. When there exists F −1 : T → S such that F ◦ F −1 = id and F −1 ◦ F = id,
+then ∆F ◦ ∆F −1 ∼= id, ΣF ◦ ΣF −1 ∼= id, and ΠF ◦ ΠF −1 ∼= id. In general F
+need not have an inverse, but when S and T have finite initial algebras / term
+models (which is a priori undecidable, and implies decidability of S and T) it is
+possible to construct F −1 whenever it exists by considering all possible functors
+T → S. When F has a right adjoint G : T → S, a weaker condition than having
+an inverse, there are canonical morphisms ΣF → ∆G and ∆F → ΠG.
+In practice “round-tripping” [5] of data is desirable even when inverses do not
+exist. For example, projection, because it forgets information, typically cannot
+be inverted, but we may want to remember where the projected data originated.
+In our formalism the adjunctions between Σ,∆,Π provide round-tripping. For
+example, for every F : S → T and S-instance I there is a canonical morphism
+I → ∆F (ΣF (I)), the unit of the ΣF ⊣ ∆F adjunction, which describes where
+each ID in I is sent to by ΣF (and similarly for ΠF ). Dually, for every T-
+instance J there is a canonical morphism ΣF (∆F (J)) → J, the co-unit of the
+ΣF ⊣ ∆F adjunction, which describes where the IDs in ∆F (J) originate (and
+similarly for ΠF ). The unit and co-unit can be used to obtain, for every morphism
+h : ΣF (I) → J, a mate h′ : I → ∆F (J) and vice-versa (and similarly for
+ΠF ). Relating adjointness to existing relaxed notions of inverse such as quasi-
+inverse [9] is an important area for future work.
+3.6
+Schema matching
+Given database schemas S and T, the schema matching problem (Section 2.5)
+is to automatically suggest schema mappings S → T to the user.
+
+In this section, we define two schema matching techniques used by CQL.
+Our techniques compare entities, and foreign keys and attributes (“symbols”)
+by name, as strings, and so our techniques depend on having (probably user-
+provided) names whose similarity as strings reflects their semantic similarity. Let
+σ : String, String → [0, 1] be any string similarity function [5] where a value of 1
+indicates a “good” match and a value of 0 indicates a “bad” match.
+– The first technique attempts to infer a schema mapping F : S → T. For each
+entity s ∈ S, we define F(s) := t where t ∈ T is an entity that maximizes
+σ(s, t). For each symbol f : s → s′ ∈ S, we then consider the set X of
+symbols F(s) → F(s′). If X is non-empty, we choose a symbol g ∈ X that
+maximizes σ(f, g) and set F(f) := g. If X is empty but there is a shortest
+path p from F(s) to F(s′), we set F(f) := p. If no shortest path exists, the
+match fails. The F so constructed is only a candidate schema mapping: CQL
+must verify that F preserves provable equality in S.
+– The second technique attempts to infer a schema A and schema mappings
+F : A → S and G : A → T. Such a span of mappings can be interpreted
+as a query of the form ΣF ◦ ∆G or ΠF ◦ ∆G. Let c be some user-provided
+string similarity cutoff. The entities of A are those pairs of S-entities and
+T-entities (s, t) such that σ(s, t) > c. The symbols (s, t) → (s′, t′) of A are
+those pairs of S-symbols and T-symbols (f : s → s′, g : t → t′) such that
+σ(f, g) > c. The mappings F and G are projections.
+4
+Conclusion
+When comparing our algebraic approach to model management with other ap-
+proaches originating in relational database theory [1] it is important to note that
+our databases are “deductive databases” [1]. That is, we define databases “inten-
+sionally”, as sets of equations, rather than as sets of tables. As such, care must
+be taken when mediating between our definitions and relational definitions. For
+example, our instances can be “inconsistent” in the sense that an instance can
+prove 1 = 2 for two distinct constant symbols 1 and 2. Such situations are often,
+but not always [12], errors, and the CQL tool checks for such situations using
+standard techniques based on “conservative theory extensions” [12]. In addition,
+our schemas do not define a set of constants (a “domain”) that all the instances
+on that schema share, as is customary in relational database theory [7]. Hence
+our approach is closer in spirit to traditional logic [6] than database theory [1].
+There are many connections between our algebraic approach to model man-
+agement and the ED-based approach described in Section 2. EDs are more ex-
+pressive than our purely equational data integrity constraints and can be added
+to our formalism in a simple way, described in [22] (although in [22], EDs are
+called “lifting problems”). In ED-based approaches the “chase” [5] operation has
+a similar semantics to our Σ operation, and a formal comparison between the
+chase and Σ is forthcoming.
+
+Acknowledgements.
+The authors thank Lucian Popa, Eswaran Subrah-
+manian, and Peter Gates and were supported by NIST SBIR grant 70NANB
+16H178, AFOSR grant FA9550–14–1–0031 and NASA grant NNL14AA05C.
+This paper appears in WADT 2016: Recent Trends in Algebraic
+Development Techniques, pp 56–69.
+References
+1. Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley-
+Longman (1995)
+2. Alagic, S., Bernstein, P.: A model theory for generic schema management. DBPL
+(2001)
+3. Barr, M., Wells, C.: Category Theory for Computing Science. Prentice Hall Inter-
+national (1995)
+4. Bernstein, P.A., Melnik, S.: Model management 2.0: Manipulating richer mappings.
+ICMD (2007)
+5. Doan, H., Halevy, A., Ives, Z.: Principles of Data Integration. Morgan Kaufmann
+(2012)
+6. Enderton, H.B.: A Mathematical introduction to logic. Academic Press (2001)
+7. Fagin, R., Kolaitis, P.G., Miller, R.J., Popa, L.: Data exchange: Semantics and
+query answering. Theoretical Computer Science (2005)
+8. Fagin, R., Kolaitis, P.G., Popa, L., Tan, W.: Composing schema mappings: Second-
+order dependencies to the rescue. TODS (2005)
+9. Fagin, R., Kolaitis, P.G., Popa, L., Tan, W.: Quasi-inverses of schema mappings.
+TODS (2008)
+10. Fagin, R.: Inverting schema mappings. TODS (2007)
+11. Fleming, M., Gunther, R., Rosebrugh, R.: A database of categories. Journal of
+Symbolic Computation 35(2) (2003)
+12. Ghilardi, S., Lutz, C., Wolter, F.: Did I damage my ontology? Principles of Knowl-
+edge Representation and Reasoning (2006)
+13. Goguen,
+J.:
+Information
+integration
+in
+institutions
+(unpublished).
+http://cseweb.ucsd.edu/˜goguen/pps/ifi04.pdf (2004)
+14. Haas, L.M., Hern´andez, M.A., Ho, H., Popa, L., Roth, M.: Clio grows up: From
+research prototype to industrial tool. ICMD (2005)
+15. Hodges, W.: A Shorter Model Theory. Cambridge University Press (1997)
+16. Melnik, S.: Generic Model Management: Concepts And Algorithms (Lecture Notes
+in Computer Science). Springer-Verlag (2004)
+17. Mitchell, J.C.: Foundations of Programming Languages. MIT Press (1996)
+18. Mossakowski, T., Krumnack, U., Maibaum, T.: What is a derived signature mor-
+phism? RTADT (2014)
+19. Schultz, P., Spivak, D.I., Vasilakopoulou, C., Wisnesky, R.: Algebraic databases.
+Theory and Applications of Categories (2017)
+20. Schultz,
+P.,
+Wisnesky,
+R.:
+Algebraic
+data
+integration
+(unpublished).
+http://arxiv.org/abs/1503.03571 (2016)
+21. Spivak, D.I.: Functorial data migration. Information and Computation (2012)
+22. Spivak, D.I.: Database queries and constraints via lifting problems. Mathematical
+Structures in Computer Science (2014)
+23. Spivak, D.I., Wisnesky, R.: Relational foundations for functorial data migration.
+DBPL (2015)
+

4NE4T4oBgHgl3EQfAwuD/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf,len=505
+page_content='Algebraic Model Management: A Survey  Patrick Schultz1, David I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Spivak1, and Ryan Wisnesky2  1 Massachusetts Institute of Technology  2 Conexus AI  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' We survey the field of model management and describe a  new model management approach based on algebraic specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  1  Introduction  In this paper we survey the field of model management and describe a new  model management approach based on techniques from the field of algebraic  specification, with the hope of establishing an interlingua between the two fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  By “model management” we mean “meta-data intensive” database management  in the sense of Bernstein & Melnik [4], which we define in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' By “a new  algebraic model management approach” we mean our particular way [19] [20] of  specifying database schemas and instances using algebraic (equational) theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  We first noticed a connection between model management and algebraic spec-  ification while investigating applications of category theory [3] to data integra-  tion [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' These investigations are described in [19] and [20], and we present no  substantial new results in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' We assume readers have basic proficiency  with category theory [3], algebraic specification [17], and SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' In Section 2 we describe the traditional approach to model manage-  ment and in Section 3 we describe our algebraic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Also in Section 3 we  describe the open-source CQL (Categorical Query Language) tool, available for  download at http://categoricaldata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='net, which implements our approach in  software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' We conclude in Section 4 by comparing our approach with the tradi-  tional approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  2  Model Management  To quote from Melnik [16]:  Many challenging problems facing information systems engineering in-  volve the manipulation of complex metadata artifacts, or models, such  as database schemas, interface specifications, or object diagrams, and  mappings between models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The applications that solve metadata ma-  nipulation problems are complex and hard to build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The goal of generic  model management is to reduce the amount of programming needed  to develop such applications by providing a database infrastructure in  which a set of high-level algebraic operators, such as Match, Merge, and  Compose, are applied to models and mappings as a whole rather than  to their individual building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='04846v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='LO]  12 Jan 2023  In the paragraph above the word “model” is defined to mean a metadata ar-  tifact such as a schema, which conflicts with the definition of the word “model”  as a structure satisfying a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' In this paper, we use the phrase “model man-  agement” to mean the field identified above, and use the word “model” to mean  a structure satisfying a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Today model management is a large sub-field of information management  with a research literature containing hundreds of published articles [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' There is  a consensus in that literature [4] that model management is concerned with at  least the problems described in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='1  Schema mapping  Given two database schemas S and T, the schema mapping problem [7] is to  construct a “mapping” F : S → T that captures some user-specified relationship  between S and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Different model management systems use different notions  of schema, including SQL, XML, and RDF [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The most common mapping  formalism studied in the literature is that of “embedded dependencies” (EDs) [5]:  formulae in a fragment of first-order logic with useful computational properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  We will use SQL schemas and EDs in our examples in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Consider  the following SQL schema S, consisting of two tables connected by a foreign key:  CREATE TABLE N2(ID INT PRIMARY KEY, age INT)  CREATE TABLE N1(ID INT PRIMARY KEY, name STRING, salary INT,  f INT FOREIGN KEY REFERENCES N2(ID))  and the following SQL schema T, consisting of one table:  CREATE TABLE N(ID INT PRIMARY KEY, age INT,  name STRING, salary INT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  These two SQL schemas are displayed graphically in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  An example schema mapping F : S → T expressing that the target table N  is the join of source tables N1 and N2 along the column f is:  ∀id1, id2, a, n, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' N1(id1, n, s, id2) ∧ N2(id2, a) → N(id1, a, n, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Two instances satisfying the above ED are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' In general, many  EDs can map between two SQL schemas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='2  Query generation  Given a schema mapping F : S → T, the query generation problem [4] is to  construct a query which converts databases on S to databases on T in a way  that satisfies F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The query languages typically studied include SQL, XQuery,  and various comprehension- and λ-calculi [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  A SQL query to implement the example mapping from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='1 is:  String N1•  name  �  salary  �  f  � N2•  age  � Int  F  −−−→  String N•  name  �  age  �  salary � ◦  Int  N1  ID name salary f  1 Alice $100 1  2  Bob $250 2  3  Sue  $300 3  N2  ID age  1 20  2 20  3 30  �∆F �  ←−−−−−−  �ΠF �,�ΣF �  −−−−−−−−−−→  N  ID name salary age  1 Alice $100 20  2  Bob $250 20  3  Sue  $300 30  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Example Data Migrations, with Foreign Keys (see Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='2)  INSERT INTO N  SELECT N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='ID, N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='age, N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='name, N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='sal  FROM N1, N2  WHERE N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='f = N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='ID  Technically, the INSERT portion of the above SQL code is not a “query”, but  rather an “update”, and in practice the code generated from a query generation  task will often store the results of the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' An example of running the above  SQL is shown as the left-to-right direction of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' In general, many or no  SQL queries may implement a set of EDs [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' EDs can also be directly executed  by an algorithm called “the chase” [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='3  Mapping Inversion  Given a schema mapping F : S → T, the mapping inversion problem [10] is to  construct a schema mapping F −1 : T → S that undoes F with respect to query  generation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' the queries generated from F and F −1 should be inverses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  The natural candidate ED to invert the schema mapping of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='1 ex-  presses that N projects onto N1 and N2:  ∀id1, a, n, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' N(id1, a, n, s) → ∃id2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' N1(id, n, s, id2) ∧ N2(id2, a)  and a possible SQL implementation of this ED is:  INSERT INTO N1  SELECT ID, name, sal, ID  FROM N  INSERT INTO N2  SELECT ID, age  FROM N  However, the above ED is not an inverse to the ED of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='1, as is seen by  taking ∅ = N1 ̸= N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Indeed, it is rare for an ED, or set of EDs, to be invertible,  and weaker notions of inverse, such as “quasi-inverse” [10], are common in the  literature [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' An example of running the above SQL is shown as the right-to-left  direction of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='4  Mapping Composition  Given schema mappings F : S → T and G : T → U, the mapping composition  problem [8] is to construct a schema mapping G ◦ F : S → U that is equivalent  with respect to the query generation problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' running the query generated  from G ◦ F should have the same effect as running the query generated from G  on the results of the query generated from F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  The composition of the ED from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='1 with the ED from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='3 is  ∀id1, id2, n, s, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' N1(id1, n, s, id2) ∧ N2(id2, a) → ∃x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' N1′(id, n, s, x) ∧ N2′(x, a)  where N1’, N2’ are target “copies” of source tables N1, N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' This composed ED is  not the identity, thereby showing that the ED from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='3 does not invert the  ED from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' In the case of EDs, composed mappings may not exist [8],  but some restrictions and extensions of EDs are closed under composition [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='5  Schema matching  Given two database schemas S and T, the schema matching problem [5] is to au-  tomatically find “correspondences” between S and T and to automatically infer  schema mappings S → T from these correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' In general, inference of en-  tire mappings cannot be fully automated and the focus of the matching problem  is to reduce the human effort required to construct a schema mapping by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=',  suggesting partial mappings that can be completed by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' There are many  techniques for schema matching ranging from comparison of column names by  string similarity to machine learning algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' for an overview, see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' In the  example from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='1, two correspondences that are easy to automatically  find are (N1, N) and (N2, N) and tools such as Clio [14] can create the ED from  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='1 from these two correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='6  Further References  In this paper we will focus on the problems described in the previous sec-  tions, but many other problems are studied in the model management litera-  ture [4], and many of these problems are related to algebraic specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' For  example, schema/instance merge problems [4], which arise often in data inte-  gration scenarios [2], can be formalized as pushouts in suitable categories of  schemas/instances [20], and such pushouts are related to model-theoretic con-  cepts such as model amalgamation [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Many software products solve model management problems [4], including  ETL (Extract, Transform, Load) tools [5], which extract data from separate  databases, apply user-specified transformations, and then load the result into  a target system such as a data warehouse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' query mediators [5], which answer  queries about a “virtual” integrated database by combining queries about sep-  arate source databases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' and visual schema mapping tools [14] which allow users  to create schema mappings by visually connecting related schema elements with  lines, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  There have been at least two attempts to provide a “meta semantics” for  model management operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' In [16] Melnik gives a “state based” meta se-  mantics to some of the above operations by defining a schema mapping S → T  to be an arbitrary binary relation between instances on S and instances on T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  the ED-based semantics described above is an instantiation of this meta seman-  tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' In [2] and [13] the authors give an “institution theoretic” meta semantics  to some of the above operations by defining a schema mapping S → T to be a  morphism in a suitable category of schemas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' CQL’s semantics is an instantiation  of this meta semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' A schema mapping in Clio [14]  file:students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='xsml (managed)  口  Source  Target  S File:StudentsSource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='xsd  S File:StudentsTarget,xsd  qa  e?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' targetDB  e gradEnrolls  e Evaluations  gradEnroll [o,*]  [* eval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' *]  e: sid (xsistring)-  e: name (xsistring)--  e: grade (xsiint)  (buuisisn) pp a  e?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' File (μsistring)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='.  @ Enrollment  e: File (xsistring)-  e* Student [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' *]  e* underGrad [o,*]  e: sid (xsistring)  e: name (rsistring)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  (ouuisisi) pis a  e Courses  e: name (xsistring)  e: address (xsistring)  [* course [o,*]  e enrolls  e eid (xsint) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='.-  [ enroll [o,*]  e: addr (xsistring)  e cid (xsistring)  e: sid (rsistring)3  Algebraic Model Management  Our approach to model management is based on the algebraic approach to  databases, data migration, and data integration we describe in [19] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Those works, and hence this work, extend a particular category-theoretic data  model that originated in the late 1990s [11] and was later extended in [21] and [23]  and implemented in CQL (http://categoricaldata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='net).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  In the next section we describe our formalism for database schemas and  instances and introduce CQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The subsequent sections implement the model  management operations from Section 2 using our formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' In this section we  abbreviate “algebraic theory” as “theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='1  Algebraic Databases  In our formalism [20], database schemas and instances are defined as theories of  a certain kind, which we describe in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' For ease of exposition, we  will sometimes conflate schemas and instances as defined in our formalism with  their CQL equivalents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Type sides We first fix a theory, Ty, called the type side of our formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The  sorts of Ty are called types and the functions of Ty are the functions that can  appear in schemas and instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  CQL allows arbitrary theories to be used as type sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' But we have found  that in practice, CQL users almost always want to use the theory of an existing  programming language, say java, for their type side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The ability to “bind” CQL  to an existing language is particularly important in model management because  input data may only be accessible through, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=', a java API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' For this reason,  CQL allows a type side to be defined by specifying, for each sort s, a java class  Cs and a java function String → Cs that tells CQL how to interpret the strings  it encounters in CQL programs as objects of Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  An example CQL type side about integers and strings is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  This type side defines a theory with two sorts and infinitely many constants –  all the java strings and integers – and no equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The java code for Int says  that whenever a string x is encountered in an CQL program and a term of sort  Int is required, that java’s parseInt function should be applied to x to yield  the desired Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The keyword literal, used in many places in CQL, indicates  a literal (user-defined constant) definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Schemas A schema on type side Ty is a theory extending Ty with new sorts  (called entities), new unary functions from entities to types (called attributes),  new unary functions from entities to entities (called foreign keys), and new equa-  tions (called data integrity constraints) of the form ∀v : s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' t = t′, where s is an  entity and t, t′ are terms of the same type, each containing a single free variable v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  The restrictions in the preceding sentence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=', no functions from types to enti-  ties) are necessary to use our formalism for model management purposes [19] [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Figure 4 shows the CQL schemas corresponding to Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' These schemas  contain no equations and are both on the type side Ty defined in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  typeside Ty = literal {  java_types  String = "java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='String"  Int = "java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='Integer"  java_constants  String = "return input[0]"  Int = "return java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='Integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='parseInt(input[0])"  }  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' CQL type side Ty  schema S = literal : Ty {  entities  N1  N2  foreign_keys  f : N1 -> N2  attributes  name : N1 -> String  salary : N1 -> Int  age : N2 -> Int  }  schema T = literal : Ty {  entities  N  attributes  name : N -> String  salary : N -> Int  age : N -> Int  }  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' CQL schemas S and T on type side Ty  instance I = literal : S {  generators  1 2 3 : N1  equations  name(1) = Alice  salary(1) = 100  age(f(1)) = 20  name(2) = Bob  salary(2) = 250  age(f(2)) = 20  name(3) = Sue  salary(3) = 300  age(f(3)) = 30  }  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' CQL instance I on schema S  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Initial algebra for CQL instance I  Delta - 9:36:11 PM 1 (exec: 0s)(gui: 0s)  Select:  Tables  Type Algebra  DP  Text  typeside Ty  N1 (3)  N2 (3)  schema S  ID  name  salary  f  ID  age  schema T  [1]  Alice  100  [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='f]  [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='f]  20  mapping F : S -> T  [2]  Bob  250  [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='f]  [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='f]  20  instance I : S  [3]  Sue  300  [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='f]  [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='f]  30Instances An instance I on schema S is a theory extending S with new 0-ary  function (constant) symbols called generators and non-quantified equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' An  example CQL instance on schema S (Figure 4) is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  The intended meaning of an instance I, written �I�, is the term model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=',  initial algebra) for I which contains, for each sort s, a carrier set consisting of the  closed terms of sort s modulo provability in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' A morphism of instances I → J  is a homomorphism (natural transformation) of algebras �I� → �J�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Figure 6 shows the meaning of the instance I from Figure 5 in the CQL tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  The CQL tool visually displays term models as sets of tables, one per entity e,  each with an ID column corresponding to the carrier set for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The tables in  Figure 6 are isomorphic to the left tables in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  In the following sections we implement the model management operations  from Section 2 using the preceding definitions of schema and instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='2  Schema mapping  Given schemas S, T, the schema mapping problem (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='1) is to construct  a “mapping” F : S → T that captures some relationship between S and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Let S and T be CQL schemas on the same type side Ty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' An CQL schema  mapping F : S → T is defined as a “derived signature morphism” [18] from  S to T that is the identity on Ty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' That is, F : S → T assigns to each entity  e ∈ S an entity F(e) ∈ T, and to each attribute / foreign key f : s → s′ a  term F(f), of type F(s′) and with one free variable of type F(s), in a way that  respects equality: if S ⊢ t = t′, then T ⊢ F(t) = F(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' We have found that  many mappings arising in practice cannot be expressed using plain signature  morphisms and require the more general notion of “derived” signature morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Whereas a schema mapping in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='1 was an ED (formula in a fragment  of first-order logic), which induces a single binary satisfaction relation between  instances, CQL schema mappings are derived signature morphisms and induce  three relations between instances, which we will describe in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  An example CQL schema mapping F : S → T is shown in Figure 7, where  CQL schemas S and T are defined in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' This mapping is also shown  graphically in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='3  Query generation  Given a mapping F : S → T, the query generation problem (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='2) is to  use F to construct a query which converts databases on S to databases on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  In our formalism, the database instances and morphisms on a schema S  constitute a category, denoted S–Inst, and a schema mapping F : S → T induces  a functor ΣF : S–Inst → T–Inst defined by substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The functor ΣF has a  right adjoint, ∆F : T–Inst → S–Inst, which corresponds to the “model reduct  functor” when our formalism is described in institution-theoretic terms [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The  functor ∆F has a right adjoint, ΠF : S–Inst → T–Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' See [19] for proof  that ∆F always has left and right adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' As adjoints, ∆F , ΠF preserve limits  and ∆F , ΣF preserve colimits, implying many useful properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' for example,  ΣF (I + J) ∼= ΣF (I) + ΣF (J) and ΠF (I × J) ∼= ΠF (I) × ΠF (J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  mapping F = literal : S -> T {  entities  N1 -> N  N2 -> N  foreign_keys  f -> lambda x:N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' x  attributes  name -> lambda x:N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' name(x)  salary -> lambda x:N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' salary(x)  age -> lambda x:N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' age(x)  }  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' CQL schema mapping F : S → T  Note that unlike Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='1, where there was a single query associated with a  schema mapping (ED), in our algebraic approach there are three queries, one for  each of ∆F , ΣF , ΠF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The conditions under which ∆F ,ΣF , ΠF can be expressed  in SQL and vice-versa are characterized in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Although it is possible to give explicit formulae to define ∆F , ΣF , ΠF [19]  we instead give examples in Figures 1 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Note that in these examples we are  not showing instances (theories) as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' we are showing term  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' For this reason, we surround ∆F , ΣF , ΠF with denotation brackets �� in  these examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' In addition, as adjoints ∆, Σ, Π are only defined up to unique  isomorphism, so we arbitrarily make up names for IDs and in these examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Figures 1 and 8 show an CQL schema mapping F which takes two distinct source  entities, N1 and N2, to the target entity N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The �∆F � functor projects in the  opposite direction of F: it projects columns from the single table for N to two  separate tables for N1 and N2, similar to FROM N AS N1 and FROM N AS N2 in  SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' When there is a foreign key from N1 to N2, the �∆F � functor populates it  so that N can be recovered by joining N1 and N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The �ΠF � functor takes the  cartesian product of N1 and N2 when there is no foreign key between N1 and  N2, and joins N1 and N2 along the foreign key when there is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The �ΣF � functor  disjointly unions N1 and N2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' because N1 and N2 are not union compatible (have  different columns), �ΣF � creates null values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' When there is a foreign key between  N1 and N2, �ΣF � merges the tuples that are related by the foreign key, resulting  in a join.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' As these examples illustrate, ∆F can be thought of as projection,  ΠF can be thought of as a product followed by a filter (which can result in a  join), and ΣF can be thought of as a disjoint union (which does not require  union-compatibility) followed by a merge (which can also result in a join).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
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+page_content=' Example Data Migrations (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='2)  this solution is not wholly satisfactory because in practice a mixture of ∆, Σ, Π  functors may be needed to accomplish any particular task (similarly, in SQL a  mixture of joins and unions may be needed to accomplish any particular task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  The following results are proved in [19] and [23]:  – Every composition ΣF ◦∆G is isomorphic to ∆F ′ ◦ΣG′ for some F ′, G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' This  statement is also true if ΣF is replaced with ΠF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  – Pairs of the form (F, G), denoting ΣF ◦ ∆G, are closed under composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  This statement is also true if ΣF is replaced with ΠF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Such pairs can be spec-  ified in an intuitive “select-from-where” syntax, described in [19] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  – Triples of the form (F, G, H), denoting ΣF ◦ ΠG ◦ ∆H, are closed under  composition, provided that F is a discrete op-fibration [3], which is exactly  the “union compatibility” condition [5] that ΣF performs unions over tables  whose columns match;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Figure 1 is not a discrete op-fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='5  Mapping Inversion  Given a schema mapping F : S → T, the mapping inversion problem (Sec-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='3) is to construct a mapping F −1 : T → S that somehow “undoes” F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Our formalism has strong inversion properties but does not have inverses per  se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' When there exists F −1 : T → S such that F ◦ F −1 = id and F −1 ◦ F = id,  then ∆F ◦ ∆F −1 ∼= id, ΣF ◦ ΣF −1 ∼= id, and ΠF ◦ ΠF −1 ∼= id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' In general F  need not have an inverse, but when S and T have finite initial algebras / term  models (which is a priori undecidable, and implies decidability of S and T) it is  possible to construct F −1 whenever it exists by considering all possible functors  T → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' When F has a right adjoint G : T → S, a weaker condition than having  an inverse, there are canonical morphisms ΣF → ∆G and ∆F → ΠG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  In practice “round-tripping” [5] of data is desirable even when inverses do not  exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' For example, projection, because it forgets information, typically cannot  be inverted, but we may want to remember where the projected data originated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  In our formalism the adjunctions between Σ,∆,Π provide round-tripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' For  example, for every F : S → T and S-instance I there is a canonical morphism  I → ∆F (ΣF (I)), the unit of the ΣF ⊣ ∆F adjunction, which describes where  each ID in I is sent to by ΣF (and similarly for ΠF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Dually, for every T-  instance J there is a canonical morphism ΣF (∆F (J)) → J, the co-unit of the  ΣF ⊣ ∆F adjunction, which describes where the IDs in ∆F (J) originate (and  similarly for ΠF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The unit and co-unit can be used to obtain, for every morphism  h : ΣF (I) → J, a mate h′ : I → ∆F (J) and vice-versa (and similarly for  ΠF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Relating adjointness to existing relaxed notions of inverse such as quasi-  inverse [9] is an important area for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='6  Schema matching  Given database schemas S and T, the schema matching problem (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='5)  is to automatically suggest schema mappings S → T to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  In this section, we define two schema matching techniques used by CQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Our techniques compare entities, and foreign keys and attributes (“symbols”)  by name, as strings, and so our techniques depend on having (probably user-  provided) names whose similarity as strings reflects their semantic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Let  σ : String, String → [0, 1] be any string similarity function [5] where a value of 1  indicates a “good” match and a value of 0 indicates a “bad” match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  – The first technique attempts to infer a schema mapping F : S → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' For each  entity s ∈ S, we define F(s) := t where t ∈ T is an entity that maximizes  σ(s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' For each symbol f : s → s′ ∈ S, we then consider the set X of  symbols F(s) → F(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' If X is non-empty, we choose a symbol g ∈ X that  maximizes σ(f, g) and set F(f) := g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' If X is empty but there is a shortest  path p from F(s) to F(s′), we set F(f) := p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' If no shortest path exists, the  match fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The F so constructed is only a candidate schema mapping: CQL  must verify that F preserves provable equality in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  – The second technique attempts to infer a schema A and schema mappings  F : A → S and G : A → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Such a span of mappings can be interpreted  as a query of the form ΣF ◦ ∆G or ΠF ◦ ∆G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Let c be some user-provided  string similarity cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The entities of A are those pairs of S-entities and  T-entities (s, t) such that σ(s, t) > c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The symbols (s, t) → (s′, t′) of A are  those pairs of S-symbols and T-symbols (f : s → s′, g : t → t′) such that  σ(f, g) > c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' The mappings F and G are projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  4  Conclusion  When comparing our algebraic approach to model management with other ap-  proaches originating in relational database theory [1] it is important to note that  our databases are “deductive databases” [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' That is, we define databases “inten-  sionally”, as sets of equations, rather than as sets of tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' As such, care must  be taken when mediating between our definitions and relational definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' For  example, our instances can be “inconsistent” in the sense that an instance can  prove 1 = 2 for two distinct constant symbols 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Such situations are often,  but not always [12], errors, and the CQL tool checks for such situations using  standard techniques based on “conservative theory extensions” [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' In addition,  our schemas do not define a set of constants (a “domain”) that all the instances  on that schema share, as is customary in relational database theory [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' Hence  our approach is closer in spirit to traditional logic [6] than database theory [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  There are many connections between our algebraic approach to model man-  agement and the ED-based approach described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' EDs are more ex-  pressive than our purely equational data integrity constraints and can be added  to our formalism in a simple way, described in [22] (although in [22], EDs are  called “lifting problems”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content=' In ED-based approaches the “chase” [5] operation has  a similar semantics to our Σ operation, and a formal comparison between the  chase and Σ is forthcoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  The authors thank Lucian Popa, Eswaran Subrah-  manian, and Peter Gates and were supported by NIST SBIR grant 70NANB  16H178, AFOSR grant FA9550–14–1–0031 and NASA grant NNL14AA05C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
+page_content='  This paper appears in WADT 2016: Recent Trends in Algebraic  Development Techniques, pp 56–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'}
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+A Characterization of Complexity in Public Goods Games
+MATAN GILBOA
+We complete the characterization of the computational complexity of equilibrium in public goods games on
+graphs by proving that the problem is NP-complete for every finite non-monotone best-response pattern. This
+answers the open problem of [Gilboa and Nisan, 2022], and completes the answer to a question raised by
+[Papadimitriou and Peng, 2021], for all finite best-response patterns.
+Manuscript submitted for review to the 24nd ACM Conference on Economics & Computation (EC'23).
+arXiv:2301.11580v1  [cs.GT]  27 Jan 2023
+
+M. Gilboa
+1
+1
+INTRODUCTION
+Public goods games describe scenarios where multiple agents face a decision of whether or not
+to produce some "good", such that producing this good benefits not only themselves, but also
+other (though not necessarily all) agents. Typically, we consider the good to be costly to produce,
+and therefore an agent might choose not to produce it, depending on the actions of the agents
+that affect her. This type of social scenarios can be found in various real-life examples, such as
+vaccination efforts (an individual pays some personal cost for being vaccinated but she and other
+people in her proximity gain from it) and research efforts (a research requires many resources,
+but the researcher benefits from the result along with other researchers in similar areas). As is
+common in the literature, to model this we use an undirected graph, where each node represents
+an agent and an edge between two nodes captures the fact that these nodes directly affect one
+another by their strategy. As in [Kempe et al., 2021, Maiti and Dey, 2022, Papadimitriou and Peng,
+2021, Yang and Wang, 2020, Yu et al., 2021], in our model the utility of an agent is completely
+determined by the number of productive neighbors she has, as well as by her own action. We focus
+on a specific version of the game which has the following characteristics. Firstly, our strategy space
+is binary, i.e. an agent can only choose whether or not to produce the good, rather than choose
+a quantity (we call an agent who produces the good a productive agent); secondly, our game is
+fully-homogeneous, meaning that all agents share the same utility function and cost of producing
+the good; and thirdly, our game is strict, which means that an agent has a single best response to
+any number of productive neighbors she might have (i.e. we do not allow indifference between the
+actions).
+The game is formally defined by some fixed cost 𝑐 of producing the good, and by some "social"
+function 𝑋 (𝑠𝑖,𝑛𝑖), which takes into account the boolean strategy of agent 𝑖 and the number of
+productive neighbors she has (marked as 𝑠𝑖 and 𝑛𝑖 respectively), and outputs a number representing
+how much the agent gains. The utility 𝑢𝑖 of agent 𝑖 is then given by the social function 𝑋 (𝑠𝑖,𝑛𝑖),
+reduced by the cost 𝑐 if the agent produces the good, i.e. 𝑢𝑖 (𝑠𝑖,𝑛𝑖) = 𝑋 (𝑠𝑖,𝑛𝑖) −𝑐 ·𝑠𝑖. However, since
+any number of productive neighbors yields a unique best response (i.e. the game is strict), we can
+capture the essence of the utility function and the cost using what we call (as in [Gilboa and Nisan,
+2022]), a Best-Response Pattern𝑇 : IN → {0, 1}. We think of the Best-Response Pattern as a boolean
+vector in which the 𝑘𝑡ℎ entry represents the best response to exactly 𝑘 productive neighbors. We
+are interested in the problem of determining the existence of a non-trivial pure Nash equilibrium
+in these games, which is defined as follows.
+Equilibrium decision problem in a public goods game: For a fixed Best-Response Pattern
+𝑇 : IN → {0, 1}, and with an undirected graph 𝐺 = (𝑉, 𝐸) given as input, determine whether there
+exists a pure non-trivial Nash equilibrium of the public goods game defined by 𝑇 on 𝐺, i.e. an
+assignment 𝑠 : 𝑉 → {0, 1} that is not all 0, such that for every 1 ≤ 𝑖 ≤ |𝑉 | we have that
+𝑠𝑖 = 𝑇 [
+∑︁
+𝑗
+𝑠.𝑡 (𝑣𝑖,𝑣𝑗) ∈𝐸
+𝑠𝑗].
+The first Best-Response Pattern for which this problem was studies was the so-called Best-Shot
+pattern (where an agent’s best response is to produce the good only if she has no productive
+neighbors), which was shown in [Bramoullé and Kranton, 2007] to have a pure Nash equilibrium
+in any graph. In [Bramoullé and Kranton, 2007], they also show algorithmic results for "convex"
+patterns, which are monotonically increasing (best response is 1 if you have at least 𝑘 productive
+neighbors). The question of characterizing the complexity of this problem for all possible patterns
+
+M. Gilboa
+2
+was first raised by [Papadimitriou and Peng, 2021], where they manage to fully answer an equivalent
+problem on directed graphs, showing NP-completeness for most patterns, and algorithmic results
+for the remaining few. The open question on undirected graphs was then partially answered in
+[Gilboa and Nisan, 2022], where they show NP-completeness for several classes of patterns, and a
+polynomial-time algorithm for one other pattern. There have been several studies concerning other
+versions of this problem as well. In [Yang and Wang, 2020], the general version of this problem
+(where the pattern is part of the input rather than being fixed) was shown to be NP-complete
+when removing the strictness assumption, (i.e. allowing indifference between actions, such that
+both 0 and 1 are best responses in certain cases) 1. In [Yu et al., 2021], NP-completeness is shown
+for the general version of the problem in the heterogeneous public goods game, in which the
+utility function varies between agents. In [Kempe et al., 2021], they show NP-completeness of the
+equilibrium problem when restricting the equilibrium to have at least 𝑘 productive agents, or at
+least some specific subset of agents. In [Maiti and Dey, 2022], the parameterized complexity of the
+equilibrium problem is studied, for a number of parameters of the graph on which the game is
+defined.
+Papadimitirou and Peng raised the problem of characterizing all Best-Response Patterns, and
+Gilboa and Nisan suggested two specific open problems regarding two specific patterns. One of
+these patterns has been recently solved by Max Klimm (personal communication) who showed
+that all monotonically decreasing patterns can be viewed as potential games, and thus always have
+a pure Nash equilibrium2.
+Our main contribution is completing the characterization of the equilibrium decision problem
+for all finite patterns, by showing that for all non-monotone patterns the problem is NP-complete.
+Theorem: For any Best-Response Pattern that is non-monotone and finite (i.e., has a finite number
+of entries with value 1), the equilibrium decision problem in a public goods game is NP-complete
+(under Turing reductions).
+The first step along this way was to prove NP-completeness for the specific open problem by
+[Gilboa and Nisan, 2022]. It has come to our attention that an alternative proof to this specific open
+problem was obtained independently and concurrently by Max Klimm and Maximilian Stahlberg
+(private communication).
+We note that we only focus on finite patterns, which we believe to be more applicable to real-life
+problems that can be modeled by this game. We believe that the characterization for all infinite
+patterns is of interest, and remains open, though some results can be found in Corollary 3.9.
+The rest of this paper is organized as follows. In Section 2 we introduce the formal model and
+some relevant definitions. We then set out to show hardness of all remaining patterns, dividing
+them into classes. In Section 3 we present a solution for an open question from [Gilboa and Nisan,
+2022], showing hardness of a pattern we call the 0-Or-2-Neighbors Best Response Pattern, and
+expanding the result to a larger sub-class of patterns that begin with 1,0,1. In Section 4 we show
+hardness of all patterns beginning with 1,0,0 (where we also have a slightly more subtle division
+into sub-classes), and in Section 5 we show hardness of all patterns beginning with 1,0,1 that were
+not covered in Section 3, thus completing the characterization for all finite patterns. The outline of
+this paper is also depicted3 in Figure 1.
+1The paper [Yu et al., 2020] had an earlier version [Yu et al., 2021] which presented a proof for this case as well, but an error
+in the proof was pointed out by [Yang and Wang, 2020], who then also provided an alternative proof.
+2Alternatively, Sigal Oren (personal communication) observed that known results about 𝑘-Dominating and 𝑘-independent
+sets [Chellali et al., 2012] (Theorem 19) can be used to prove this.
+3Some patterns which start with 1,0 were solved in [Gilboa and Nisan, 2022], though for simplicity we omit them from
+Figure 1.
+
+M. Gilboa
+3
+Fig. 1. Outline of this paper.
+2
+MODEL AND DEFINITIONS
+A Public Goods Game (PGG) is defined on an undirected graph 𝐺 = (𝑉, 𝐸), 𝑉 = {𝑣1, ..., 𝑣𝑛}, where
+each node represents an agent. The strategy space, which is identical for all agents, is 𝑆 = {0, 1},
+where 1 represent producing the good and 0 represents not producing it. The utility of node
+𝑣𝑖 (which is assumed to be the same for all agents) is completely determined by the number of
+productive neighbors 𝑣𝑖 has, as well as by 𝑣𝑖’s own strategy. Moreover, our model is restricted to
+utility functions where an agent always has a single best response to the strategies of its neighbors,
+i.e. there is no indifference between actions in the game. Therefore, rather than defining a PGG
+with an explicit utility function and cost for producing the good, we can simply consider the best
+response of an agent for any number of productive agents in its neighborhood. Essentially, this can
+be modeled as a function 𝑇 : IN → {0, 1}, which, as in [Gilboa and Nisan, 2022], we represent in
+the form of a Best Response Pattern:
+Definition 2.1. A Best-Response Pattern (BRP) of a PGG, denoted by 𝑇, is an infinite boolean
+vector in which the 𝑘𝑡ℎ entry indicates the best response for each agent 𝑣𝑖 given that exactly 𝑘
+neighbors of 𝑣𝑖 (excluding 𝑣𝑖) produce the good:
+∀𝑘 ≥ 0 𝑇 [𝑘] = best response to 𝑘 productive neighbors.
+Definition 2.2. Given a Public Goods Game defined on a graph 𝐺 = (𝑉, 𝐸) with respect to a
+BRP 𝑇, a strategy profile 𝑠 = (𝑠1, ...,𝑠𝑛) ∈ 𝑆𝑛 (where 𝑠𝑖 ∈ {0, 1} represents the strategy of node
+𝑣𝑖 ∈ 𝑉 ) is a pure Nash equilibrium (PNE) if all agents play the best response to the strategies of
+their neighbors:
+∀1 ≤ 𝑖 ≤ 𝑛 𝑠𝑖 = 𝑇 [
+∑︁
+𝑗
+𝑠.𝑡 (𝑣𝑖,𝑣𝑗) ∈𝐸
+𝑠𝑗].
+
+finite,non-monotonepatterns
+starts with 0
+starts with 1,0
+starts with 1,1
+Solved
+Solved
+[GilboaandNisan,2022]
+[GilboaandNisan,2022]
+starts with1,0,1
+starts with1,0,0
+of the form:
+all other forms
+has"isolated" odd 1
+doesn't have
+1,0,1,0,1,0,..,1,0,0,0,...
+"isolated"odd 1
+or
+1,0,1,0,1,0, .,1,1,?,?..
+Section 3
+Section 5
+Section 4.1
+Section 4.2M. Gilboa
+4
+In addition, if there exists 1 ≤ 𝑖 ≤ 𝑛 s.t 𝑠𝑖 = 1, then 𝑠 is called a non-trivial pure Nash equilibrium
+(NTPNE).
+We note that throughout the paper we also use the notation 𝑣𝑖 = 0 and 𝑣𝑖 = 1 to indicate the
+strategy of some node 𝑣𝑖, rather than use 𝑠𝑖 = 0 and 𝑠𝑖 = 1, respectively.
+Definition 2.3. For a fixed BRP 𝑇, the non-trivial4 pure Nash equilibrium decision problem
+corresponding to 𝑇, denoted by NTPNE(𝑇), is defined as follows: The input is an undirected graph
+𝐺. The output is ’True’ if there exists an NTPNE in the PGG defined on 𝐺 with respect to 𝑇, and
+’False’ otherwise.
+Definition 2.4. A BRP 𝑇 is called monotonically increasing (resp. decreasing) if ∀𝑘 ∈ IN, 𝑇 [𝑘] ≤
+𝑇 [𝑘 + 1] (resp. 𝑇 [𝑘] ≥ 𝑇 [𝑘 + 1]).
+Definition 2.5. A BRP 𝑇 is called finite if it has a finite number of entries with value 1:
+∃𝑁 ∈ IN 𝑠.𝑡 ∀𝑛 > 𝑁 𝑇 [𝑛] = 0
+As seen in Figure 1, the only patterns for which the equilibrium decision problem remains open
+are patterns that begin with 1,0. We divide those into the two following classes of patterns.
+Definition 2.6. A BRP 𝑇 is called semi-sharp if:
+(1) 𝑇 [0] = 1
+(2) 𝑇 [1] = 𝑇 [2] = 0
+i.e. 𝑇 begins with 1, 0, 0.
+Definition 2.7. A BRP 𝑇 is called spiked if:
+(1) 𝑇 [0] = 𝑇 [2] = 1
+(2) 𝑇 [1] = 0
+i.e. 𝑇 begins with 1, 0, 1.
+3
+HARDNESS OF THE 0-OR-2-NEIGHBORS PATTERN
+In this section we show that the equilibrium problem is NP-complete for the 0-Or-2-Neighbors
+pattern, and provide some intuition about the problem. This result answers an open question by
+Gilboa and Nisan [Gilboa and Nisan, 2022]. We then expand this to show hardness of a slightly
+more general class of patterns. In the 0-Or-2-Neighbors BRP the best response is 1 only to zero or
+two productive neighbors, as we now define.
+Definition 3.1. The 0-Or-2-Neighbors Best Response Pattern is defined as follows:
+∀𝑘 ∈ IN 𝑇 [𝑘] =
+�
+1
+if 𝑘 = 0 𝑜𝑟 𝑘 = 2
+0
+otherwise
+i.e.
+𝑇 = [1, 0, 1, 0, 0, 0, ...].
+Theorem 3.2. Let 𝑇 be the 0-Or-2 Neighbors BRP. Then NTPNE(𝑇) is NP-complete.
+4In this paper, we only study BRPs where the best response for zero productive neighbors is 1, for which there never exists
+a trivial all-zero PNE (as these are the only BRPs left to solve). However, we sometimes reduce from patterns where this is
+not the case, and therefore include the non-triviality restriction in our problem definition, in order to correspond with the
+literature.
+
+M. Gilboa
+5
+Before proving the theorem, we wish to provide basic intuition about the 0-Or-2-Neighbors BRP,
+by examining several simple graphs. Take for example a simple cycle graph. Since𝑇 [2] = 1 (i.e. best
+response for two productive neighbors is 1), we have that any simple cycle admits a non-trivial pure
+Nash equilibrium, assigning 1 to all nodes (see Figure 2. However, looking at a simple path with 𝑛
+nodes, we see that the all-ones assignment is never a pure Nash equilibrium. The reason for this is
+that 𝑇 [1] = 0 (i.e. best response for one productive neighbors is 0), and so the two nodes at both
+edges of the path, having only one productive neighbor, do not play best response. Nevertheless,
+any simple path does admit a pure Nash equilibrium. To see why, let us observe the three smallest
+paths, of length 2,3 and 4. Notice that in a path of length two a PNE is given by the assignment 0,1;
+in a path of length three a PNE is given by the assignment 0,1,0; and in a path of length four a PNE
+is given by the assignment 1,0,0,1. We can use these assignment to achieve a PNE in any simple
+path: given a simple path of length 𝑛, if 𝑛 ≡ 0 (mod 3) we use the path of length three as our basis,
+adding 0,1,0 to it as many times as needed; if 𝑛 ≡ 1 (mod 3) we use the path of length four as our
+basis, adding 0,0,1 to it as many times as needed; and if 𝑛 ≡ 2 (mod 3) we use the path of length
+two as our basis, adding 0,0,1 to it as many times as needed (see example in Figure 3).
+Fig. 2. PNE in cycles.
+Fig. 3. PNE in paths of lengths 2 and 5.
+In contrast to the graphs we have discussed so far, there are graphs in which a pure Nash
+equilibrium doesn’t exist for the 0-Or-2-Neighbors pattern. An example of this can be seen in a
+graph composed of four triangles, connected as a chain where each two neighboring triangles have
+a single overlapping vertex, as demonstrated in Figure 4. One may verify that no PNE exists in this
+graph. This specific graph will also be of use to us during our proof5.
+Fig. 4. No PNE exists in this graph.
+Having provided some intuition regarding the problem, we move on to prove Theorem 3.2. The
+reduction is from ONE-IN-THREE 3SAT, which is a well known NP-complete problem [Schaefer,
+1978]. In ONE-IN-THREE 3SAT, the input is a CNF formula with 3 literals in each clause, and the
+5The Negation Gadget defined throughout the proof of Theorem 3.2 is constructed similarly to the graph described here.
+
+10
+0
+0
+0M. Gilboa
+6
+Fig. 5. Clause Gadget.
+goal is to determine whether there exists a boolean assignment to the variables such that in each
+clause exactly one of the literals is assigned True. We begin by introducing our Clause Gadget,
+which is a main component of the proof. Given a CNF formula, for each of its clauses we construct
+a 21-nodes Clause Gadget, in which three of the nodes, denoted 𝑙1,𝑙2,𝑙3 (also referred to as the
+literal nodes) represent the three literals of the matching clause. The purpose of this gadget is to
+enforce the property that in any NTPNE, exactly one literal node in the gadget will be assigned 1,
+which easily translates to the key property of a satisfying assignment in the ONE-IN-THREE 3SAT
+problem. The three literal nodes are connected to one another, forming a triangle. Additionally, for
+each 𝑖 ∈ {1, 2, 3}, 𝑙𝑖 is connected to two other nodes 𝑥𝑖,𝑦𝑖, which are also connected to one another,
+forming another triangle. Lastly, 𝑥𝑖 and 𝑦𝑖 each form yet another triangle, along with nodes 𝑎𝑖,𝑏𝑖
+and 𝑐𝑖,𝑑𝑖 respectively. We refer to 𝑥𝑖,𝑦𝑖,𝑎𝑖,𝑏𝑖,𝑐𝑖,𝑑𝑖 as the sub-gadget of 𝑙𝑖. We note that out of the
+nodes of the Clause Gadget, only the literal nodes may be connected to other nodes outside of their
+gadget, a property on which we rely throughout the proof. The structure of the Clause Gadget is
+demonstrated in Figure 5, where each sub-gadget is colored differently.
+The next four lemmas lead us to the conclusion that the gadget indeed has the desired property
+mentioned above.
+Lemma 3.3. In any NTPNE in a graph 𝐺 which includes a Clause Gadget 𝑐𝑔, if a literal node 𝑙𝑖 of 𝑐𝑔
+is assigned 1 then so are its two neighbors from its respective sub-gadget, 𝑥𝑖,𝑦𝑖.
+Proof. Divide into cases.
+Case 1: If 𝑥𝑖 = 𝑦𝑖 = 0, then if 𝑎𝑖 ≠ 𝑏𝑖 (meaning only one of them is assigned 1) then 𝑥𝑖 would
+have two productive neighbors and would not be playing its best response. However, if 𝑎𝑖 = 𝑏𝑖 then
+𝑎𝑖 and 𝑏𝑖 would not be playing their best response, and we reach a contradiction.
+Case 2: If 𝑥𝑖 = 1,𝑦𝑖 = 0 (the case where 𝑥𝑖 = 0,𝑦𝑖 = 1 is, of course, symmetrical) then 𝑥𝑖 must
+have exactly one more productive neighbor (either 𝑎𝑖 or 𝑏𝑖) in order to be playing best response.
+But then that node would not be playing best response, in contradiction.
+
+C1
+d1
+a1
+X1
+Y1
+V3
+3
+3
+V2
+a3M. Gilboa
+7
+Case 3: We are left with the option where 𝑥𝑖 = 𝑦𝑖 = 1, where it is easy to verify that all nodes of
+the sub-gadget of 𝑙𝑖 are playing their best response if we set 𝑎𝑖 = 𝑏𝑖 = 𝑐𝑖 = 𝑑𝑖 = 0.
+□
+Lemma 3.4. In any NTPNE in a graph 𝐺 which includes a Clause Gadget 𝑐𝑔, if one of the literal
+nodes 𝑙𝑖 of 𝑐𝑔 is assigned 1 then the other two literal nodes of 𝑐𝑔 must be assigned 0.
+Proof. Since 𝑙𝑖 = 1, from Lemma 3.3 we have that 𝑥𝑖 = 𝑦𝑖 = 1. Therefore, 𝑙𝑖 has two productive
+neighbors and cannot have any more, and so we have that the other two literal nodes must play
+0.
+□
+Lemma 3.5. In any graph 𝐺 which includes a Clause Gadget 𝑐𝑔, if exactly one of the literal nodes of
+𝑐𝑔 is assigned 1 then there exists a unique assignment to the other nodes of 𝑐𝑔 such that they all play
+their best response.6
+Proof. W.l.o.g assume that 𝑙1 = 1,𝑙2 = 𝑙3 = 0. Then, focusing first on the sub-gadget of 𝑙1,
+according to Lemma 3.3 we have that 𝑥1 = 𝑦1 = 1. Since 𝑥1,𝑦1 already have two productive
+neighbors, they mustn’t have any others, and so it must be that 𝑎1 = 𝑏1 = 𝑐1 = 𝑑1 = 0. We move
+on to the sub-gadget of 𝑙2. If 𝑥2 ≠ 𝑦2 then 𝑙2 would have 2 productive neighbors and would not be
+playing its best response. If 𝑥2 = 𝑦2 = 1 then there is no assignment to 𝑎2,𝑏2 s.t 𝑎2,𝑏2,𝑥2 all play
+their best response. Therefore 𝑥2 = 𝑦2 = 0. We are left only with the option of setting 𝑎2 ≠ 𝑏2 and
+𝑐2 ≠ 𝑑2 (for instance 𝑎2 = 𝑐2 = 1,𝑏2 = 𝑑2 = 0). The sub-gadget of 𝑙3 is symmetrical to that of 𝑙2. One
+may verify that in this assignment all nodes of 𝑐𝑔 indeed play their best response.
+□
+Lemma 3.6. In any graph 𝐺 which includes a Clause Gadget 𝑐𝑔, if all three of the literal nodes of
+𝑐𝑔 are assigned 0, and the literal nodes do not have any productive neighbors outside of 𝑐𝑔, then the
+assignment is not a PNE.
+Proof. Assume by way of contradiction that there exists a PNE where 𝑙1 = 𝑙2 = 𝑙3 = 0, and all
+three of them have no productive neighbors outside 𝑐𝑔. It must be that the other two neighbors of
+𝑙1, 𝑥1,𝑦1, are assigned with different values (otherwise 𝑙1 is not playing its best response). W.l.o.g
+assume 𝑥1 = 1,𝑦1 = 0. Now, If the remaining neighbors of 𝑦1 (𝑐1 and 𝑑1) are both assigned with 0 or
+both assigned with 1, then they themselves would not be playing their best response. On the other
+hand, if we assign them with different values then 𝑦1 would not be playing its best response, and
+so we have reached a contradiction.
+□
+So far, we have seen that in any PNE which includes a Clause Gadget, it must be that exactly
+one of the literal nodes of that gadget is assigned with 1, as long as the literal nodes don’t have
+productive neighbors outside of their Clause Gadget. As we introduce the external nodes that will
+be connected to the literal nodes, we will show that they all must be assigned with 0 in any PNE,
+and thus a literal node cannot have any productive neighbor outside of its Clause Gadget, which
+will finalize the property we were looking to achieve with the Clause Gadget.
+Our next goal is to make sure the translation between solutions from one domain to the other is
+always valid. Specifically, we wish to ensure that in any PNE in our constructed graph, if any two
+literal nodes represent the same variable in the CNF formula then they will be assigned the same
+value, and if they represent a variable and its negation then they will be assigned opposite values.
+We begin with the latter, introducing our Negation Gadget. The goal of the Negation Gadget is to
+force opposite assignments to two chosen nodes, in any Nash equilibrium. The Negation Gadget
+is composed of 9 nodes: five ’bottom’ nodes 𝑏1,𝑏2,𝑏3,𝑏4,𝑏5, and four ’top’ nodes 𝑡1,𝑡2,𝑡3,𝑡4, and
+for each 𝑘 ≤ 4 we create the edges (𝑏𝑖,𝑏𝑖+1), (𝑡𝑖,𝑏𝑖) and (𝑡𝑖,𝑏𝑖+1). It can intuitively be described as
+6We ignore the possibility of changing between the assignments of 𝑎𝑖 and 𝑏𝑖, or 𝑐𝑖 and 𝑑𝑖 for 𝑖 ∈ {1, 2, 3}, as it does not
+affect anything in our proof.
+
+M. Gilboa
+8
+four triangles that are connected as a chain. Say we have two nodes 𝑢, 𝑣 which we want to force
+to have opposite assignments, we simply connect them both to node 𝑡2 of a Negation Gadget, as
+demonstrated in Figure 6.
+Lemma 3.7. In any Nash equilibrium in a graph 𝐺 which includes two nodes 𝑢, 𝑣 connected through
+a Negation Gadget 𝑛𝑔, 𝑢 and 𝑣 must have different assignments. In addition, the node 𝑡2 of 𝑛𝑔, to which
+𝑢 and 𝑣 are connected, must be assigned 0.
+Proof. We first show that 𝑢 and 𝑣 must have different assignments, dividing into two cases.
+Case 1: Assume by way of contradiction that 𝑢 = 𝑣 = 0. We divide into two sub-cases, where in
+the first one 𝑡2 = 0; in this case, exactly one of the two remaining neighbors of 𝑡2 must be assigned
+1 in order for 𝑡2 itself to be playing best response. If 𝑏2 = 0 then 𝑏3 = 1 and so, looking at 𝑡1,𝑏1 (the
+remaining neighbors of 𝑏2), we see that any assignment to them results either in 𝑏2 not playing
+best response, or in 𝑡1,𝑏1 not playing best response, in contradiction. If, however, 𝑏3 = 0, then
+𝑏2 = 1, and so, looking at 𝑡3,𝑏4 (the remaining neighbors of 𝑏3), the same logic leads us to a similar
+contradiction. In the second sub-case, where 𝑡2 = 1, we have that its two remaining neighbors must
+be assigned the same value in order for 𝑡2 itself to be playing best response. If 𝑏2 = 𝑏3 = 0 then
+again there is no assignment to 𝑏1,𝑡1 s.t all of 𝑏1,𝑡1,𝑏2 play best response, and if 𝑏2 = 𝑏3 = 1 then
+one may verify that there is no assignment to 𝑡3,𝑡4,𝑏4,𝑏5 s.t all of 𝑡3,𝑡4,𝑏3,𝑏4,𝑏5 play best response,
+and so we reach a contradiction.
+Case 2: Assume 𝑢 = 𝑣 = 1. Then we again divide into sub-cases according to 𝑡2’s assignment.
+If 𝑡2 = 0, it must have at least one more productive neighbor in order to play best response. The
+assignments where 𝑏2 = 𝑏3 = 1 or 𝑏2 = 0,𝑏1 = 1 are easily disqualified, seeing as there is no
+assignment to 𝑡1,𝑏1 s.t 𝑡1,𝑏1,𝑏2 all play best response. If 𝑏2 = 1,𝑏3 = 0 then it must hold that 𝑡3 = 𝑏4
+in order for 𝑏3 to play best response, but this would mean that 𝑡3 is not playing best response, in
+contradiction. If 𝑡2 = 1, then its two remaining neighbors 𝑏2,𝑏3 must be set to 0 in order for it to
+play best response, and then there is no assignment to 𝑏1,𝑡1 s.t all of 𝑡1,𝑏1,𝑏2 play best response, in
+contradiction.
+And so it cannot be that 𝑢 = 𝑣. We move on to show that 𝑡2 must play 0. Assume by way of
+contradiction that 𝑡2 = 1. Then, seeing as exactly one of 𝑢, 𝑣 is productive, 𝑡2 must have exactly one
+more productive neighbor in order to play best response. If 𝑏2 = 1,𝑏3 = 0 we reach a contradiction
+as there is no assignment to 𝑡1,𝑏1 s.t 𝑡1,𝑏1,𝑏2 all play best response. If 𝑏2 = 0,𝑏3 = 1 we reach a
+contradiction as there is no assignment to 𝑡3,𝑡4,𝑏4,𝑏5 s.t all of 𝑡3,𝑡4,𝑏3,𝑏4,𝑏5 play best response.
+Lastly, one may verify that in the assignment where 𝑡1 = 𝑏4 = 1,𝑏1 = 𝑏2 = 𝑏3 = 𝑏5 = 𝑡2 = 𝑡3 = 𝑡4 = 0
+all nodes of the gadget play best response.
+□
+Now, for each variable that appears in the CNF formula, we choose one instance of it and one
+instance of its negation7 and connect the literal nodes representing these instances via a Negation
+Gadget, thus ensuring they are assigned opposite values in any PNE, according to Lemma 3.7. We
+note that this is not the only place where we use this gadget, as we will see shortly.
+We move on to introduce our Copy Gadget, which we will use to force literal nodes which
+represent the same variable to have the same assignment in any PNE. The Copy Gadget is composed
+of two negation gadgets 𝑛𝑔1,𝑛𝑔2, and two additional nodes 𝑥,𝑦 which have an edge between them.
+Say we have two nodes 𝑢, 𝑣 which we want to force to have the same assignment in any PNE, then
+we simply connect 𝑢 and 𝑥 to 𝑛𝑔1, and we connect 𝑣 and 𝑥 to 𝑛𝑔2. The gadget is demonstrated in
+Figure 7.
+7We will soon ensure that instances of the same variable would get the same assignment in any PNE, and thus it is sufficient
+to negate the assignments of only one instance of a variable and its negation.
+
+M. Gilboa
+9
+Fig. 6. Negation Gadget connecting 𝑢 and 𝑣.
+Fig. 7. Copy Gadget connecting 𝑢 and 𝑣.
+Lemma 3.8. In any Nash equilibrium in a graph 𝐺 which includes two nodes 𝑢, 𝑣 connected through
+a Copy Gadget 𝑐𝑝𝑔, 𝑢, 𝑣 must have the same assignment, and must have no productive neighbors from
+𝑐𝑝𝑔. In addition, if 𝑢 = 𝑣 then there exists an assignment to the nodes of 𝑐𝑝𝑔 s.t all of them play best
+response.
+Proof. We first show that 𝑢 and 𝑣 must have the same assignment. This follows directly from
+the fact that 𝑥 is connected to both 𝑢 and 𝑣 via a Negation Gadget. Therefore, from Lemma 3.7
+we have that 𝑥 ≠ 𝑢 and 𝑥 ≠ 𝑣, and so 𝑢 = 𝑣. Lemma 3.7 also tells us that the Negation Gadget
+cannot add productive neighbors to the nodes that are connected to it in any PNE, and therefore 𝑢
+and 𝑣 have no productive neighbors from 𝑐𝑝𝑔. Lastly, we show that there exists an assignment to
+the nodes of 𝑐𝑝𝑔 s.t they all play best response. From Lemma 3.7 𝑥 cannot have any productive
+neighbors from 𝑛𝑔1 or 𝑛𝑔2. Therefore, if 𝑢 = 𝑣 = 0 then we can assign 𝑥 = 1,𝑦 = 0, and if 𝑢 = 𝑣 = 1
+then we can assign 𝑥 = 0,𝑦 = 1. In both cases, we assign 𝑛𝑔1 and 𝑛𝑔2 as suggested in Lemma 3.7.
+One may verify that in this assignment indeed all nodes of 𝑐𝑝𝑔 play best response.
+□
+Now, for each variable in the CNF formula, we connect all the literal nodes representing its
+different instances via a chain of copy gadgets, thus (transitively) ensuring they are all assigned the
+same value in any PNE, according to Lemma 3.8.
+Given these lemmas and the graph we constructed, we can now prove Theorem 3.2.
+Proof. (Theorem 3.2) The problem is in NP, since an assignment to the nodes can be easily
+verified as a NTPNE by iterating over the nodes and checking whether they all play their best
+response. It is left to show the problem is NP-hard. Given a ONE-IN-THREE 3SAT instance, we
+construct a graph 𝐺 as described previously. If there exists a satisfying assignment to the 3SAT
+problem, we can set all literal nodes according to the assignment of their matching variable, and
+set all other nodes as described throughout lemmas 3.5, 3.7, 3.8, and according to those lemmas,
+we get a pure Nash equilibrium. On the opposite direction, if there exists a non-trivial pure Nash
+equilibrium, then by lemmas 3.6,3.4 in each clause exactly one literal node is assigned 1, and by
+lemmas 3.7,3.8 we have that literal nodes have the same assignment if they represent the same
+variable, and opposite ones if they represent a variable and its negation. Thus we can easily translate
+the NTPNE into a satisfying ONE-IN-THREE 3SAT assignment, assigning ’True’ to variables whose
+literal nodes are set to 1, and ’False’ otherwise.
+□
+
+u
+V
+t2
+t3
+b1
+b2
+b3
+b4
+b5u
+x
+ng1
+ng2M. Gilboa
+10
+We now wish to expand this result to two slightly more general classes of patterns. Firstly, we
+notice that the graph constructed throughout the proof of Theorem 3.2 is bounded8 by a maximum
+degree of 6. Therefore, the proof is indifferent to entries of the pattern from index 7 onward, which
+means it holds for any pattern that agrees with the first 7 entries of the 0-Or-2-Neighbors pattern.
+Corollary 3.9. Let 𝑇 be a BRP such that:
+• 𝑇 [0] = 𝑇 [2] = 1
+• ∀𝑘 ∈ {1, 3, 4, 5, 6} 𝑇 [𝑘] = 0
+Then NTPNE(𝑇) is NP-complete.
+Secondly, according to Theorem 7 in [Gilboa and Nisan, 2022], adding 1,0 at the beginning of a
+hard pattern that begins with 1 yields yet another hard pattern. Using this theorem recursively on
+the patterns of Corollary 3.9, we have that the equilibrium decision problem is hard for any pattern
+of the form:
+𝑇 = [1, 0, 1, 0, 1, 0, ..., 1, 0
+��������������������������������������
+𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′
+, 0, 0, 0, ?, ?, ...].
+Corollary 3.10. Fix 𝑚 ≥ 1, and let 𝑇 be a BRP such that:
+• ∀0 ≤ 𝑘 ≤ 𝑚
+(1) 𝑇 [2𝑘] = 1
+(2) 𝑇 [2𝑘 + 1] = 0
+• 𝑇 [2𝑚 + 2] = 𝑇 [2𝑚 + 3] = 𝑇 [2𝑚 + 4] = 0
+Then NTPNE(𝑇) is NP-complete.
+We will see later on that this result will also be of use during the proof of Theorem 5.1.
+There is one very similar class of patterns on which the proofs throughout the paper rely. This is
+the class of all finite patterns that start with a finite number of 1,0, followed by 1,1, i.e. all patterns
+of the form:
+𝑇 = [1, 0, 1, 0, 1, 0, ..., 1, 0
+��������������������������������������
+𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′
+, 1, 1, ?, ?, ..., 0, 0, ...].
+The complexity of those patterns was already discussed and solved in Section 5.4 of [Gilboa and
+Nisan, 2022], but was not formalized and so we state it here in the following lemma.
+Lemma 3.11. Fix 𝑚 ≥ 2, and let 𝑇 be a BRP s.t:
+• 𝑇 is finite
+• ∀𝑘 ∈ IN 𝑠.𝑡 2𝑘 ≤ 𝑚 𝑇 [2𝑘] = 1
+• 𝑇 [1] = 0
+• ∃1 ≤ 𝑛 𝑤ℎ𝑒𝑟𝑒 2𝑛 + 1 ≤ 𝑚 + 1, 𝑠.𝑡 𝑇 [2𝑛 + 1] = 1
+Then NTPNE(𝑇) is NP-complete under Turing reduction.
+Proof. The proof follows directly from Theorems 6 and 7 from [Gilboa and Nisan, 2022]9.
+□
+8A literal node is connected to 4 nodes within its clause gadget, and possibly 2 nodes from copy gadgets or 1 node from a
+negation gadget and 1 node from a copy gadget (assuming we connect the negation gadgets at the end of their respective
+Copy-Gadget-chains).
+9The reader who has read the details of Section 5.4 of [Gilboa and Nisan, 2022] may notice that in fact the use of Theorems
+6 and 7 from [Gilboa and Nisan, 2022] covers a slightly more general class of patterns, but this entire class is not needed
+currently, and is covered in Section 5.
+
+M. Gilboa
+11
+4
+HARDNESS OF SEMI-SHARP PATTERNS
+In this section we show hardness of semi-sharp Best-Response Patterns, beginning with a specific
+sub-class of those patterns in Section 4.1, and expanding to all other semi-sharp patterns in Section
+4.2. We remind the reader that semi-sharp patterns are patterns that begin with 1,0,0.
+4.1
+Semi-Sharp Patterns with Isolated Odd 1
+In this section we prove that any finite, semi-sharp pattern such that there exists some ’isolated’
+1 (meaning it has a zero right before and after it) at an odd index, presents a hard equilibrium
+decision problem. Those patterns can be summarized by the following form:
+𝑇 = [1, 0, 0, ?, ?, ..., 0
+1
+����
+𝑜𝑑𝑑 𝑖𝑛𝑑𝑒𝑥
+, 0, ?, ?, ..., 0, 0, 0, ...]
+Theorem 4.1. Let 𝑇 be a BRP which satisfies the following conditions:
+• 𝑇 is finite
+• 𝑇 is semi-sharp
+• ∃𝑚 ≥ 1
+s.t:
+(1) 𝑇 [2𝑚] = 𝑇 [2𝑚 + 2] = 0
+(2) 𝑇 [2𝑚 + 1] = 1
+Then NTPNE(𝑇) is NP-complete under Turing reduction.
+Before proceeding to the proof, we introduce two gadgets and prove two lemmas regarding their
+functionality.
+Force-1-Gadget: The first gadget is denoted the Force-1-Gadget, and it will appear in several parts
+of the graph we construct for the reduction. The goal of this gadget is to enable us to force any
+node to be assigned 1 in any Nash equilibrium in a PGG defined by 𝑇. This gadget is composed
+primarily of a triangle 𝑥,𝑦,𝑧, where the triangle’s nodes have also several ’Antenna’ nodes, which
+are connected only to their respective node from the triangle. Specifically, 𝑥 will have 2𝑚 + 1
+Antenna nodes, and 𝑦 and 𝑧 will each have 2𝑚 Antenna nodes. Say we have some node 𝑢, whose
+assignment we wish to force to be 1, then we simply connect 𝑢 to one of of the Antenna nodes of 𝑥,
+denoted 𝑎. The gadget is demonstrated in Figure 8.
+Add-1-Gadget: our second gadget of this proof is denoted the Add-1-Gadget, and its goal is
+to enable us to assure the existence of (at least) a single productive neighbor to any node in a
+Nash equilibrium of a PGG defined by 𝑇. Say we have a node 𝑣, to which we wish to add a single
+productive neighbors, in any equilibrium. We construct the Add-1-Gadget as follows. We create
+𝑚 + 1 nodes denoted 𝑥1, ...,𝑥𝑚+1, 𝑚 + 1 nodes denoted 𝑦1, ...𝑦𝑚+1, and an additional ’bridge’ node,
+denoted 𝑏. We connect 𝑥1 and 𝑦1 to all of the other 𝑥𝑖 and 𝑦𝑖 nodes. For all 𝑖, 𝑗 ≥ 2 s.t 𝑖 ≠ 𝑗, we
+create the edges (𝑥𝑖,𝑥𝑗), (𝑦𝑖,𝑦𝑗), (𝑥𝑖,𝑦𝑗) (the 𝑥𝑖,𝑦𝑖 nodes almost form a clique, except that for each
+𝑖 ≥ 2 we omit the edge (𝑥𝑖,𝑦𝑖)). Additionally, For all 𝑖 ≥ 2 the bridge node 𝑏 is connected to 𝑥𝑖 and
+to 𝑦𝑖. To 𝑏 we attach a Force-1-Gadget, and we also connect 𝑏 to 𝑣. The gadget is demonstrated in
+Figure 9.
+The following lemmas formalize the functionality of the two gadgets, beginning with the Force-
+1-Gadget in Lemma 4.2.
+Lemma 4.2. In any PNE in a graph 𝐺 corresponding to the BRP 𝑇 (from Theorem 4.1), where 𝐺 has
+a node 𝑢 that is connected to a Force-1-Gadget 𝑓 𝑔 as described, 𝑢 must be assigned 1, and its neighbor
+
+M. Gilboa
+12
+Fig. 8. Force-1-Gadget with 𝑚 = 2, attached to 𝑢.
+Fig. 9. Add-1-Gadget with 𝑚 = 2, attached to 𝑣.
+from 𝑓 𝑔, 𝑎, must be assigned 0.10 Furthermore, if 𝑢 = 1 there exists an assignment to the nodes of 𝑓 𝑔
+such that they each play their best response.
+Proof. First we show that 𝑢 must be assigned 1. Assume by way of contradiction that 𝑢 = 0.
+Divide into the following two cases. If 𝑥 = 1, then all of its Antenna nodes must be assigned 0
+(according to 𝑇). Additionally, 𝑦 and 𝑧 must also be assigned 0, as otherwise 𝑥 wouldn’t be playing
+best response, since 𝑇 is semi-sharp. Therefore, the best response of all of the Antenna nodes of 𝑦
+and 𝑧 is to play 1, which leaves 𝑦 and 𝑧 with 2𝑚 + 1 productive neighbors each, and so they are
+not playing best response, in contradiction. If 𝑥 = 0, then all of its Antenna nodes must play 1.
+Therefore, 𝑥 must have at least one other productive neighbor, as otherwise it would have 2𝑚 + 1
+productive neighbors and wouldn’t be playing best response; w.l.o.g assume 𝑦 = 1. Then all of
+𝑦’s Antenna nodes must play 0. Therefore, 𝑧 must play 0, as otherwise 𝑦 wouldn’t be playing best
+response. This means the best response for 𝑧’s Antenna nodes is to play 1, which leaves 𝑧 with
+2𝑚 + 1 productive neighbors, and so it isn’t playing best response, in contradiction. We move on
+to showing that 𝑎 must play 0. This follows directly from the fact that 𝑢 = 1. Since 𝑎 only has
+one other neighbor (𝑥), regardless of its strategy the best response for 𝑎, according to 𝑇, would be
+playing 0. It is left to show that when 𝑢 = 1 and 𝑎 = 0, there exists an assignment to the nodes
+of 𝑓 𝑔 s.t they all play best response. One may verify that when we set 𝑥 = 𝑦 = 𝑧 = 0 and set all
+the Antenna nodes in 𝑓 𝑔 (except for 𝑎) to 1, then all nodes of 𝑓 𝑔 play best response (specifically,
+𝑥,𝑦,𝑧 would each have exactly 2𝑚 productive neighbors, which, by definition of 𝑇, means they are
+playing best response).
+□
+We move on to proving the following Lemma, which formalizes the functionality of the Add-1-
+Gadget.
+Lemma 4.3. Lemma 8 In any graph 𝐺 corresponding to the BRP 𝑇 (from Theorem 4.1), where 𝐺 has
+a node 𝑣 that is connected to an Add-1-Gadget 𝑎𝑔 as described, there always exists an assignment to
+the nodes of 𝑎𝑔 s t they all play best response, regardless of 𝑣’s strategy. In addition, the bridge node 𝑏
+of 𝑎𝑔 must be assigned 1 in such an assignment.
+Proof. The claim that 𝑏 must play 1 follows directly from the fact that it has a Force-1-Gadget
+attached to it, i.e. from Lemma 4.2. Additionally, all the nodes of the Force-1-Gadget attached to 𝑏
+10The property that 𝑎 = 0 allows us to use the Force-1-Gadget without risking potentially adding productive neighbors to
+the respective node.
+
+a
+u
+X
+ZX1
+Y1
+X2
+Y2
+X3
+Y3
+b
+Force- 1-GadgetM. Gilboa
+13
+can be assigned as suggested in Lemma 4.2. It is left to show a possible assignment to the rest of the
+nodes of 𝑎𝑔. We divide into cases. If 𝑣 = 0, then we set 𝑥1 = 1 and all other 𝑥𝑖,𝑦𝑖 nodes we set to 0.
+If 𝑣 = 1, then we set 𝑥𝑖 = 𝑦𝑖 = 1 for all 1 ≤ 𝑖 ≤ 𝑚 + 1. One may verify that given these assignments
+all nodes of 𝑎𝑔 play their best response.
+□
+In addition to these two gadgets, we wish to introduce the following definition, after which we
+will proceed to the proof of Theorem 4.1, which we can now prove.
+Definition 4.4. Let 𝑇,𝑇 ′ be two BRPs. We say that 𝑇 ′ is shifted left by 𝑡 from 𝑇 if
+∀𝑘 ≥ 0 𝑇 ′[𝑘] = 𝑇 [𝑘 + 𝑡].
+Proof. (Theorem 4.1) Denote by 𝑇 ′ the pattern which is shifted left by 1 from T, i.e.:
+∀𝑘 ≥ 0 𝑇 ′[𝑘] = 𝑇 [𝑘 + 1].
+Notice that𝑇 ′ is flat, non-monotonic and finite, and therefore NTPNE(𝑇 ′) is NP-complete according
+to Theorem 4 in [Gilboa and Nisan, 2022], which allows us to construct a Turing reduction from it.
+The technique of the reduction is very similar to those of the proofs of Theorems 5,6 in [Gilboa and
+Nisan, 2022]. Given any graph 𝐺 = (𝑉, 𝐸), where 𝑉 = 𝑣1, ..., 𝑣𝑛, we construct 𝑛 graphs 𝐺1, ...,𝐺𝑛,
+where for each 1 ≤ 𝑖 ≤ 𝑛 the graph 𝐺𝑖 is defined as follows. The graph 𝐺𝑖 contains the original
+input graph 𝐺, and in addition, we connect a unique Add-1-Gadget to each of the original nodes,
+and a Force-1-Gadget only to node 𝑣𝑖. If there exists some non-trivial PNE in the PGG defined on
+𝐺 by 𝑇, let 𝑣 𝑗 be some node who plays 1. Then the same NTPNE is also an NTPNE in the PGG
+defined by 𝑇 ′ on 𝐺𝑗, when we assign the nodes of the additional gadget as suggested in lemmas
+4.2,4.3. To see why, notice that 𝑇 ′ is shifted left by 1 from 𝑇, and the Add-1-Gadgets ensure that all
+nodes have exactly one additional productive neighbor than they had in 𝐺.
+In the other direction, if there exists an NTPNE in a PGG defined by 𝑇 ′ on one of the graphs 𝐺𝑗,
+then by the same logic this is also a PNE in the game defined by 𝑇 on 𝐺 (ignoring the assignments
+of the added nodes). Moreover, the Force-1-Gadget ensures this assignment is non-trivial even after
+removing the added nodes, since 𝑣 𝑗 must play 1 in this assignment.
+□
+4.2
+All Semi-Sharp Patterns
+In this section we show that any finite, non-monotone, semi-sharp pattern presents a hard equilib-
+rium problem.
+Theorem 4.5. Let 𝑇1 be a finite, non-monotone, semi-sharp BRP. Then NTPNE(𝑇1) is NP-complete
+under Turing reduction.
+Before proceeding to the proof, we wish to introduce the following definition and prove two
+lemmas related to it.
+Definition 4.6. Let 𝑇,𝑇 ′ be two BRPs such that ∀𝑘 ∈ IN it holds that 𝑇 [𝑘] = 𝑇 ′[2𝑘]. Then we say
+that 𝑇 ′ is a double-pattern of 𝑇, and 𝑇 is the half-pattern of 𝑇 ′. Notice that a pattern has a unique
+half-pattern, whereas, since the definition does not restrict 𝑇 ′ in the odd indices, any pattern has
+infinite double-patterns.
+The first lemma is very simple and intuitive, stating that the largest index with value 1 in a half
+pattern is strictly smaller than the largest index with value 1 in its original pattern. This is true
+since for any index 𝑖 s.t the value of the half pattern is 1 in that index, the original pattern has a
+value of 1 in index 2𝑖.
+Lemma 4.7. Let 𝑇,𝑇 ′ be two finite BRPs such that 𝑇 is the half-pattern of 𝑇 ′. Denote by 𝑖 the largest
+index s.t 𝑇 [𝑖] = 1 and denote by 𝑗 the largest index s.t 𝑇 ′[𝑗] = 1. Then if 𝑗 > 0 we have that 𝑖 < 𝑗.
+
+M. Gilboa
+14
+Proof. The proof is trivially given by the definition of a half pattern, since 𝑇 ′[2𝑖] = 𝑇 [𝑖].
+□
+The next lemma is less trivial, stating the relation between hardness of a pattern and its double-
+pattern.
+Lemma 4.8. Let 𝑇 be a BRP such that NTPNE(𝑇) is NP-complete, and let 𝑇 ′ be a double-pattern of
+𝑇. Then NTPNE(𝑇 ′) is NP-complete.
+Proof. We use a specific case of the same reduction that was used to prove Theorem 4 in [Gilboa
+and Nisan, 2022]. Given a graph 𝐺1 = (𝑉1, 𝐸1) as input, where 𝑉1 = 𝑣1
+1, ..., 𝑣1
+𝑛, we create another
+replica of it 𝐺2 = (𝑉2, 𝐸2), where 𝑉2 = 𝑣2
+1, ..., 𝑣2
+𝑛. For each node (from both graphs), we add edges
+connecting it to all replicas of its neighbors from the opposite graph. That is, the following group
+of edges is added to the graph:
+𝐸 = {(𝑣1
+𝑖 , 𝑣2
+𝑗)|(𝑣1
+𝑖 , 𝑣1
+𝑗) ∈ 𝐸1}.
+A demonstration of the reduction can be seen in Figure 10.
+Fig. 10. Example of the reduction of Lemma 4.8’s proof.
+Denote by 𝑃 the PGG defined on 𝐺1 by 𝑇, and by 𝑃 ′ the PGG defined by 𝑇 ′ on 𝐺 ′ = (𝑉 ′, 𝐸′)
+where 𝑉 ′ = 𝑉1 ∪ 𝑉2, 𝐸′ = 𝐸 ∪ 𝐸1 ∪ 𝐸2. We show that there exists an NTPNE in 𝑃 iff there exists
+one in 𝑃 ′. If there exists an NTPNE in 𝑃, we simply give the nodes of 𝐺2 the same assignment as
+those of 𝐺1. Since 𝑇 ′ is a double pattern of 𝑇, any node 𝑣 ′ ∈ 𝑉 ′ must play best response, having
+exactly twice as many supporting neighbors than it had (or its replica had) in 𝑃.
+In the opposite direction, if there exists an NTPNE in 𝑃 ′, notice that for all 1 ≤ 𝑖 ≤ 𝑛 it must
+be that 𝑣1
+𝑖 and 𝑣2
+𝑖 have identical assignments, since they both share exactly the same neighbors,
+and thus have identical best responses. Therefore, any node 𝑣 ′ ∈ 𝑉 must have an even number
+of productive neighbors, half of which are in 𝑉1 and the other half in 𝑉2 (as for each productive
+neighbor from 𝑉1 there is a respective productive neighbor from 𝑉2). We then simply ignore 𝐺2, and
+leave the assignment of 𝐺1 as it is, and each node shall now have exactly half as many productive
+neighbors as it had in the original assignment. Since 𝑇 is a half pattern of 𝑇 ′, we get an NTPNE in
+𝑃.11
+□
+Given lemmas 4.7,4.8, we are now able to prove Theorem 4.5. The intuitive idea of the proof is
+that we halve the pattern 𝑇1 (i.e. find its half-pattern) repeatedly, until eventually we reach some
+pattern for which we already know the equilibrium problem is hard, which, as we will see, must
+happen at some point. Then, by applying Lemma 4.8 recursively, we have that 𝑇1 is hard.
+11In both directions of this proof, the non-triviality comes from the fact that 𝑇 [0] = 𝑇 ′[0], by definition. Therefore, a
+non-trivial PNE in one domain must translate to a non-trivial one in the other.
+
+Vi
+2
+V2
+2
+V
+3M. Gilboa
+15
+Proof. (Theorem 4.5) From Lemma 4.7 we have that if we halve a non-flat pattern enough times,
+we will eventually reach the Best-Shot pattern: 𝑇𝐵𝑒𝑠𝑡−𝑆ℎ𝑜𝑡 [0] = 1 and ∀𝑘 ≥ 1 𝑇𝐵𝑒𝑠𝑡−𝑆ℎ𝑜𝑡 [𝑘] = 0.
+Divide into two cases.
+In the first case assume that ∀𝑘 ∈ IN it holds that𝑇1[2𝑘] = 0. In this case, we know that no matter
+how many times we halve 𝑇1 into patterns 𝑇2,𝑇3, ..., for each 𝑖 we will have that 𝑇𝑖 [1] = 0, i.e. the
+value in index 1 of all these half-patterns will always be 0, i.e. 𝑇𝑖 [1] = 0 for all 𝑖. Assume that we
+halve𝑇1 repeatedly into patterns𝑇2,𝑇3, ...,𝑇𝑚 (where𝑇𝑖 is the half pattern of𝑇𝑖−1) such that𝑇𝑚 is the
+first time that we reach the Best-Shot pattern. Observe 𝑇𝑚−1. For any even index 𝑘 ≠ 0 it must hold
+that 𝑇𝑚−1[𝑘] = 0, otherwise 𝑇𝑚 would not be the Best-Shot pattern. Additionally, there must exist
+at least one odd index 𝑗 s.t 𝑇𝑚−1[𝑗] = 1, since 𝑇𝑚 is the first time we reach the Best-Shot pattern.
+For these two reasons, we have that 𝑇𝑚−1 satisfies the conditions of Theorem 4.1 and therefore
+NTPNE(𝑇𝑚−1) is NP-complete under Turing reduction. From Lemma 4.8 (used inductively), we
+have that ∀1 ≤ 𝑖 ≤ 𝑚 − 2 NTPNE(𝑇𝑖) is also NP-complete under Turing reduction, and specifically
+NTPNE(𝑇1).
+In the second case, assume that there exists some 𝑘 ∈ IN s.t 𝑇1[2𝑘] = 1. In that case, after at most
+𝑘 halvings, we reach some pattern for which the value of index 1 is 1. Assume that we halve 𝑇1
+repeatedly into patterns 𝑇2,𝑇3, ...,𝑇𝑛 (where 𝑇𝑖 is the half pattern of 𝑇𝑖−1) such that 𝑇𝑛 is the first
+time that we reach a pattern for which index 1 is 1, i.e. ∀1 ≤ 𝑖 ≤ 𝑛 − 1 𝑇𝑖 [1] = 0 and 𝑇𝑛[1] = 1.
+Notice that, additionally, by definition of a half-pattern for each 𝑖 it holds that 𝑇𝑖 [0] = 1 (since
+𝑇1[0] = 1). If 𝑇𝑛 is non-monotone, then by Theorem 5 in [Gilboa and Nisan, 2022] we have that
+NTPNE(𝑇𝑛) is NP-complete under Turing reduction, and from Lemma 4.8 (used inductively), we
+have that ∀1 ≤ 𝑖 ≤ 𝑛 − 1 NTPNE(𝑇𝑖) is also NP-complete under Turing reduction, and specifically
+NTPNE(𝑇1). Otherwise (i.e. 𝑇𝑛 is monotone), denote by 𝑙 the largest index s.t 𝑇𝑛[𝑙] = 1, and observe
+𝑇𝑛−1. By definition of double-patterns, we have that:
+∀𝑗 ∈ IN 𝑇𝑛−1[2𝑗] =
+�
+1
+if 𝑗 ≤ 𝑙
+0
+otherwise
+i.e. the value in the even indices is 1 until 2𝑙, and 0 afterwards. Since 𝑇𝑛 is defined to be the first
+halving of 𝑇1 s.t its value in index 1 is 1, we have that 𝑇𝑛−1[1] = 0. However, since the definition of
+a double-pattern does not restrict it in the odd indices, there might be some odd indices (strictly
+larger than 1) for which the value of 𝑇𝑛−1 is 1. Divide into 3 sub-cases:
+Sub-case 1: If there exists some 𝑧 ≤ 𝑙 s.t 𝑇𝑛−1[2𝑧 + 1] = 1, then by Lemma 3.11, we have that
+NTPNE(𝑇𝑛−1) is NP-complete under Turing reduction.
+Sub-case 2: Otherwise, if there exists some 𝑧 > 𝑙 s.t 𝑇𝑛−1[2𝑧 + 1] = 1, then observe the pattern
+𝑇 ′
+𝑛−1, which we define as the pattern shifted left by 2𝑙 from 𝑇𝑛−1 i.e.:
+∀𝑗 ∈ IN 𝑇 ′
+𝑛−1[𝑗] = 𝑇𝑛−1[𝑗 + 2𝑙]
+Notice that this pattern satisfies the conditions of Theorem 4.1, and therefore NTPNE(𝑇 ′
+𝑛−1) is
+NP-complete under Turing reduction. Then, by applying Theorem 7 from [Gilboa and Nisan, 2022]
+𝑙 times, we have that NTPNE(𝑇𝑛−1) is also NP-complete under Turing reduction.
+Sub-case 3: Otherwise (i.e. there is no odd index whatsoever in which the value of 𝑇𝑛−1 is 1), then
+by Corollary 3.10 we have that NTPNE(𝑇𝑛−1) is NP-complete under Turing reduction.
+And so, in either case we have that NTPNE(𝑇𝑛−1) is NP-complete under Turing reduction, and
+therefore from Lemma 4.8 (used inductively), we have that ∀1 ≤ 𝑖 ≤ 𝑛 − 1 NTPNE(𝑇𝑖) is also
+NP-complete under Turing reduction, and specifically NTPNE(𝑇1).
+□
+
+M. Gilboa
+16
+5
+HARDNESS OF ALL SPIKED PATTERNS
+There are several finite, spiked patterns that we have not yet proved hardness for, and we now have
+enough tools to close the remaining gaps. We remind the reader that spiked patterns are patterns
+that begin with 1,0,1. The following theorem formalizes the result of this section, and completes
+the characterization of all finite patterns.
+Theorem 5.1. Let𝑇 be a finite, spiked BRP. Then NTPNE(𝑇) is NP-complete under Turing reduction.
+The intuitive idea of the proof is as follows. If the pattern simply alternates between 1 and 0 a
+finite amount of times, followed infinite 0’s, i.e. the pattern is of the form
+𝑇 = [1, 0, 1, 0, 1, 0, ..., 1, 0
+��������������������������������������
+𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′
+, 0, 0, 0, ...]
+then the problem12 is already shown to be hard by Corollary 3.10. Otherwise, we wish to look at
+the first "disturbance" where this pattern stops alternating from 1 to 0 regularly. Either the first
+"disturbance" is a 1 at an odd index, i.e. the pattern is of the form
+𝑇 = [1, 0, 1, 0, 1, 0, ..., 1, 0
+��������������������������������������
+𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′
+, 1, 1, ?, ?, ...]
+or the first "disturbance" is a 0 at an even index, i.e. the pattern is of the form
+𝑇 = [1, 0, 1, 0, 1, 0, ..., 1, 0
+��������������������������������������
+𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′
+, 0, ?, ?, ..., 1, ?, ?, ...]
+(in the latter option, after the first "disturbance" there must be some other index with value 1, since
+the pattern does not fit the form of Corollary 3.10). The first option was solved in Lemma 3.11, and
+the second option can be solved using our previous results, as we shall now formalize in the proof.
+Proof. (Theorem 5.1) If𝑇 satisfies the conditions of Corollary 3.10 or Lemma 3.11 then NTPNE(𝑇)
+is NP-complete under Turing reduction according to them. Otherwise, let 𝑘 be the smallest integer
+such that 𝑇 [2𝑘] = 0. Denote by 𝑇 ′ the pattern which is shifted left by 2𝑘 − 2 from T, i.e.:
+∀𝑗 ≥ 0 𝑇 ′[𝑗] = 𝑇 [𝑗 + 2𝑘 − 2]
+Notice that from definition of 𝑘 (being the first even index such that 𝑇 [2𝑘] = 0) we have that for
+all 𝑗 < 𝑘 it holds that 𝑇 [2𝑗] = 1. Moreover, since 𝑇 does not satisfy the conditions of Lemma 3.11
+it must hold for all 𝑗 ≤ 𝑘 that 𝑇 [2𝑗 − 1] = 0, i.e. the value of 𝑇 in the odd indices until 2𝑘 is 0
+(since otherwise𝑇 would start with a finite number of 1,0, followed by 2 consecutive 1’s, and would
+satisfy the conditions of Lemma 3.11). Thus, we have that
+∀𝑗 < 2𝑘 𝑇 [𝑗] =
+�
+1 if 𝑗 is even
+0 if 𝑗 is odd
+(1)
+In particular, we have that𝑇 [2𝑘 − 2] = 1, 𝑇 [2𝑘 − 1] = 0, which implies that𝑇 ′[0] = 1, 𝑇 ′[1] = 0;
+as 𝑇 [2𝑘] = 0 we have that 𝑇 ′[2] = 0, and thus we conclude that 𝑇 ′ is semi-sharp. In addition, since
+𝑇 does not satisfy the conditions of Corollary 3.10, there must be some other index 𝑥 > 2𝑘 such
+that 𝑇 [𝑥] = 1, and therefore we have that 𝑇 ′ is non-monotone. Therefore, by Theorems 4.1, 4.5, we
+have that NTPNE(𝑇 ′) is NP-complete under Turing reduction. We now wish to use this in order to
+prove that NTPNE(𝑇) is also hard.
+12In fact, Corollary 3.10 gives a more general result, but we currently only need the private case where the pattern ends
+with infinite 0’s.
+
+M. Gilboa
+17
+From Equation 1, we can apply Theorem 7 of [Gilboa and Nisan, 2022] (𝑘 − 1) times, and we
+have that NTPNE(𝑇) is NP-complete under Turing reduction.
+□
+ACKNOWLEDGMENTS
+I would like to thank Noam Nisan for many useful conversations throughout the work, and for
+suggesting the Copy Gadget seen in the proof of Theorem 3.2.
+I would like to thank Roy Gilboa for many useful conversations throughout the work and for
+adjusting the Copy Gadget seen in the proof of Theorem 3.2.
+I would like to thank Noam Nisan for communicating to me the solution of the monotone case by
+Max Klimm, and the alternative derivation by Sigal Oren.
+This project has received funding from the European Research Council (ERC) under the European
+Union’s Horizon 2020 Research and Innovation Programme (grant agreement no. 740282).
+REFERENCES
+Yann Bramoullé and Rachel Kranton. 2007. Public Goods in Networks. Journal of Economic Theory 135, 1 (2007), 478–494.
+https://doi.org/10.1016/j.jet.2006.06.006
+Mustapha Chellali, Odile Favaron, Adriana Hansberg, and Lutz Volkmann. 2012. k-Domination and k-Independence in
+Graphs: A Survey. Graphs and Combinatorics 28, 1 (2012), 1–55. https://doi.org/10.1007/s00373-011-1040-3
+Matan Gilboa and Noam Nisan. 2022. Complexity of Public Goods Games on Graphs. In Proceedings of the 15th International
+Symposium on Algorithmic Game Theory (Lecture Notes in Computer Science), Panagiotis Kanellopoulos, Maria Kyropoulou,
+and Alexandros Voudouris (Eds.). Springer Cham, Colchester UK, 151–168.
+David Kempe, Sixie Yu, and Yevgeniy Vorobeychik. 2021. Inducing Equilibria in Networked Public Goods Games through
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+Arnab Maiti and Palash Dey. 2022. On Parameterized Complexity of Binary Networked Public Goods Game. In proceedings of
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+for Autonomous Agents and Multiagent Systems (IFAAMAS), Auckland New Zealand, 871–879.
+Christos Papadimitriou and Binghui Peng. 2021. Public Goods Games in Directed Networks. In EC ’21: Proceedings of the
+22nd ACM Conference on Economics and Computation. Association for Computing Machinery, New York, NY, United
+States, Budapest Hungary, 745–762.
+Thomas J. Schaefer. 1978. The Complexity of Satisfiability Problems. In STOC ’78: Proceedings of the tenth annual ACM
+symposium on Theory of computing. Association for Computing Machinery, New York, NY, United States, San Diego
+California USA, 216–226.
+Yongjie Yang and Jianxin Wang. 2020. A Refined Study of the Complexity of Binary Networked Public Goods Games. (2020).
+https://doi.org/10.48550/arXiv.2012.02916
+Sixie Yu, Kai Zhou, Jeffrey Brantingham, and Yevgeniy Vorobeychik. 2020. Computing Equilibria in Binary Networked
+Public Goods Games. In Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 34(2). Association for the
+Advancement of Artificial Intelligence (AAAI), New York, NY, USA, 2310–2317. https://doi.org/10.1609/aaai.v34i02.5609
+Sixie Yu, Kai Zhou, Jeffrey Brantingham, and Yevgeniy Vorobeychik. 2021. Computing Equilibria in Binary Networked
+Public Goods Games. (2021). https://doi.org/10.48550/arXiv.1911.05788
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+A Note On Acyclic Token Sliding Reconfiguration
+Graphs of Independent Sets
+David Avis1
+Duc A. Hoang2
+1 Graduate School of Informatics, Kyoto University, Japan
+School of Computer Science, McGill University, Canada
+[email protected]
+2 Graduate School of Informatics, Kyoto University, Japan
+[email protected]
+Abstract
+We continue the study of token sliding reconfiguration graphs of independent sets initiated by
+the authors in an earlier paper (arXiv:2203.16861). Two of the topics in that paper were to study
+which graphs G are token sliding graphs and which properties of a graph are inherited by a token
+sliding graph. In this paper we continue this study specializing on the case of when G and/or its
+token sliding graph TSk(G) is a tree or forest, where k is the size of the independent sets considered.
+We consider two problems. The first is to find necessary and sufficient conditions on G for TSk(G)
+to be a forest. The second is to find necessary and sufficient conditions for a tree or forest to be
+a token sliding graph. For the first problem we give a forbidden subgraph characterization for the
+cases of k = 2, 3. For the second problem we show that for every k-ary tree T there is a graph G for
+which TSk+1(G) is isomorphic to T. A number of other results are given along with a join operation
+that aids in the construction of TSk(G)-graphs.
+1
+Introduction
+In a reconfiguration variant of a computational problem (e.g., Satisfiability, Independent Set,
+Vertex-Coloring, etc.), a transformation rule that describes an adjacency relation between feasi-
+ble solutions (e.g., satisfying truth assignments, independent sets, proper vertex-colorings, etc.) of the
+problem is given. One of the main goals is to decide whether there is a sequence of adjacent feasible
+solutions that “reconfigures” one given solution into another. Another way of looking at these reconfig-
+uration problems is via the so-called reconfiguration graph—a graph whose nodes are feasible solutions
+and two nodes are adjacent if one can be obtained from the other by applying the given rule exactly once.
+The mentioned question now becomes deciding whether there is a path between two given nodes in the
+reconfiguration graph. Recently, reconfiguration problems have been intensively studied from different
+perspectives [2, 6–8].
+One of the most well-studied reconfiguration variants of Independent Set is the so-called Token
+Sliding problem, which was first introduced by Hearn and Demaine [4] in 2005.
+We refer readers
+to [2, 7, 8] and the references therein for more details. Surprisingly, though Token Sliding has been
+well-investigated, the realizability and structural properties of its corresponding reconfiguration graph—
+the one which we will refer to as the TSk-graph (which stands for Token Sliding (Reconfiguration)
+graph)—have not been studied until recently [1]. On the other hand, when considering either general
+vertex subsets, dominating sets, or proper vertex-colorings of a graph as the “input feasible solutions”,
+their corresponding reconfiguration graphs have been very well-characterized [5, 6].
+For any graph-theoretic terminology and notation not defined here, we refer readers to [3]. Given a
+graph G = (V, E) and an integer k ≥ 2. For two sets X, Y , we sometimes use X + Y and X − Y to
+indicate X ∪ Y and X \ Y . We abbreviate X ∪ {u} (resp., X \ {u}) by X + u (resp., X − u). We use
+NG(u), or simply just N(u) when the graph G is clear from the context, to denote the (open) neighbors
+of u, i.e., set of all vertices in G that are adjacent to u. The closed neighbors of u, denoted by NG[u]
+or simply N[u], is the set NG(u) + u. The degree of u, denoted by degG(u), is nothing but the size of
+NG(u). An independent set (or stable set) of G is a vertex subset I such that for every u, v ∈ I we
+have uv /∈ E(G). The TSk-graph of G, denoted by TSk(G), takes all size-k independent sets of G as its
+1
+arXiv:2301.00317v1  [math.CO]  1 Jan 2023
+
+nodes and two nodes I, J are adjacent (under Token Sliding (TS)) if there exist two vertices u, v ∈ V (G)
+such that I − J = {u}, J − I = {v}, and uv ∈ E(G). Two graphs G and H are isomorphic, denoted
+by G ≃ H, if there exists a bijective mapping f : V (G) → V (H) such that uv ∈ E(G) if and only if
+f(u)f(v) ∈ E(H). A graph G is called a TSk-graph if there exists a graph H such that G ≃ TSk(H). A
+forest is a graph having no cycles (i.e., it is acyclic) and a connected forest is a tree. A TSk-tree/forest
+is a TSk-graph which is also a tree/forest. Figure 1 illustrates a TS2-tree on six vertices (right). In [1],
+ab
+ac
+bd
+ae
+ef
+ce
+TS2(G)
+a
+b
+c
+d
+e
+f
+G
+Figure 1: A graph G with TS2(G) = D1,3,2. Each node ab represents a size-2 stable set of G.
+the authors studied various properties of the family of TSk-graphs. For a graph G, two of the questions
+studied were:
+(Q1) What are necessary and sufficient conditions for G so that TSk(G) is a forest?
+(Q2) What are necessary and sufficient conditions for G to be a TSk-graph?
+In this paper, we study these two questions for the case when G is a tree or a forest.
+The union G ∪ H of two (labelled) graphs G and H is the graph with V (G ∪ H) = V (G) ∪ V (H) and
+E(G ∪ H) = E(G) ∪ E(H). When vertices and edges of G and H are considered distinct regardless of
+their labels, we say that G ∪ H is the disjoint union of G and H, and write G + H instead of G ∪ H to
+distinguish from their union. We respectively denote by Kn, Pn, and Cn the complete graph, path, and
+cycle on n vertices. Km,n (m ≤ n) is the complete bipartite graph whose two partite sets are of sizes m
+and n respectively. K1,n is also called a star—a tree obtained by attaching n leaves to a central vertex.
+A family of graphs that we will use in the sequel generalizes stars and paths. For fix integers n, r, s ≥ 1,
+let Dr,n,s be the tree obtained from Pn by appending r leaves at one end and s leaves at the other. Note
+that D1,1,s is the star K1,s+1 and D1,n,1 is the path Pn+2. Figure 1 illustrates D1,3,2 (right). An n-ary
+tree is a rooted tree in which each node has at most n children. Any tree with maximum degree at most
+n+1 can be rooted at a vertex with degree at most n (e.g., a leaf) to produce a n-ary tree. In particular,
+a 2-ary tree is nothing but the well-known binary tree.
+In the next section, we begin by partially answering (Q1) when G is a tree/forest and k ∈ {2, 3} and
+conclude the section by conjecturing for k ≥ 4. Then, before addressing (Q2) for some trees/forests, in
+particular k-ary trees and Dr,n,s, we define an important graph operation which, under certain conditions,
+can be used for combining two TSk-graphs by taking their union to obtain a new one. The final section
+of the paper gives some concluding remarks.
+2
+Results on (Q1)
+In this section, we prove the necessary and sufficient conditions on a tree/forest G such that TSk(G) is
+acyclic for k ∈ {2, 3}, partially answering (Q1).
+We begin with some definitions and observations. The complement G of a graph G is the graph
+with V (G) = V (G) and E(G) = {uv : uv /∈ E(G)}. The size-m matching, denoted by mK2, is the
+graph obtained by taking the disjoint union of m copies of K2. Observe that TS2(2K2) ≃ C4. We
+label vertices in a Dr,n,s (r, s ≥ 1) as follows: Vertices of Pn are labelled p1, . . . , pn.
+The r leaves
+attached to p1 are u1, . . . , ur and the s leaves attached to pn are v1, . . . , vn. D2,2,2 is shaped like an
+2
+
+G
+TS2(G)
+Figure 2: A list G of n-vertex graphs G (4 ≤ n ≤ 7) excluding Cn (n ≥ 5) such that if TS2(G′) has no
+cycle then G′ does not contain any member G of G as an induced subgraph.
+3
+
+H and TS2(D2,2,2) contains a cycle C8 whose vertex-set is {u1v1, u1p2, u1v2, p1v2, u2v2, u2p2, u2v1, p1v1}.
+Indeed, respectively from Lemma 1 of [1] and Figure 2, if a n-vertex graph G is either Cn (n ≥ 5) or a
+graph in the list G described in Figure 2 (which includes 2K2 and D2,2,2), the graph TS2(G) contains a
+cycle. Additionally, we have:
+Lemma 1.
+(a) For k ≥ 2, TSk(2K2 + nK1) contains a cycle C4 if n ≥ k − 2 otherwise it is acyclic.
+(b) For k ∈ {2, 3}, s ≥ 1, TSk(D1,n,s) contains a cycle C4 if n ≥ 2k − 1 otherwise it is acyclic.
+(c) For k ∈ {2, 3} and r, s ≥ 2, TSk(Dr,n,s) contains a cycle C8 if n ≥ 2k − 2 otherwise it is acyclic.
+Proof.
+(a) If n < k − 2, there is no size-k independent set in 2K2 + nK1, thus its TSk-graph is
+obviously acyclic. Otherwise, let I ⊆ V (nK1) be an arbitrary independent set of size k − 2, and let
+E(2K2) = {ab, cd}. Then, {I +a+c, I +a+d, I +b+c, I +b+d} induce a C4 in TSk(2K2 +nK1).
+(b) Observe that if n ≥ 2k − 1, D1,n,s contains an induced 2K2 + (k − 2)K1, which can be obtained
+by taking u1p1 and pnv1 as edges of 2K2 and the remaining k − 2 independent vertices from the
+path D1,n,s − {u1, p1, p2, pn−1, pn, v1, . . . , vs} on n − 4 vertices. (Since n ≥ 2k − 1, this path has
+an independent set of size at least ⌈(n − 4)/2⌉ ≥ ⌈(2k − 5)/2⌉ = k − 2.) Then, using a similar
+argument as in (a) we have TSk(D1,n,s) contains a C4.
+On the other hand, if n ≤ 2k − 2 for k ∈ {2, 3}, since D1,n−1,s is always an induced subgraph of
+D1,n,s for n ≥ 2, it follows that if TS2(D1,n−1,s) has a cycle then so is TS2(D1,n,s). Therefore,
+it suffices to show that TSk(D1,2k−2,s) is acyclic for k ∈ {2, 3}. Indeed, based on the number of
+tokens placed on the path u1p1 . . . pn (which is at most three), one can verify that each component
+of TSk(D1,2k−2,s) is either an isolated vertex, a path, or a star.
+(c) Observe that if n ≥ 2k −2, Dr,n,s contains the independent sets I +u1 +v1, I +u1 +pn, I +u1 +vs,
+I + p1 + v1, I + p1 + vs, I + ur + v1, I + ur + pn, and I + ur + vs, where I = ∅ when n = 2 and
+otherwise I is an independent set of the path p2 . . . pn−1 of size k − 2. (Note that p2 . . . pn−1 has
+an independent set of size at most ⌈(n − 2)/2⌉ ≥ k − 2.) They indeed induce a C8 in TSk(Dr,n,s).
+On the other hand, if n ≤ 2k−3 for k ∈ {2, 3}, using a similar case-analysis as in (b), one can verify
+that each component of TSk(Dr,n,s) is either an isolated vertex, a path, or a star, and therefore it
+is acyclic.
+We are now ready to show the necessary and sufficient conditions for a tree/forest G such that TSk(G)
+is acyclic, where k ∈ {2, 3}.
+Proposition 2. Let T be a tree. Then TS2(T) is acyclic if and only if T is {2K2, D2,2,2}-free.
+Proof. (⇒) Suppose to the contrary that either 2K2 or D2,2,2 is an induced subgraph of T. In the first
+case it follows from the discussion above that TS2(T) contains a C4 and in the second case that it
+contains a C8.
+(⇐) We assume that TS2(T) contains a cycle and show that it must contain one of the two forbidden
+subgraphs. Firstly, suppose that T is a path Pn. Since TS2(T) contains a cycle, it follows from
+Lemma 1(b) that n ≥ 5 and so T contains an induced 2K2.
+We now assume T has a vertex of at least degree 3. We will construct a copy T ′ of T by initially
+choosing a vertex a of maximum degree in T and letting T ′ = N[a]. Note that TS2(T ′) is acyclic.
+We add edges from T to T ′ and show after each addition that either T ′ contains a forbidden
+subgraph, so we are done, or that TS2(T ′) remains acyclic so that T ̸= T ′.
+Let b be a child of a of highest degree, c be a child of next highest degree, and d be any other
+child. Since TS2(T ′) is acyclic T ̸= T ′ and b must have r ≥ 1 children. Let e be a child of b with
+maximum degree. We add N[b] to T ′ obtaining a copy of Dr,2,s, where s = degT (a) − 1 ≥ 2. If
+r ≥ 2, we have the required forbidden induced subgraph. If r = 1 then by Lemma 1(b) TS2(T ′)
+is acyclic, so there must be extra edges to add to T ′. If c has a child y then {b, c, e, y} induce a
+2K2. Otherwise, e must have at least one child g. Adding eg to T ′ we obtain 2K2 as an induced
+subgraph on {a, d, e, g}. This completes the proof.
+4
+
+a
+b
+c
+d
+e
+a
+d
+e
+b
+c
+y
+a
+d
+b
+c
+e
+g
+r ≥ 2
+r = 1
+Figure 3: Illustration for Proposition 2: Some trees T ′ containing N[b] whose TS2-graphs have a cycle.
+Here r is the number of children of b. Copies of 2K2 and D2,2,2 are marked by red color.
+Corollary 3. Let T be a tree. Then TS2(T) is acyclic if and only if T is either K1,s or D1,2,s for some
+positive integer s.
+Proof. The proof of Proposition 2 can be viewed as an algorithm that takes a tree T and either terminates
+with T = T ′ being one of the trees in the corollary or finds a forbidden induced graph in T.
+Corollary 4. Let F be a forest. Then TS2(F) is a acyclic if and only if F is {2K2, D2,2,2}-free.
+Proof. We prove that TS2(F) contains a cycle if and only if F contains one of the graphs in {2K2, D2,2,2}
+as an induced subgraph.
+Suppose that TS2(F) contains a cycle. Since the independent sets have size two, both vertices of
+each independent set must lie in the same connected component T of F. By Proposition 2, the tree T
+must have either 2K2 or D2,2,2 as an induced subgraph.
+Conversely if F contains 2K2 or D2,2,2 as an induced subgraph then TS2(F) contains respectively a
+C4 or a C8.
+Moving to the case of stable sets of size three, the conditions for trees and forests differ slightly. We
+deal with the tree case first.
+Proposition 5. Let T be a tree. Then TS3(T) is acyclic if and only if T is {2K2 + K1, D2,4,2}-free.
+Proof. The structure of the proof is the same as for Proposition 2. However, there are more cases to
+consider.
+(⇒) Suppose to the contrary that either 2K2 + K1 or D2,4,2 is an induced subgraph of T. In the first
+case it follows that TS3(T) contains a C4 and in the second case that it contains a C8.
+(⇐) We assume that TS3(T) contains a cycle and show that it must contain one of the two forbidden
+subgraphs. The first part of the proof is essentially the same as for Proposition 2 with minor
+modifications. Firstly suppose that T is a path Pn. Since TS3(T) contains a cycle it follows from
+Lemma 1(b) that n ≥ 7 and so T contains an induced 2K2 + K1.
+We now assume T has a vertex of at least degree 3. We will construct a copy T ′ of T by initially
+choosing a vertex a of maximum degree in T and letting T ′ = N[a]. Note that TS3(T ′) is acyclic.
+We add edges from T to T ′ showing after each addition that either T ′ contains a forbidden subgraph,
+so we are done, or that TS3(T ′) remains acyclic so that T ̸= T ′.
+Let b be a child of a of highest degree, c be a child of next highest degree, and d be any other child.
+Since TS3(T ′) is acyclic T ̸= T ′ and b must have r ≥ 1 children. Let e be a child of b with maximum
+degree. If c has a child y then {b, c, d, e, y} induce a 2K2 + K1 and we are done. Otherwise we
+5
+
+add N[b] to T ′ obtaining a copy of Dr,2,s, where s = degT (a) − 1 ≥ 2. By Lemma 1(c), TS3(T ′) is
+acyclic and so T ̸= T ′. There are two cases:
+(r ≥ 2) Let f be a second child of b and let g be a child of e. Adding eg to T ′ we obtain 2K2 + K1
+as an induced subgraph on {a, d, e, f, g}.
+(r = 1) Since e is the only child of b it must have children. Let t ≥ 1 be the number of children of e
+and let h be the child of e of maximum degree. We add N[e] to T ′ obtaining a copy of Dt,3,s
+and TS3(T ′) is acyclic by Lemma 1(c). There are two subcases:
+(t ≥ 2) Let i be any other child of e. Since TS3(T ′) is acyclic h must have at least one child j.
+We have now constructed an induced 2K2 + K1 on {a, d, h, i, j}.
+(t = 1) If h has a single child k add hk to T ′ which is a copy of D1,4,s and again by Lemma 1(c)
+TS3(T ′) is acyclic. So k has a child l. Adding kl to T ′ it contains an induced P7 and we
+find the forbidden subgraph 2K2 + K1 on vertices {a, d, e, k, l}. Otherwise, h has at least
+two children including vertices k and m. Adding edges hk and hm to T ′ we obtain the
+forbidden subgraph D2,4,2. This completes the proof.
+a
+b
+c
+d
+e
+f
+g
+a
+c
+d
+b
+e
+h
+i
+j
+a
+c
+d
+b
+e
+h
+k
+l
+a
+c
+d
+b
+h
+i
+j
+r ≥ 2
+r = 1, t ≥ 2
+r = 1, t = 1
+e
+Figure 4: Illustration for Proposition 5: Some trees T ′ containing N[b] whose TS3-graphs have a cycle.
+Here r and t are respectively the number of children of b and its child e. Copies of 2K2 + K1 and D2,4,2
+are marked by red color.
+Corollary 6. Let T be a tree. Then TS3(T) is a acyclic if and only if for some positive integer s, T is
+either K1,s, D1,n,s where n ≤ 4, or Dr,n,s where r ≥ 2 and n ≤ 3.
+Proof. The proof of Proposition 5 can be viewed as an algorithm that takes a tree T and either terminates
+with T = T ′ being one of the trees in the corollary or finds a forbidden induced graph in T showing that
+TS3(T) has a cycle.
+6
+
+Corollary 7. Let F be a forest.
+Then TS3(F) is a forest if and only if F is {2K2 + K1, D2,2,2 +
+K1, D2,4,2}-free.
+Proof. We prove that TS3(F) contains a cycle if and only if F contains one of the graphs in {2K2 +
+K1, D2,2,2 + K1, D2,4,2} as an induced subgraph.
+Suppose that TS3(F) contains a cycle C. Since the independent sets have size three, there are three
+cases to consider. Firstly, if the three vertices of each independent set in C lie in the same connected
+component T of F, by Proposition 5, the tree T must have either 2K2 + K1 or D2,4,2 as an induced
+subgraph. Secondly, suppose two of the vertices of each stable set lie in the same connected component
+T of F, which must have at least two connected components. Thus, C induces a cycle in TS2(T). So by
+Proposition 2, the tree T must have either 2K2 or D2,2,2 as an induced subgraph. Since F has at least
+two components, F contains 2K2 + K1 or D2,2,2 + K1. Finally, suppose each vertex of each stable set
+lies in a different component of F, which therefore has at least three components. At least two of these
+components must be non-trivial, i.e., contain an edge. Therefore, F contains an induced 2K2 + K1.
+Conversely, suppose F contains 2K2 + K1, D2,2,2 + K1 or D2,4,2 as an induced subgraph. Then
+TS3(F) contains a C4 in the first instance or a C8 in the other two.
+For k ≥ 4, we have the following proposition.
+Proposition 8. Let F be a forest. For k ≥ 4, if F contains either 2K2+(k−2)K1, or D2,2,2+(k−2)K1,
+or D2,4,2 + (k − 3)K1 as an induced subgraph, TSk(F) has a cycle.
+Proof. One can verify that TS2(2K2) contains a C4, and TS2(D2,2,2) and TS3(D2,4,2) both contain a
+C8. As a result, so do TSk(2K2 + (k − 2)K1), TSk(D2,2,2 + (k − 2)K1), and TSk(D2,4,2 + (k − 3)K1),
+respectively. Consequently, TSk(F) has a cycle, as desired.
+We conclude this section with the following conjecture for k ≥ 4.
+Conjecture 9. Let F be a forest. For k ≥ 4, if TSk(F) is a forest, F is {2K2 + (k − 2)K1, D2,2,2 + (k −
+2)K1, D2,4,2 + (k − 3)K1}-free.
+3
+H-join and H-decomposition
+Before considering (Q2), in this section, we describe an operation for combining TSk-graphs to produce
+new ones. We first define a family of base graphs as follows. Let V be a set of k + 1 vertices including
+two labelled u and v. Then Bk(V, uv) is the graph with vertex set V and single edge uv. We have
+TSk(Bk(V, uv)) = K2 whose two vertices are labelled by the independent sets V − u and V − v. Next,
+we define the H-join operation and its inverse.
+Definition 10. Vertex-labelled graphs G1 and G2 are H-consistent if the (possibly empty) intersection
+of their vertex sets define the same (possibly empty) common induced subgraph H. The H-join of H-
+consistent graphs G1 and G2 is the graph H(G1, G2) with V (H(G1, G2)) = V (G1) ∪ V (G2). The edges
+E(H(G1, G2)) consist of E(G1)∪E(G2) plus all edges vw with v ∈ V (G1)\V (H) and w ∈ V (G2)\V (H).
+Recall that a (vertex) cut-set in a connected graph G is a vertex set W such that G−W is disconnected.
+We extend this definition to the case where G is disconnected by allowing W = ∅. We say that W
+decomposes G into two (not necessarily connected) induced subgraphs G1 and G2 for which V (G1) ∩
+V (G2) = W and V (G1) ∪ V (G2) = V (G). If G − W has more than two (connected) components, the
+decomposition is not unique.
+Definition 11. Let G be a vertex-labelled graph. Let W ⊂ V (G) = V (G) decompose the complement
+G into G1 and G2. Let H be the subgraph of G induced by W. We say that G can be H-decomposed
+into G1 and G2.
+It follows from the definitions that if G = H(G1, G2) then G can be H-decomposed into G1 and G2,
+and vice versa. It is easy to verify that the size-k independent sets of H(G1, G2) are the union of those
+of G1 and those of G2.
+As an example consider the two 4-vertex graphs G1 and G2 that are paths with edge sets E(G1) =
+{ad, bc, cd} and E(G2) = {ad, ae, eb}. These share a common induced subgraph H with V (H) = {a, b, d}
+and E(H) = {ad}. We have V (H(G1, G2)) = {a, b, c, d, e} and E(H(G1, G2)) = {ad, ae, bc, cd, ce, be}.
+Note that TS2(G1) is the path with edges {ac − ab, ab − bd} and that TS2(G2) is the path with edges
+7
+
+a
+b
+c
+d
+G1
+a
+b
+e
+d
+G2
+c
+e
+a
+b
+d
+H(G1, G2)
+ab
+ac
+bd
+TS2(G1)
+ab
+de
+bd
+TS2(G2)
+ab
+ac
+de
+bd
+TS2(H(G1, G2))
+Figure 5: The graphs G1, G2, H(G1, G2), and their corresponding TS2-graphs. Here TS2(H(G1, G2)) =
+TS2(G1) ∪ TS2(G2).
+{ab−bd, bd−de}. It can be verified that TS2(H(G1, G2)) is the path with edges {ac−ab, ab−bd, bd−de}
+which is the union of two paths TS2(G1) and TS2(G2). (See Figure 5.)
+Now consider the graph G3 which is the path with edges {ad, cd, ce}.
+G1 and G3 share a com-
+mon induced subgraph H with V (H) = {a, c, d} and E(H) = {ad, cd}.
+We have E(H(G1, G3)) =
+{ad, bc, be, cd, ce}. Note that TS2(G3) is the path with edges {ac−ae, ae−de}. In this case, TS2(H(G1, G3))
+is the graph with edges {ab − ac, ac − ae, ae − de, de − bd, bd − ab, ab − ae} which is the union of TS2(G1),
+TS2(G3), and the two additional edges de − bd, ab − ae. (See Figure 6.)
+a
+b
+c
+d
+G1
+a
+e
+c
+d
+G3
+b
+e
+a
+c
+d
+H(G1, G3)
+ab
+ac
+bd
+TS2(G1)
+ac
+ae
+ed
+TS2(G3)
+ab
+ac
+ae
+bd
+ed
+TS2(H(G1, G3))
+Figure 6: The graphs G1, G3, H(G1, G3), and their corresponding TS2-graphs. Here TS2(H(G1, G3)) ̸=
+TS2(G1) ∪ TS2(G3).
+As the last example in this section, consider the graphs G4 and G5 as follows.
+G4 is the cycle
+with edges {ae, eb, bc, cd, ad} and G5 is the graph with edges {ae, eb, bc, ag, eg, bg}. G4 and G5 shares a
+common induced subgraph H with V (H) = {a, e, b, c} and E(H) = {ae, eb, bc}. We have E(H(G4, G5)) =
+{ae, eb, bc, cd, ad, ag, eg, bg, dg}. In this case, TS2(H(G4, G5)) is the (non-acyclic) graph with edges {ab−
+ac, ac − ce, ce − de, de − bd, ab − bd, ac − cg, ce − cg} which is the union of TS2(G4) and TS2(G5). (See
+Figure 7.)
+In the next proposition, we show how to compute the TSk-graph of an H-join, generalizing the
+examples given above.
+Proposition 12. Let k ≥ 2 and let G1 and G2 be two H-consistent graphs. TSk(H(G1, G2)) is the
+union of TSk(G1), TSk(G2) and for every pair of k-element independent sets S1 in G1 and S2 in G2
+satisfying
+|S1 ∩ V (H)| = |S2 ∩ V (H)| = |S1 ∩ S2| = k − 1,
+(1)
+the edge between S1 and S2.
+8
+
+a
+e
+b
+c
+d
+G4
+a
+e
+b
+c
+g
+G5
+d
+g
+a
+e
+b
+c
+H(G4, G5)
+ab
+ac
+ce
+de
+bd
+TS2(G4)
+ab
+ac
+ce
+cg
+TS2(G5)
+ab
+ac
+ce
+de
+bd
+cg
+TS2(H(G4, G5))
+Figure 7: The graphs G4, G5, H(G4, G5) and their corresponding (non-acyclic) TS2-graphs.
+Here
+TS2(G4, G5) = TS2(G4) ∪ TS2(G5).
+Proof. As remarked, the k-element independent sets of H(G1, G2) are the same as the union of those
+of G1 and G2. Therefore, V (TSk(H(G1, G2))) = V (TSk(G1)) ∪ V (TSk(G2)). Next, consider an edge in
+E(TSk(G1)) (respectively, E(TSk(G2))). It is a token-slide between two independent sets S1 and S2 in G1
+(respectively, G2). This remains as a token-slide in H(G1, G2). Therefore, E(TSk(G1)) ∪ E(TSk(G2)) ⊆
+E(TSk(H(G1, G2))). Now, consider an edge in E(TSk(H(G1, G2))) between two independent sets S1
+and S2. If both of these are independent sets are in G1 (respectively, G2) then the edge is also present
+in E(TSk(G1)) (respectively, E(TSk(G2))). Otherwise, we may assume the edge in E(TSk(H(G1, G2)))
+has as endpoints an independent set S1 in G1 (but not G2) and an independent set S2 in G2 (but not
+G1). We have S1 ∩ S2 ⊂ V (H) and since S1 and S2 are adjacent |S1 ∩ S2| = k − 1. It follows that
+|S1 ∩ V (H)| = |S2 ∩ V (H)| = k − 1 and so condition (1) is satisfied. We have shown that each edge in
+E(TSk(H(G1, G2))) is either in TSk(G1), TSk(G2) or satisfies condition (1), proving the proposition.
+For two H-consistent graphs G1 and G2, we say that H(G1, G2) is k-crossing free if there are no
+k-element independent sets satisfying condition (1) of Proposition 12. For example, one can verify that
+the graphs H(G1, G2) in Figure 5 and H(G4, G5) in Figure 7 are both k-crossing free, while the graph
+H(G1, G3) in Figure 6 is not. The following result will be used for constructing TSk-trees/forests.
+Corollary 13. Let k ≥ 2 and let G1 and G2 be two H-consistent graphs. H(G1, G2) is k-crossing free
+if and only if
+TSk(H(G1, G2)) = TSk(G1) ∪ TSk(G2).
+(2)
+Proof. If H(G1, G2) is k-crossing free then (2) follows from Proposition 12. Otherwise their exist k-
+element independent sets S1 is in G1 and S2 is in G2 satisfying (1). This implies that TSk(H(G1, G2))
+contains an additional edge between S1 and S2.
+Therefore, if H(G1, G2) is k-crossing free and both TSk(G1) and TSk(G2) are acyclic, then so is
+TSk(H(G1, G2)). The reason for allowing H to be empty in defining an H-join is that the corollary
+then applies to vertex disjoint graphs G1 and G2, since in this case H(G1, G2) is trivially k-crossing free.
+Therefore, we can create reconfiguration graphs that are forests from reconfiguration graphs that are
+trees (or forests).
+The following result follows from the relationship between H-join and H-decomposition discussed
+above.
+Corollary 14. If G can be H-decomposed into G1 and G2 and H(G1, G2) is k-crossing free then TSk(G)
+can be decomposed into TSk(G1) ∪ TSk(G2).
+9
+
+4
+Results on (Q2)
+We currently have no general necessary and sufficient conditions for when a forest F is a TSk-graph,
+but we present some partial results in this section. Firstly, we recall that in [1] it is shown that Pn is
+a TSk-graph for all n ≥ 1 and k ≥ 2 and K1,n is a TSk-graph if and only if n ≤ k. In this section, we
+show how to construct acyclic TSk-graphs from graphs that have a single edge using the join operation
+that was introduced in Section 3. We show that it gives an alternate method of constructing TSk-graphs
+which are paths and stars. Moreover, this operation can also be applied to construct more general TSk
+trees/forests, especially members of the classes k-ary trees and Dr,n,s.
+4.1
+Paths and stars revisited
+Using just the base graphs and the H-join operation defined in Section 3, we can obtain large families
+of TSk trees/forests. We begin with paths. For any k ≥ 2, let Jk = {b1, . . . , bk} be an independent set
+of size k and define the base graph Bi
+k = Bk(Jk−2 ∪ {ai, ai+1, ai+2}, aiai+2) and let G2 = Bi
+k.
+Proposition 15. For i ≥ 2, Gi and Bi
+k are H-consistent with H being the independent set Jk−2 ∪
+{ai, ai+1}. Define Gi+1 := H(Gi, Bi
+k). Then
+TSk(Gi+1) = TSk(Gi) ∪ TSk(Bi
+k) ≃ Pi+1.
+Proof. We will prove by induction, for i ≥ 2, that TSk(Gi) is the path Pi with vertices labelled Jk−2 ∪
+{aj, aj+1}, j = 1, . . . , i. For the base case i = 2, we observe that indeed TSk(Bi
+k) is a P2 with vertices
+labelled Jk−2 ∪ {a1, a2} and Jk−2 ∪ {a2, a3}.
+For the inductive step we observe that, for i ≥ 2, Gi and Bi
+k are H-consistent with H the independent
+set Jk−2 ∪ {ai, ai+1}. To verify that H(Gi, Bi
+k) is k-crossing free, note that the only independent set we
+need to consider in Bi
+k is Jk−2∪{ai+1, ai+2}. In the path Pi which is TSk(Gi), the candidate independent
+sets are Jk−2 ∪ {aj, aj+1}, j = 1, . . . , i. Their intersection with Bi
+k is Jk−2 which has cardinality k − 2.
+Therefore condition (1) of Proposition 12 is not satisfied, which indeed confirms that H(Gi, Bi
+k) is k-
+crossing free. We define Gi+1 := H(Gi, Bi
+k). By Corollary 13, TSk(Gi+1) is the union of the above
+labelled Pi with a P2 with endpoints Jk−2 ∪ {ai, ai+1} and Jk−2 ∪ {ai+1, ai+2}. This is the required
+Pi+1.
+An easy inductive argument based on the H-join in the proposition shows that, for i ≥ 2, Gi is
+isomorphic to P n+1 ∪ Jk−2, a result proved in Corollary 5(a) of [1]. (Observe that the vertex ai+1 in Gi
+is adjacent to every aj for 1 ≤ j ≤ i − 1.)
+Next we consider graphs Gi such that TSk(Gi) is the star K1,i.
+For k ≥ 2 and 1 ≤ i ≤ k, let
+Ik = {a1, . . . , ak} be an independent set of size k, define the base graph Ci
+k = Bk(Ik + bi, aibi) and let
+G1 = C1
+k.
+Proposition 16. For k ≥ 2 and 1 ≤ i ≤ k, Gi and Ci+1
+k
+are H-consistent with H being the independent
+set Ik. Define Gi+1 := H(Gi, Ci+1
+k
+). Then
+TSk(Gi+1) = TSk(Gi) ∪ TSk(Ci+1
+k
+) ≃ K1,i+1.
+Proof. We will prove by induction, for i ≥ 1, that TSk(Gi) is the star K1,i with centre labelled Ik and
+leaves labelled Ik + bj − aj, j = 1, . . . , i. For the base case i = 1, we observe that indeed TSk(Ci
+k) is a
+K1,1 with centre labelled Ik and leaf labelled Ik + b1 − a1.
+For the inductive step we observe that, for i ≥ 1, Gi and Ci+1
+k
+are H-consistent with H the in-
+dependent set Ik. To verify that H(Gi, Ci+1
+k
+) is k-crossing free, note that the only independent set
+we need to consider in Ci+1
+k
+is Ik + bi+1 − ai+1.
+In the above labelled K1,i which is TSk(Gi), the
+candidate independent sets for condition (1) of Proposition 12 are Ik + bj − aj, j = 1, . . . , i. Their inter-
+section with Ik + bi+1 − ai+1 has cardinality k − 2. Therefore, condition (1) is not satisfied. We define
+Gi+1 := H(Gi, Ci+1
+k
+). By Corollary 13, TSk(Gi+1) is the union of the above labelled K1,i and a K1,1
+with centre also labelled Ik and leaf labelled Ik + bi+1 − ai+1. This is the required K1,i+1.
+4.2
+k-ary trees
+In this section, we show that for each k ≥ 2, every k-ary tree is a TSk+1-graph (Proposition 19). Next,
+we show that any tree T is an induced subgraph of some TS2-forest (Proposition 22). Moreover, we
+state and prove the necessary and sufficient conditions for T to be an induced subgraph of some TS2-tree
+10
+
+(Proposition 23). Additionally, when T = K1,n, we describe a sufficient condition for T to be an induced
+subgraph of some TSk-tree (Proposition 24).
+We begin by defining a canonical vertex labelling.
+In this subsection, for any integer n, define
+In := {a1, . . . , an} and Jn := {b1, . . . , bn}.
+Definition 17. Let k ≥ 2 and G be a graph for which T := TSk+1(G) is a k-ary tree. We say that G
+and T are canonically labelled if
+(a) the root of T is labelled Ik+1,
+(b) the d ≤ k children of the root are labelled Ik+1 − ai + bi, i = 1, . . . , d,
+(c) the labels bj, j = d + 1, . . . , k (if any) are not used, and
+(d) all other nodes in T receive a label S such that |Ik+1 ∩ S| ≤ k − 1.
+It is clear that labelling K1,d, d ≤ k according to (a) and (b) with root the centre of the star is a
+canonical labelling. In this subsection, we will show that every k-ary tree has canonical labelling hence
+proving it is a TSk+1-graph. First, we give a lemma that shows how to combine canonically labelled
+k-ary trees to get a larger k-ary tree that is canonically labelled.
+Lemma 18. For integers k ≥ 2 and 1 ≤ i ≤ d ≤ k, let Gi be a graph for which TSk+1(Gi) a canonically
+labelled k-ary tree. We can construct a canonically labelled k-ary tree T isomorphic to the tree formed
+by choosing a new root and adjoining it to the root of each Ti.
+Proof. The proof consists of showing that we can make a series of H-joins between the leaves of a
+canonically labelled K1,d and the roots of the canonically labelled trees Ti, i = 1, . . . , d, after a suitable
+relabelling. Suppose the root of Ti has ni ≤ k children. We relabel the vertices in the underlying graphs
+as follows:
+(i) relabel vertices of the Gi not in Ik+1 ∪ Jk to be distinct, ie, for 1 ≤ i ≤ j ≤ d, we have V (Gi) ∩
+V (Gj) ⊆ Ik+1 ∪ Jk,
+(ii) for i = 1, . . . d, j = 1, . . . , ni set bj ← bi
+j, where the bi
+j were previously unused, and
+(iii) for i = 1, . . . d, set ai ← ak+1 and ak+1 ← bi.
+By an abuse of notation, for simplicity we let for i = 1, . . . , d, Gi and Ti refer to the relabelled graphs
+and trees. Item (i) ensures that the only labels shared between two trees are in Ik+1 ∪ Jk, (ii) ensures
+that all labels from Jk in the Ti are given unique labels to avoid clashes, and (iii) gives the root of Ti a
+correct label to be a child of a new root labelled Ik. We note that after relabelling bi only appears in
+Ti, ai does not appear in Ti and the only labels shared between the Ti are in Ik. Furthermore all tree
+vertices have unique labels.
+Next take a canonically labelled graph G0 such that TSk+1(G0) ≃ K1,d, with the centre of the star
+labelled Ik+1. For i = 1, . . . , d, we claim that the H-join Gi := H(Gi−1, Gi) is well-defined, k-crossing
+free, and TSk+1(Gi) is canonically labelled. To see this, note at that iteration i, V (Gi−1) ∩ V (Gi) =
+Ik+1 −ai +bi which is the label of the root of Ti and a leaf of TSk+1(Gi−1). Definition 17(d) implies that
+condition (1) of Proposition 12 is not satisfied. Therefore by Corollary 13, TSk+1(Gi) is obtained from
+TSk+1(Gi−1) by appending Ti to the corresponding leaf in TSk+1(Gi−1). The conditions of Definition
+17 are satisfied so TSk+1(Gi) is canonically labelled. At the end of iteration d, T := TSk+1(Gd) is the
+required tree.
+The construction described in the proof is illustrated in Figure 8. We may now prove the main result
+of this section.
+Proposition 19. For every k-ary tree T, there is a canonically labelled graph G such that T ≃ TSk+1(G).
+Proof. Suppose that the root r of T has d ≤ k children. We prove the proposition by induction on the
+height t of T, which is the length of the longest path to a leaf from the root. If t = 1 then T ≃ K1,d
+and so has a canonically representation as described following Definition 17. Otherwise, by deleting r
+we obtain d subtrees Ti, i = 1, . . . , d, which are also k-ary trees, with height less than t. Therefore, by
+induction each Ti can be represented by a canonically labelled graph Gi. It follows from Proposition 18
+that we can perform d H-joins to obtain a canonically labelled graph G for which T ≃ TSk+1(G).
+11
+
+a1a2a3
+b1a2a3
+a1b2a3
+a1a2a3
+b1a2a3
+a1b2a3
+T1
+T2
+Relabel
+a3a2b1
+b1
+1a2a3
+a3b1
+2b1
+a1a3b2
+b2
+1a2a3
+a1b2
+2b2
+k = 2
+a1a2a3
+K1,2
+Figure 8: Construction of D2,3,2 from two K1,2.
+As noted in Section 4 of [1], K1,k+1 is an example of a k-ary tree that is not an TSk-graph so the
+proposition is tight. Nevertheless, if we add a sufficient number of isolated vertices to K1,t, for t > k, it
+becomes a TS2-graph—a result we will now prove in general. We will need a special labelling of a tree
+that will be defined next.
+Definition 20. A tree T is well-labelled if
+(a) the root r of T is labelled ab,
+(b) the d children of r have roots labelled ri = bci, i = 1, . . . , d − 1 and rd = acd,
+(c) the only labels containing a and b are ab, acd, bci, 1 ≤ i ≤ d − 1, and
+(d) for i = 1, . . . , d label ci only occurs in the subtree with root ri.
+We note that there is nothing special about the ordering of the subtrees of r. The subtree rooted
+at ri can play the role of rd by relabelling those two subtrees with the exchanges a ↔ b and ci ↔ cd,
+which leaves T well-labelled. As an example, for d ≥ 1 we can well-label K1,d simply by using (a) and
+(b). Consider the graph G defined by V (G) = {a, b} ∪ {ci : 1 ≤ i ≤ d} and E(G) = {aci, cicd : 1 ≤
+i ≤ d − 1} ∪ {bcd}. Furthermore let J = {cicj : 1 ≤ i < j ≤ d − 1}. Then it is not hard to verify that
+TS2(G) ≃ K1,d + (d − 1)(d − 2)K1, where the K1,d is well-labelled and the K1 are labelled by the set J.
+This motivates the following definition.
+Definition 21. A tree T is well-labelled by a labelled graph G if there is an integer n such that TS2(G) ≃
+T + nK1 and T is well-labelled.
+We now show the following general result.
+Proposition 22. For every tree T there is a graph G and integer n such that T is well-labelled by G
+and TS2(G) ≃ T + nK1.
+Proof. The proof is by induction on N, the number of nodes in a given tree T. As noted above, the
+proposition is true for all stars K1,t and these act as base cases. For the inductive step, assume the
+proposition is true for all trees on N nodes and consider a tree T with N + 1 nodes. If T is a star
+we are done. Otherwise, let r be the root of T and assume r has degree d with its children ri being
+roots of subtrees Ti, 1, . . . , d. We may also assume that Td is a subtree of T with height at least one.
+We now construct two trees from T. The first, T 1 consists of T with subtree Td deleted and a pendant
+vertex added to its root r.
+The second, T 2 consists of Td with a pendant vertex added to its root
+rd. By induction, there are integers n1, n2 and graphs G1, G2 which well-label T 1 and T 2 such that
+TS2(G1) ≃ T 1 + n1K1 and TS2(G2) ≃ T 2 + n2K1. Apart from the vertex labels used in Definition 20,
+we may assume the vertex labels in G1 and G2 are different.
+We will show that G1 and a relabelled G2 can be H-joined and that this will identify the pendant
+edges added to T 1 and T 2 to give us back T. In T 1 we note that root r is labelled ab, and by relabelling
+subtree roots if necessary, that the added pendent vertex can be labelled acd. In T 2 the root rd is also
+12
+
+labelled ab and we can again assume the added pendant vertex is labelled acd. In T 2 we interchange the
+labels b ↔ cd and set ci ← c′
+i, i = 1, . . . , d−1, for labels c′
+i that are unused in either T 1 or T 2. Let G3 and
+T 3 denote the relabelled G2 and T 2. Setting H = {a, b, cd}, we have V (G1) ∩ V (G3) = H. H induces
+the same subgraph, containing the single edge bcd, in both G1 and G3. G1 and G3 are H-consistent and
+since k = 2 and their vertex sets are otherwise disjoint, condition (1) of Proposition 12 is not satisfied.
+Let G4 = H(G1, G3). Applying Corollary 13 we have that
+T 4 := TS2(G4) ≃ TS2(G1) ∪ TS2(G3) ≃ {T 1 + n1K1} ∪ {T 3 + n2K1} ≃ T + (n1 + n2)K1.
+is well-labelled by G4. This proves the proposition.
+The proof of the proposition is illustrated in Figure 9. The proposition tells us that for every tree
+r
+r1
+r2
+r3
+r4
+T
+r
+r1
+r2
+r3
+r4
+new pendant edges
+bc1
+bc2
+bc3
+ab
+ac4
+c4c′
+1
+c4c′
+2
+c4c′
+3
+ab
+a
+c4
+c′
+1
+c′
+2
+c′
+3
+a
+b
+c1
+c2
+c3
+c4
+(relabelled) G2
+G1
+(relabelled) T 2
+T 1
+ac4
+b
+Figure 9: Illustrating Proposition 22.
+T there is a graph G for which TS2(G) is forest containing T as an induced subgraph. Therefore, there
+can be no forbidden induced subgraph characterization of which forests are TS2-graphs. However, this
+does not imply that there can be no forbidden induced subgraph characterization of which trees are
+TS2-graphs. Indeed, in the next propositions, we present some of such characterizations.
+Proposition 23. Let T be a tree. Then there exists a TS2-tree containing T if and only if T is a 3-ary
+tree.
+Proof. (⇐) In the proof of Proposition 22, we see that isolated vertices are only added when the base
+case of a star appears as a subproblem. Therefore, it suffices to consider only the case T = K1,t, 1 ≤
+t ≤ 4. As we have noted, neither K1,3 nor K1,4 are TS2-graphs. It is not hard to see that there is
+a G1 such that TS2(G1) ≃ K1,3 + K1. However, by adding an extra vertex to G1, we can construct
+a graph G2 such that TS2(G2) ≃ D1,3,2. Furthermore, we can construct a graph G3 by applying
+H-join to two copies of G2 with slightly different vertex-labellings such that TS2(G3) is isomorphic
+to a P7 with two pendant vertices attached to the midpoint of the path. (See Figure 10.) Thus,
+if follows that when T = K1,t, 1 ≤ t ≤ 4, we can embed it as an induced subgraph of a tree
+T ′ = TS2(G), for some graph G (see Figure 10). Our proof of the if direction is complete.
+13
+
+a
+c
+d
+e
+f
+b
+ab
+ae
+bd
+ac
+b
+g
+c
+d
+h
+a
+ab
+bd
+ac
+bg
+ce
+ef
+dh
+dg
+ab
+ae
+bd
+ac
+ce
+ef
+dh
+dg
+G2
+TS2(G2)
+bg
+TS2(G3)
+Figure 10: Taking H-join of two copies of G2, where H is the path adcb, results a graph G3 such that
+TS2(G3) is isomorphic to a P7 with two pendant vertices attached to the midpoint of the path.
+(⇒) We show that if T is a k-ary tree but not a 3-ary tree for k ≥ 4 then there does not exist any
+TS2-tree T ′ containing T (as an induced subgraph). (By definition, any k-ary tree is also a ℓ-ary
+tree for ℓ ≥ k.) Let x be a vertex of T whose degree is at least five. (Since T is a k-ary tree but
+not a 3-ary tree, such a vertex x exists.)
+Suppose to the contrary that T ′ exists, i.e., there exists a graph G′ such that T ′ ≃ TS2(G′) contains
+T. Without loss of generality, assume that x is labelled by ab, where {a, b} is a size-2 stable set
+of G′. By the pigeonhole principle, we may further assume that three neighbors x1, x2, and x3
+of x are labelled ac, ad, and ae, respectively. Since T ′ is a tree, it follows that cd, ce, and de are
+respectively the labels of y1, y2, and y3 where yi is not adjacent to any of �
+j{xj}+x+�
+j̸=i{yj} for
+1 ≤ i, j ≤ 3. It follows that T ′ contains the labelled graph F ≃ K1,3 + 3K1 and therefore G′ must
+ab
+ac
+ad
+ae
+cd
+ce
+de
+F
+a
+b
+c
+d
+e
+G
+Figure 11: The graphs F and G in the proof of Proposition 23.
+contain the labelled graph G ≃ K1,3 + K1, both described in Figure 11, as an induced subgraph.
+Since T ′ ≃ TS2(G′) is a tree and G′ contains G, it follows that G′ has exactly one non-trivial
+component C (having more than two vertices) and C contains G, otherwise G′ must contain an
+14
+
+induced 2K2 and by Proposition 2 its TS2-graph is not a tree, a contradiction.
+– Case 1: a ∈ V (C). By definition, the distance from a to any of b, c, d, e in G′ must be at
+least two. If there is a path of length at least two between a and one of c, d, e not passing
+through b, the graph G′ contains a 2K2, a contradiction. Thus, any path between a and one
+of c, d, e must go through b. Moreover, if there is a path of length at least three between a
+and b not passing through any of c, d, e, again the graph G′ contains a 2K2, a contradiction.
+Since a ∈ V (C), it follows that a and b must have a common neighbor in G′, say f. Observe
+that for each y ∈ V (C) − {a, b, c, d, e, f}, y must be adjacent to b in G′, otherwise G′ either
+contains 2K2 or D2,2,2 and again by Proposition 2 its TS2-graph is not a tree, a contradiction.
+However, this implies that TS2(C) must be a forest and since G′ has exactly one non-trivial
+component C, we have TS2(G′) is also a forest, a contradiction.
+– Case 2:
+a /∈ V (C).
+In this case, there are two types of size-2 stable sets of G′: those
+containing a and those do not. Since G′ contains G, each type has at least one member.
+Moreover, since a is isolated (the only non-trivial component is C and a is not in it), no
+member from one type is adjacent to a member from another type in TS2(G′), which means
+TS2(G′) is indeed disconnected, a contradiction.
+In the above cases, we proved that some contradiction must happen. Our proof is complete.
+Indeed, for K1,n, in general we have
+Proposition 24. There exists a TSk-ary tree T containing K1,n if n ≤ 2k.
+Proof. From either [1] or Proposition 16, the proposition holds for n ≤ k. (Indeed, in this case, T = K1,n.)
+Thus, it suffices to consider k + 1 ≤ n ≤ 2k. For each i ∈ {1, . . . , n − k}, let Ai = {1, . . . , k} − i.
+Let Ik = {a1, . . . , ak} and Bn = {b1, . . . , bn}.
+We construct a graph G0 such that TSk(G0) ≃
+K1,n + (n − k)(k − 1)K1. Let Ik = {a1, . . . , ak} and Bn = {b1, . . . , bn}. Let V (G) = Ik + Bn. Vertices in
+Bn form a graph Kn − M where M is the matching that contains bibk+i for 1 ≤ i ≤ n − k. Additionally,
+for each i ∈ {1, . . . , k}, we add an edge in G0 between ai and both bi and bk+i. Observe that V (TSk(G0))
+consists of Ik, the sets Ik −ai +bi (1 ≤ i ≤ k), Ik −ai +bk+i (1 ≤ i ≤ n−k), and (Ik −ai +bi)−aj +bk+i
+(1 ≤ i ≤ n − k and j ∈ Ai). Moreover, one can verify that the independent sets (Ik − ai + bi) − aj + bk+i
+are isolated in TSk(G0) and the remaining independent sets form a K1,n in which Ik is adjacent to every
+other set. In short, G0 is indeed our desired graph.
+For each i ∈ {1, . . . , n − k}, we construct a graph Gi whose TSk-graph is a star K1,k−1 as follows.
+Let V (Gi) = (Ik − ai + bi) + �
+j∈{1,...,k}−i{ci
+j}. Vertices in �
+j∈Ai{ci
+j} form a clique in Gi of size k − 1.
+We also add an edge in Gi between aj and ci
+j for each j ∈ Ai. From either [1] or Proposition 16, one can
+verify that TSk(Gi) ≃ K1,k−1 as desired. For each i ∈ {1, . . . , n − k} and j ∈ Ai, we construct a graph
+Gi
+j whose TSk-graph is a K2 as follows. Let V (Gi
+j) = (Ik − ai + bi) − aj + bk+i + ci
+j. The only edge in
+Gi
+j is the one joining ci
+j and bk+i. From either [1] or Proposition 15, one can verify that TSk(Gi
+j) ≃ K2
+as desired.
+Now, we construct a graph G whose TSk-graph is a tree containing K1,n as follows. For convenience,
+we assume that for each i ∈ {1, . . . , n−k} the set Ai = {1, . . . , k}−i can be enumerated as {j1, . . . , jk−1}.
+We define Ki
+j0 = Gi and Ki
+jp = Hjp(Ki
+jp−1, Gi
+jp) for jp ∈ Ai where Hjp is the stable set (Ik − ai + bi) −
+ajp +ci
+jp for p ∈ {1, . . . , k −1}. Observe that the graphs Ki
+jp−1 and Gi
+jp are Hjp-consistent, which implies
+that Ki
+jp are well-defined. Moreover, one can also directly verify that the sets (Ik − ai + bi) − aj + ci
+j
+and (Ik − ai′ + bi′) − aj′ + ci′
+j′ always differ in at least two members, which means the condition (1) of
+Proposition 12 is not satisfied. In short, for each i ∈ {1, . . . , n − k}, we obtain the graph Ki
+jk−1 whose
+TSk-graph is isomorphic to the one obtained from K1,k−1 by replacing each edge with a P3. Next, we
+define K0 = G0 and Ki = Hi(Ki−1, Gi) where i ∈ {1, . . . , n − k} and Hi is the subgraph induced by
+(Ik −ai +bi)+bk+i. Observe that the graphs Ki are well-defined because Ki−1 and Gi are Hi-consistent.
+Moreover, we have Ik and each (Ik − ai + bi) − aj + ci
+j for 1 ≤ i ≤ n − k and j ∈ Ai always differ in
+at least two members. It follows that the condition (1) of Proposition 12 is not satisfied. In short, we
+finally obtain the graph G = Kn−k whose TSk-graph is indeed a tree containing K1,n as desired.
+Unfortunately, we have not been able to show whether the reverse statement of Proposition 24 also
+holds. We conclude this section with the following open problems:
+15
+
+b5a2a3a4
+b1a2a3a4
+a1a2a3b8
+b1c1
+2a3a4
+b1a2c1
+3a4
+a1a2a3b4
+a1c4
+2a3b4
+a1a2a3a4
+TS4(G)
+b2
+b1
+b3
+b4
+b5
+b6
+b7
+b8
+c1
+2
+c1
+3
+c1
+4
+c4
+1
+c4
+2
+c4
+3
+a1
+a2
+a3
+a4
+G
+b1a2a3c1
+4
+c4
+1a2a3b4
+a1a2c4
+3b4
+b1b5a3a4
+b1a2b5a4
+b1a2a3b5
+b8a2a3b4
+a1b8a2b4
+a1a2b8b4
+Figure 12: Construction of a graph G such that TS4(G) is a tree containing K1,8. Vertices of G in the
+yellow box form a clique having all dashed edges removed. The red induced subgraph of G forms a graph
+G0 whose TS4(G0) ≃ K1,8 + 12K1.
+Problem 25. For every k ≥ 3 and tree T, is there a graph G such that TSk(G) is a forest containing T
+as an induced subgraph?
+Problem 26. For every k ≥ 3 and (k + 1)-ary tree T, is there a graph G such that TSk(G) is a tree
+containing T as an induced subgraph?
+Problem 27. Does there exist a TSk-tree T containing K1,n for n > 2k?
+4.3
+Dr,n,s
+We now consider graphs in the Dr,n,s family for whose TSk-graphs are trees and show how they can
+be constructed by the H-join operation. We remark that when n = 1, Dr,n,s is nothing but a star
+K1,r+s and this case was considered in [1] and revisited in Proposition 16. Furthermore, it follows from
+Proposition 19 that for n, k ≥ 2 and 1 ≤ r ≤ s ≤ k − 1, Dr,n,s is a k-ary tree and so by Proposition 19
+it is a TSk-graph. The reverse statement does not hold in general: there exists a TSk-graph Dr,n,s even
+when s ≥ k. For example, one of such graphs, as already proved in [1], is D1,3,2 (r = 1, s = k = 2, and
+n = 3). (See also Figure 1.) Indeed, as we will see in Proposition 29, it is the unique TS2-graph among
+all trees D1,n,2 for n ≥ 1. Additionally, for the sake of completeness, we will also show in Proposition 30
+that the reverse statement indeed holds when n = 2.
+We are now characterizing which D1,n,2-graphs are TS2-graphs and show that this property is non-
+hereditary for this simple class of trees. We then consider the Dr,2,s-graphs characterizing those that are
+TSk-graphs.
+Assume for some G, TS2(G) is a forest containing a K1,3. There are four stable sets in G corresponding
+to the vertices of the K1,3. There are two ways of labelling the K1,3 but in each case there are five vertices,
+say a, . . . , e, of G involved. Up to permutations of the labels, the corresponding stable sets in G are
+either {ab, ac, bd, ae} or {ab, ac, ad, ae}. Using these definitions we have the following lemma.
+Lemma 28. Let H be the subgraph of G induced by a, b, . . . , e. The edges of H are
+(a) ad, de, eb, bc, cd, if the K1,3 is labelled {ab, ac, bd, ae}, or
+(b) bc, bd, be if the K1,3 is labelled {ab, ac, ad, ae}.
+Proof.
+(a) This labelling of K1,3 immediately gives edges ad, bc, be and non-edges ab, ac, ae, bd. That
+leaves three edges of H to be decided:
+(i) ce must be a non-edge else there is an edge ae, ac in the K1,3.
+(ii) cd is an edge else there is a cycle ab, bd, cd, ad in TS2(G), so it is not a tree.
+16
+
+ab
+ac
+bd
+ae
+ab
+ac
+ad
+ae
+a
+b
+c
+d
+e
+b
+c
+d
+e
+(a)
+(b)
+K1,3
+H
+a
+Figure 13: If TS2(G) is a forest containing a K1,3 then G must contain one of the induced subgraphs H.
+(iii) de is an edge else there is a cycle de, bd, ab, ae in TS2(G).
+Note that ce must also be a vertex in TS2(G).
+(b) This labelling of K1,3 immediately gives edges bc, bd, be and non-edges ab, ac, ad, ae. There are no
+other edges in H as c, d, e form a stable set. This implies that TS2(G) must also contain vertices
+cd, ce and de.
+Using the lemma we show that precisely one of the D1,n,2-graphs is a TS2-graph, incidentally proving
+the non-hereditary property mentioned above for this class of graphs.
+Proposition 29. D1,n,2 is a TS2-graph if and only if n = 3.
+Proof. We first consider 1 ≤ n ��� 3 and show that D1,3,2 is a TS2-graph while D1,1,2 = K1,3 and D1,2,2
+are not. (We note that the results for the first two graphs have also been proved in [1].) According to
+Lemma 28, if D1,n,2 is a TS2-graph of some graph G, the unique star K1,3 in D1,n,2 can be labelled
+in one of two ways. However, we may immediately eliminate the possibility of the labelling in Lemma
+28(b). This is because, as pointed out in the proof, there must be additional vertices in D1,n,2 = TS2(G)
+labelled cd, ce and de which are non-adjacent since c, d, e form a stable set in G.
+This implies that
+n ≥ 6. So we may assume that if D1,n,2 is a TS2-graph, the K1,3 must be labelled as in Lemma 28(a)
+with corresponding induced subgraph H of D1,n,2. From the proof of Lemma 28(a) there must be an
+additional vertex ce in D1,n,2 however this cannot be adjacent to any of the other four vertices. This
+implies that n ≥ 3 and so neither D1,1,2 nor D1,2,2 can be TS2-graphs. However we may extend H to
+G by adding a vertex f adjacent to all vertices except e, as illustrated in Figure 1. This introduces the
+new stable set ef which is adjacent to both ae and ce. Therefore D1,3,2 is isomorphic to TS2(G). We
+note that G is the unique graph (up to label permutations) for which this is true, due to the uniqueness
+of the labelling of K1,3.
+It remains to consider n ≥ 4 and show that D1,n,2 is not a TS2-graph. Suppose to the contrary that
+there exists a graph G such that D1,n,2 = TS2(G). Again, D1,n,2 must contain a copy of K1,3 with
+exactly two ways of labelling (up to label permutations) by size-2 independent sets of G.
+• Case 1: K1,3 is labelled {ab, ac, bd, ae}. Since ac and ae are not adjacent, ce must be a vertex
+of D1,n,2 = TS2(G). We consider the following cases:
+– Case 1.1: the distance between ce and any vertex of {ac, bd, ae} is at least three.
+Since the roles of c and e are equal, we assume without loss of generality that ce is adjacent
+to some vertex cf. Observe that a and f are not adjacent in G, otherwise ac and cf are
+adjacent, which means the distance between ac and ce is two, a contradiction. Since ce and
+cf are adjacent, so are ae and af. Moreover, bf must be a vertex, otherwise there is an edge
+between ab and af in D1,n,2 = TS2(G) which creates a C3 having {ab, ae, af} as its vertex-set,
+a contradiction. Since ab and ac are adjacent, so are cf and bf. Now, df must be a vertex,
+17
+
+otherwise bd and bf are adjacent which contradicts D1,n,2 = TS2(G). Since ab and bd are
+adjacent, so are af and df. From the proof of Lemma 28(a)(ii) c and d are adjacent in G, so
+df and cf are adjacent, which again contradicts D1,n,2 = TS2(G).
+– Case 1.2: the distance between ce and one of {ac, bd, ae} is exactly two. Observe
+that bd and ce has no common neighbor, otherwise that neighbor must be labelled as one of
+{bc, be, dc, de}: the first two can be ignored because ab and ac (resp., ab and ae) are adjacent,
+the last two can be ignored because ab and bd are adjacent. Again, since the roles of c and
+e are equal, we assume without loss of generality that ae and ce has a common neighbor ef.
+Since n ≥ 4, ce must have another neighbor which is different from ef, which can be either
+cg or eg for some vertex g of G.
+∗ If it is cg then ag must be a vertex, otherwise cg and ac must be adjacent, which creates a
+C6 whose vertex-set is {ac, ab, ae, ef, ce, cg}, a contradiction. Since ce and cg are adjacent,
+so are ae and ag, which contradicts D1,n,2 = TS2(G).
+∗ If it is eg then ag must be a vertex, otherwise eg and ae must be adjacent, which creates
+a C4 whose vertex-set is {ae, ef, ce, eg}, a contradiction. Since ce and eg are adjacent, so
+are ag and ac, which contradicts D1,n,2 = TS2(G).
+• Case 2: K1,3 is labelled {ab, ac, ad, ae}. As before, cd, ce, and de must be vertices in D1,n,2.
+Without loss of generality, since the roles of c, d, e are equal, we may assume that only ae is
+adjacent to another vertex of D1,n,2. As shown in the proof of Lemma 28(b), D1,n,2 must also
+contain vertices cd, ce, de. Let P be the path between ae and cd. Since the roles of c and d are
+equal, we can assume without loss of generality that cd is adjacent to a vertex cf in P. Observe
+that if af is not a vertex ac and cf are adjacent contradicting the choice of ae. So af is a vertex
+and since cd and cf are adjacent so are ad and af, which contradicts D1,n,2 = TS2(G).
+We remark that if we add a vertex g to G in Figure 1 joining it to all vertices except d the corresponding
+TS2-graph is obtained by adding the edge between bd and dg to TS2(G). Note that this tree is not in
+the class Dr,n,s.
+In the next proposition we consider two arbitrary stars whose centers are connected by an edge.
+Proposition 30. Dr,2,s (1 ≤ r ≤ s) is a TSk-graph if and only if s ≤ k − 1.
+Proof. (⇐) It follows directly from Proposition 19.
+(⇒) Suppose that Dr,2,s (r ≤ s) is obtained from P2 = p1p2 by attaching r leaves u1, . . . , ur at p1
+and s leaves v1, . . . , vs at p2 for some s ≥ k. We show that this graph is not a TSk-graph for
+any fixed k ≥ 2. Suppose to the contrary that there exists a graph G such that Dr,2,s ≃ TSk(G),
+i.e., there exists a bijective mapping f : V (Dr,2,s) → V (TSk(G)) such that uv ∈ E(Dr,2,s) if and
+only if f(u)f(v) ∈ E(TSk(G)). Without loss of generality, let f(p2) = I = {a1, . . . , ak}, where I
+is a size-k independent set of G. Since p2 has s + 1 neighbors, from the pigeonhole principle, it
+follows that there must be some i ∈ {1, . . . , k} such that f(u) = I − ai + x and f(v) = I − ai + y,
+where u, v ∈ N(p2). Observe that J = (I − ai − aj) + x + y /∈ {f(p2), f(u), f(v)} must be a size-k
+independent set of G, where j ∈ {1, . . . , k} − i and therefore there exists z ∈ V (Dr,2,s) − {p2, u, v}
+such that f(z) = J. We consider the following cases:
+– Neither u nor v is p1. In this case, we must have z /∈ N(p2), otherwise it must be adjacent to
+p2, but then f(z) = J and f(p2) = I must be adjacent in TSk(G), a contradiction. It follows
+that z ∈ N(p1) − p2 and thus f(p1) must be in {I − ai + x, I − ai + y, I − aj + x, I − aj + y}.
+Since neither u nor v is p1, the first two can be ignored. Now, if f(p1) = I − aj + x, the
+vertices x and aj must be adjacent in G, which contradicts the fact that f(u) ∈ TSk(G). A
+similar contradiction can be derived for the case f(p1) = I − aj + y. Thus, f(p1) cannot be
+defined.
+– u is p1. Again, z /∈ N(p2). Thus, z ∈ N(p1) − p2, which implies that y and aj must be
+adjacent in G. This contradicts f(v) ∈ TSk(G). Thus, f(z) cannot be defined.
+In both cases, we showed that some contradiction must occur. Our proof is complete.
+18
+
+5
+Conclusions
+In this paper, we considered two token sliding problems for trees and forests. The two questions studied
+seem remarkably complicated, even for this simple class of graphs. For the first question, finding necessary
+and sufficient conditions on G for TSk(G) to be a forest, we could only get a complete solution for
+k = 2, 3. For the second question, finding necessary and sufficient conditions for a tree or forest to be
+a token sliding graph, we could get more general results. Nevertheless, as noted in Section 4 several
+interesting important questions remain. We expect the join and decomposition operations introduced
+there will be of use for similar questions for more general graphs.
+Acknowledgments
+Avis’ research is partially supported by the Japan Society for the Promotion of Science (JSPS) KAK-
+ENHI Grants JP18H05291, JP20H00579, and JP20H05965 (AFSA) and Hoang’s research by JP20H05964
+(AFSA).
+References
+[1] David Avis and Duc A. Hoang. On reconfiguration graphs of independent sets under token sliding.
+arXiv preprint, 2022. arXiv:2203.16861.
+[2] Nicolas Bousquet, Amer E. Mouawad, Naomi Nishimura, and Sebastian Siebertz. A survey on the
+parameterized complexity of the independent set and (connected) dominating set reconfiguration
+problems. arXiv preprint, 2022. arXiv:2204.10526.
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+19
+

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+arXiv:2301.00798v1  [cs.IT]  2 Jan 2023
+Timely Opportunistic Gossiping in Dense Networks
+Purbesh Mitra
+Sennur Ulukus
+Department of Electrical and Computer Engineering
+University of Maryland, College Park, MD 20742
+[email protected]
+[email protected]
+Abstract—We consider gossiping in a fully-connected wireless
+network consisting of n nodes. The network receives Poisson up-
+dates from a source, which generates new information. The nodes
+gossip their available information with the neighboring nodes
+to maintain network timeliness. In this work, we propose two
+gossiping schemes, one semi-distributed and the other one fully-
+distributed. In the semi-distributed scheme, the freshest nodes
+use pilot signals to interact with the network and gossip with the
+full available update rate B. In the fully-distributed scheme, each
+node gossips for a fixed amount of time duration with the full
+update rate B. Both schemes achieve O(1) age scaling, and the
+semi-distributed scheme has the best age performance for any
+symmetric randomized gossiping policy. We compare the results
+with the recently proposed ASUMAN scheme [1], which also gives
+O(1) age performance, but the nodes need to be age-aware.
+I. INTRODUCTION
+Gossiping is an information sharing mechanism where
+nodes transmit their own data to the neighboring nodes ran-
+domly. Gossiping does not require any centralized schedul-
+ing and is particularly suitable for communication in dense
+networks. There are many gossip algorithms in the literature,
+e.g., [2]–[4], that focus on maximizing the effectiveness of
+information dispersion. Gossip algorithms have been studied
+from a timeliness point of view in [5]. The analysis in [5]
+uses the version age metric, which is one of the measures of
+information freshness in the literature [6]–[19].
+The analysis in [5] shows that for a fully-connected network
+of n nodes, if each node gossips with a fixed rate λ, the
+average version age of any individual node scales as O(log n)
+with the network size. Subsequent works [20]–[26] show that
+there can be improvements in the age scaling by introducing
+particular network mechanisms, such as clustering, file slicing
+and network coding. In this paper, we focus on another aspect
+of existing gossip schemes, which is their uniform gossip rate
+assignment to all nodes. A main drawback of uniform rate
+gossiping is that it allocates the same gossip rate to nodes
+with relatively stale and relatively fresh information. This
+negatively impacts the timeliness performance of the network.
+Our goal in this paper is to efficiently and distributedly allocate
+the total network gossip capacity dynamically among the users,
+thereby enabling opportunistic gossiping, where fresher nodes
+gossip with higher gossip rates.
+The first paper to address the inefficiency of uniform
+rate gossiping is [1] which proposed the ASUMAN scheme,
+which is an opportunistic gossiping scheme that relies on the
+assumption that the nodes are age-aware. In ASUMAN, since
+the nodes are age-aware, whenever the source updates itself,
+all the nodes in the network get synchronized and a new
+gossiping frame starts. When a new frame starts, the nodes
+stop gossiping and send a small pilot signal after waiting for
+a back-off period proportional to their current age. In this way,
+the freshest nodes get to start gossiping first as their back-off
+period is smallest and the relatively staler nodes do not gossip
+after receiving the pilot signal from the freshest nodes. If in
+any frame, the number of fresh nodes is more than one, then
+that number is estimated from the received pilot signals and
+the total update rate B = nλ is equally divided between them.
+The analysis in [1] shows that the version age of an individual
+node scales as O(1) with the network size n.
+Although ASUMAN achieves better age performance, the
+system model poses some challenges in real-life implementa-
+tions. One such challenge is that when multiple nodes have
+the same minimum age, all of them transmit the pilot signal
+simultaneously. Thus, multiple short signals overlap over the
+air, which leads to incorrect estimation of the minimum-
+age nodes, causing interference within the gossiping nodes.
+Another downside of ASUMAN is that the nodes have to be
+age-aware. This can be achieved if the source sends a signal to
+the nodes when it updates itself, adding additional complexity
+to the simple gossiping model.
+gossiping scheme
+age scaling
+ASUMAN proposed in [1]
+2 λe
+λ + 1
+semi-distributed proposed here
+2 λe
+λ
+fully-distributed proposed here
+(1 + e) λe
+λ
+TABLE I
+AGE SCALING COMPARISON FOR DIFFERENT GOSSIPING SCHEMES.
+In this paper, we propose two new gossiping schemes, one
+semi-distributed and the other fully-distributed, that both yield
+O(1) performance. These schemes are able to circumvent
+the previously mentioned downsides. In the semi-distributed
+scheme, each time a node gets updated by the source, it
+transmits a pilot signal to the neighboring nodes and starts
+gossiping with the maximum capacity until it receives a signal
+from some other node. In the fully-distributed scheme, each
+time a node gets updated by the source, it gossips for a
+fixed duration with the maximum capacity and stops. The
+age scaling comparison of these schemes is shown in Table I.
+Further, we prove that the semi-distributed gossiping scheme
+yields the best age performance among all possible symmetric
+gossiping schemes with an upper bound on the instantaneous
+maximum gossip rate. For our analysis, we use stochastic
+
+hybrid system (SHS) formulation [27], similar to [1], [5], to
+calculate of mean steady-state version age of the nodes.
+II. SYSTEM MODEL
+We consider a gossip network consisting of a source labeled
+node 0, and a set of nodes labeled N = {1, 2, . . ., n}, as
+shown in Fig. 1. The source updates its information with
+Poisson arrivals of rate λe, and it sends Poisson updates to
+the network with a total rate of λ. For simplicity, we consider
+a symmetric network, i.e., each of the nodes receives updates
+from the source with a rate λ
+n. In the timely gossiping papers
+in the literature [5], [20]–[26], it is assumed that each node
+of the network gossips with a rate of λ; thus, on a fully
+connected network where each node is connected to (n − 1)
+other nodes, each node i gossips with a node j with a rate
+of
+λ
+n−1. Therefore, the total update capacity of the network is
+B = nλ. As in [1], in this paper, we consider allocating this
+total update rate B to users dynamically. Once a gossip rate is
+assigned to a node, it gossips with its (n − 1) neighbors with
+equal rates in the fully connected network.
+Thus, the network has an upper bound of B on the instan-
+taneous gossiping rate. If at any time, multiple nodes transmit
+and the total instantaneous gossip rate exceeds B, there will be
+interference, and the gossiped data is lost. Hence, for effective
+gossiping, at any time instant, the total instantaneous gossip
+rate has to be less than or equal to B. The goal of our work
+is to improve the timeliness of such a network. To measure
+the timeliness of the ith node, we use version age, denoted
+as ∆i(t). This measure counts how many versions the data at
+the ith node is lagging, compared to the data available at the
+source at time t. Mathematically, we write
+∆i(t) = Ns(t) − Ni(t),
+(1)
+where Ns(t) and Ni(t) are the versions of the data available
+at the source and at the ith node, respectively, at time t.
+We denote all the ages of nodes at time t as the age vector
+∆(t) = [∆1(t), ∆2(t), . . . , ∆n(t)]. When the source updates
+itself, all the ages of the nodes increase by 1. If the source
+sends an update to a node, its age becomes 0. When node
+i sends a gossip update to node j, it stores the data with
+the freshest version, i.e., the age of the jth node becomes
+ˆ∆j(t) = ∆{i,j}(t) = min{∆i(t), ∆j(t)}.
+III. SEMI-DISTRIBUTED GOSSIPING
+In this section, we introduce the semi-distributed gossiping
+scheme. The motivation for this is to allow the freshest node
+of the network to gossip with maximum capacity. Suppose
+we denote the kth source-to-ith node update as t(i)
+k . In this
+scheme, at time t(i)
+k , i transmits a small pilot signal to all the
+other nodes in the network and starts gossiping with rate B to
+the other nodes with equal rate. While gossiping, if i receives
+a pilot signal from any other node, it will stop gossiping. We
+define the gossiping node at any given time t as M(t). Since,
+the probability of two simultaneous Poisson arrivals is 0, i.e.,
+P(|M(t)| ≥ 2) = 0, here we do not face the problem of
+overlapping pilot signals like ASUMAN [1].
+0
+1
+2
+3
+4
+λe
+λ
+5
+Fig. 1. Source 0 updates itself with rate λe and sends updates to the nodes
+N = {1, 2, 3, 4, 5} uniformly with total rate λ, i.e., with rate λ/5 to each
+of the nodes. The nodes gossip with each other with total update rate B.
+We investigate the mean steady-state age of an individual
+node, denoted as,
+ai = lim
+t→∞ ai(t) = lim
+t→∞ E[∆i(t)],
+(2)
+in particular, how network size n affects ai, in Theorem 1.
+Theorem 1 If B = nλ, the average version age of a node ai
+in a semi-distributed gossip network scales as O(1).
+Proof: We use SHS formulation of [27]. Note that, for any
+time t, the gossiping node is the minimum age node in the
+network. Let us denote this minimum age as ∆min(t) =
+min{∆1(t), ∆2(t), . . . , ∆n(t)}. From [5], we know that
+limt→∞ E[∆min(t)] =
+λe
+λ . Since for any given t, only the
+node with the minimum age is gossiping, we can express
+the state transition of the system as an SHS with only one
+type of transition, i.e., Q = 0. We choose the test function
+ψi :
+Rn × [0, ∞) →
+R, where i ∈ N, as
+ψi(∆(t), t) = ∆i(t).
+(3)
+Now, following [27, Thm. 1], we evaluate the extended gen-
+erator function as
+E[(Lψi)(∆(t), t)] =
+�
+(j,ℓ)∈L
+λj,ℓ(∆(t), t)E
+�
+ψi(φj,ℓ(∆(t), t))
+− ψi(∆(t), t)
+�
+,
+(4)
+where L denotes all possible state transitions. We define the
+reset maps φj,ℓ(∆(t), t) = ˆ∆(t) = [ ˆ∆1(t), ˆ∆2(t), . . . , ˆ∆n(t)]
+as follows
+ˆ∆i(t) =
+
+
+
+
+
+
+
+∆i(t) + 1,
+if j = 0, ℓ = 0
+0,
+if j = 0, ℓ = i
+min(∆j(t), ∆ℓ(t)),
+if j ∈ N, ℓ = i
+∆i(t),
+otherwise.
+(5)
+The update rates λj,ℓ are given as
+λj,ℓ(∆(t), t) =
+
+
+
+λe,
+if j = 0, ℓ = 0
+λ
+n,
+if j = 0, ℓ = i
+B
+n−1
+1{j = M(t)},
+otherwise,
+(6)
+
+where
+1{·} denotes the indicator function. Now, we can
+rewrite (4) as
+E[(Lψi)(∆(t), t)]
+= E
+�
+λe(∆i(t) + 1 − ∆i(t)) + λ
+n(0 − ∆i(t))
++
+�
+j∈N
+B
+n − 1
+1{j = M(t)}
+�
+∆{j,i}(t) − ∆i(t)
+� �
+. (7)
+Since the gossiping node is always the minimum age node,
+we can write
+E[(Lψi)(∆(t), t)]
+= λe − λ
+nai(t) + E
+�
+�
+j=M(t)
+B
+n − 1(∆min(t) − ∆i(t))
+�
+= λe − λ
+nai(t) +
+B
+n − 1(amin(t) − ai(t)).
+(8)
+Now, since the version age is a piece-wise constant function
+of time, we obtain
+dE[ψi(∆(t), t)]
+dt
+= dE[∆i(t)]
+dt
+= 0,
+(9)
+for any continuity point t. Hence, the expected value in (8) is
+0, by Dynkin’s formula, as given in [27]. Thus, (8) becomes
+0 = λe − λ
+nai(t) +
+B
+n − 1(amin(t) − ai(t)).
+(10)
+Hence, the mean age of an individual node is expressed as
+ai(t) =
+λe +
+B
+n−1amin(t)
+λ
+n +
+B
+n−1
+.
+(11)
+To evaluate the steady-state mean age, we take t → ∞ in (11)
+which gives
+ai =
+λe +
+B
+n−1
+λe
+λ
+λ
+n +
+B
+n−1
+.
+(12)
+Finally, to calculate the scaling of the average age, we use
+B = nλ, which yields
+lim
+n→∞ ai = lim
+n→∞
+λe
+λ
+�
+1 +
+n
+n−1
+1
+n +
+n
+n−1
+�
+= 2λe
+λ ,
+(13)
+concluding the proof. ■
+Next, we show that this semi-distributed scheme gives
+the best version age performance for any possible gossip-
+ing scheme with a constraint on the instantaneous gossiping
+scheme, in Theorem 2.
+Theorem 2 For any symmetric network with maximum in-
+stantaneous gossip rate of B, the semi-distributed gossiping
+scheme yields the minimum average age for the nodes.
+Proof: Suppose we use any arbitrary gossiping policy. Since
+the total gossip rate is upper bounded by B, we have
+�
+j,i∈N,j̸=i
+λj,i(∆(t), t) ≤ B,
+∀t.
+(14)
+From the symmetry of the network, we can write
+E
+
+
+�
+j∈N,j̸=i
+λj,i(∆(t), t)
+
+ ≤
+B
+n − 1.
+(15)
+Note that the sum in (14) is over all i, j whereas the sum
+in (15) is over j only. Now, equating the extended generator
+function to 0, yields
+λ
+nai(t) + E
+
+
+�
+j∈N,j̸=i
+λj,i(∆(t), t)∆i(t)
+
+
+= λe + E
+
+
+�
+j∈N,j̸=i
+λj,i(∆(t), t)∆{j,i}(t)
+
+ .
+(16)
+Using the inequality in (15) and by definition the fact that
+∆{j,i}(t) ≥ ∆min(t), we can rewrite (16) as
+λ
+nai(t) +
+B
+n − 1ai(t) ≥ λe +
+B
+n − 1amin(t).
+(17)
+Taking t → ∞ in (17) and using the expression of amin(t),
+we obtain
+ai ≥
+λe +
+B
+n−1
+λe
+λ
+λ
+n +
+B
+n−1
+,
+(18)
+where the right-hand side of the inequality is the average
+age of a node with the proposed semi-distributed policy. This
+concludes the proof. ■
+IV. FULLY-DISTRIBUTED GOSSIPING
+In this section, we introduce a gossiping policy which is
+fully-distributed. In ASUMAN [1], the nodes need to be age-
+aware and in the semi-distributed scheme, the nodes need to
+implement a pilot-signal based communication in the network.
+We improve upon them and formulate a gossiping policy that
+does not require age-awareness or pilot-signal transmissions.
+In this scheme, whenever node i receives an update from the
+source at time t(i)
+k , it starts gossiping to all the other nodes
+with rate B for a fixed time duration δ, and then it stops, as
+shown in Fig. 2. We investigate the age performance of this
+scheme in Theorem 3.
+Theorem 3 If B = nλ, the average version age of a node in
+a fully-distributed gossip network scales as O(1).
+Proof: From Fig. 2, we observe that at any given time, if
+there is any effective gossiping, only the minimum age node
+is responsible for it. This is because, effective gossiping is
+possible only if a single node is gossiping and in that case, the
+node has to be a minimum age node. Whereas, when multiple
+nodes are gossiping with rate B, there will be no effective
+gossiping due to interference. Additionally, each update from
+the source is a Poisson arrival with rate λ, and gossiping starts
+immediately for a time duration of δ. Hence, we can model
+this process as an M/D/∞ queue. Now, from [28], [29], we
+
+version age
+t
+t(1)
+1
+t(2)
+1
+t(1)
+2
+t(2)
+2
+t(1)
+3
+t(2)
+3
+t(1)
+4
+t(2)
+4
+t(1)
+5
+t(1)
+6
+t(2)
+5
+t(2)
+6
+t(1)
+7
+δ
+δ
+number of entries in M/D/∞ queue
+effective gossiping
+interference
+1
+2
+1
+2
+t
+∆1(t)
+∆2(t)
+∆min(t)
+Fig. 2. Distributed gossiping in a 2 node network. At each t(i)
+k , ∆i(t) becomes zero and node i starts gossiping for a δ duration. The corresponding M/D/∞
+queue indicates the number of nodes gossiping simultaneously. Effective gossiping only happens when only one node is gossiping. Presence of multiple
+gossiping nodes creates interference, resulting in no net gossip.
+know that the stationary distribution for any general M/G/∞
+queue follows the Poisson distribution,
+πk = (λ/µ)ke−λ/µ
+k!
+,
+k = 0, 1, 2, . . .
+(19)
+For this M/D/∞ queue, µ = 1
+δ . Since effective gossip happens
+only when there is one entry in the queue, the effective gossip
+rate becomes
+˜B = π1B = λδe−λδB.
+(20)
+The rest of the analysis is the same as in Theorem 1. Therefore,
+we can directly substitute ˜B instead of B in (12) to obtain the
+mean age of the ith node as
+ai =
+λe +
+˜
+B
+n−1
+λe
+λ
+λ
+n +
+˜
+B
+n−1
+.
+(21)
+Using B = nλ and taking n → ∞ in (21), we get the age
+scaling as
+lim
+n→∞ ai = lim
+n→∞
+λe + λδe−λδnλ
+n−1
+λe
+λ
+λ
+n + λδe−λδnλ
+n−1
+(22)
+= λe
+λ
+�
+1 +
+1
+λδe−λδ
+�
+,
+(23)
+which concludes the proof. ■
+Finally, we note that the age expression in (23) for the fully-
+distributed gossiping scheme depends on the chosen gossiping
+duration δ. Thus, we can improve the age expression in (23)
+by choosing an optimal δ that minimizes the mean age. Since
+λδe−λδ ≤ 1
+e, the maxima being at δ∗ = 1
+λ, the lower bound
+of mean age of distributed gossiping is λe
+λ
+�
+1 +
+1
+e−1
+�
+= (1 +
+e) λe
+λ . This result matches our intuition, because if δ is too
+small, it will not allow sufficient time to gossip. On the other
+hand, if δ is too large, there will not be effective gossiping
+due to interference from simultaneous gossiping nodes. The
+minimum age is achieved when the effective gossiping rate ˜B
+is maximized, which is ˜B|δ∗ = B
+e .
+V. NUMERICAL RESULTS
+In this section, we present simulation results for the two
+proposed gossiping schemes, and compare them with the
+theoretically derived age expressions. We also show the results
+for ASUMAN [1] as a benchmark.
+In Fig. 3, we present the numerical results for λe
+λ = 0.4,
+λe
+λ
+= 1 and
+λe
+λ
+= 2 in Fig. 3(a), Fig. 3(b) and Fig. 3(c),
+respectively, with λ = 1 in all cases. From the figures,
+it is evident that all the gossiping schemes result in O(1)
+performance and the semi-distributed gossiping scheme yields
+the best performance among all.
+In Fig. 3(a), where λe
+λ = 0.4 <
+1
+e−1, ASUMAN gives the
+worst age performance among the three schemes. However,
+in Fig. 3(b) and Fig. 3(c), i.e., for
+λe
+λ
+>
+1
+e−1, ASUMAN
+performs worse than the semi-distributed scheme, but is better
+than the fully-distributed scheme. This matches our intuition
+because, in ASUMAN, we use the information about source
+self-updates to allocate gossip rate more efficiently, while in
+the fully-distributed scheme, multiple nodes gossiping together
+causes interference to lose some portion of the total gossip
+rate. This effect of interference becomes more prominent when
+the source to network update rate λ is high as compared
+to source self-update rate λe. We have chosen δ =
+1
+λ = 1
+for the simulation to get the minimum average age for fully-
+distributed gossiping.
+For
+ASUMAN,
+the
+asymptotic
+age
+scales
+as
+limn→∞ λe
+λ
+�
+1+
+n
+n−1 (1+ λ
+λe )
+1
+n +
+n
+n−1
+�
+= 2 λe
+λ + 1, while the other two
+
+0
+100
+200
+300
+400
+500
+600
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Theorem 1 formula (12)
+semi-distributed gossip
+Theorem 3 formula (21)
+fully-distributed gossip
+ASUMAN
+(a) λe
+λ = 0.4
+0
+100
+200
+300
+400
+500
+600
+1
+1.5
+2
+2.5
+3
+3.5
+4
+Theorem 1 formula (12)
+semi-distributed gossip
+Theorem 3 formula (21)
+fully-distributed gossip
+ASUMAN
+(b) λe
+λ = 1
+0
+100
+200
+300
+400
+500
+600
+2
+3
+4
+5
+6
+7
+8
+Theorem 1 formula (12)
+semi-distributed gossip
+Theorem 3 formula (21)
+fully-distributed gossip
+ASUMAN
+(c) λe
+λ = 2
+Fig. 3.
+Average version age of a single node versus the total number of
+nodes in the network n for semi-distributed, fully-distributed and ASUMAN
+schemes.
+schemes obtain 2 λe
+λ
+and (1 + e) λe
+λ , as shown in (13) and
+(23) (with optimized δ), respectively, and as listed in Table I.
+The numerical simulation results exactly match the derived
+formulas. With an increase in the ratio
+λe
+λ , the average
+age increases due to source being updated more frequently
+compared to the network for all schemes, as we observe
+going from Fig. 3(a) to Fig. 3(b) to Fig. 3(c).
+VI. CONCLUSION
+We proposed a semi-distributed and a fully-distributed
+gossiping scheme for a fully-connected network. The semi-
+distributed scheme allows the freshest node to communicate
+in the network through pilot signals and to gossip with full
+capacity. This scheme archives the lowest possible average
+age for any symmetric network, with a constraint on the
+instantaneous gossip rate. On the other hand, in the fully-
+distributed scheme, the freshest node gossips for a fixed time
+duration with full capacity. The effective gossip happens only
+a fraction of the total time, when there is no interference
+from multiple nodes gossiping. Both of the proposed schemes
+yield O(1) age performance. Compared to our previous work
+ASUMAN, which also gives O(1) age scaling, this work is an
+improvement because here we do not require the nodes to be
+age-aware or to transmit pilot signals for channel reservation.
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+in parallel relay networks. In IEEE ISIT, July 2021.
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+individuals in a population. In IEEE Infocom, May 2021.
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+in clustered gossip networks. IEEE Jour. on Selected Areas in Informa-
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+

FdAyT4oBgHgl3EQf4_rs/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf,len=392
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='00798v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='IT]  2 Jan 2023  Timely Opportunistic Gossiping in Dense Networks  Purbesh Mitra  Sennur Ulukus  Department of Electrical and Computer Engineering  University of Maryland, College Park, MD 20742  pmitra@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='edu  ulukus@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='edu  Abstract—We consider gossiping in a fully-connected wireless  network consisting of n nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' The network receives Poisson up-  dates from a source, which generates new information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' The nodes  gossip their available information with the neighboring nodes  to maintain network timeliness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' In this work, we propose two  gossiping schemes, one semi-distributed and the other one fully-  distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' In the semi-distributed scheme, the freshest nodes  use pilot signals to interact with the network and gossip with the  full available update rate B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' In the fully-distributed scheme, each  node gossips for a fixed amount of time duration with the full  update rate B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Both schemes achieve O(1) age scaling, and the  semi-distributed scheme has the best age performance for any  symmetric randomized gossiping policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' We compare the results  with the recently proposed ASUMAN scheme [1], which also gives  O(1) age performance, but the nodes need to be age-aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' INTRODUCTION  Gossiping is an information sharing mechanism where  nodes transmit their own data to the neighboring nodes ran-  domly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Gossiping does not require any centralized schedul-  ing and is particularly suitable for communication in dense  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' There are many gossip algorithms in the literature,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=', [2]–[4], that focus on maximizing the effectiveness of  information dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Gossip algorithms have been studied  from a timeliness point of view in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' The analysis in [5]  uses the version age metric, which is one of the measures of  information freshness in the literature [6]–[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  The analysis in [5] shows that for a fully-connected network  of n nodes, if each node gossips with a fixed rate λ, the  average version age of any individual node scales as O(log n)  with the network size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Subsequent works [20]–[26] show that  there can be improvements in the age scaling by introducing  particular network mechanisms, such as clustering, file slicing  and network coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' In this paper, we focus on another aspect  of existing gossip schemes, which is their uniform gossip rate  assignment to all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' A main drawback of uniform rate  gossiping is that it allocates the same gossip rate to nodes  with relatively stale and relatively fresh information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' This  negatively impacts the timeliness performance of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  Our goal in this paper is to efficiently and distributedly allocate  the total network gossip capacity dynamically among the users,  thereby enabling opportunistic gossiping, where fresher nodes  gossip with higher gossip rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  The first paper to address the inefficiency of uniform  rate gossiping is [1] which proposed the ASUMAN scheme,  which is an opportunistic gossiping scheme that relies on the  assumption that the nodes are age-aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' In ASUMAN, since  the nodes are age-aware, whenever the source updates itself,  all the nodes in the network get synchronized and a new  gossiping frame starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' When a new frame starts, the nodes  stop gossiping and send a small pilot signal after waiting for  a back-off period proportional to their current age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' In this way,  the freshest nodes get to start gossiping first as their back-off  period is smallest and the relatively staler nodes do not gossip  after receiving the pilot signal from the freshest nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' If in  any frame, the number of fresh nodes is more than one, then  that number is estimated from the received pilot signals and  the total update rate B = nλ is equally divided between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  The analysis in [1] shows that the version age of an individual  node scales as O(1) with the network size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  Although ASUMAN achieves better age performance, the  system model poses some challenges in real-life implementa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' One such challenge is that when multiple nodes have  the same minimum age, all of them transmit the pilot signal  simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Thus, multiple short signals overlap over the  air, which leads to incorrect estimation of the minimum-  age nodes, causing interference within the gossiping nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  Another downside of ASUMAN is that the nodes have to be  age-aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' This can be achieved if the source sends a signal to  the nodes when it updates itself, adding additional complexity  to the simple gossiping model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  gossiping scheme  age scaling  ASUMAN proposed in [1]  2 λe  λ + 1  semi-distributed proposed here  2 λe  λ  fully-distributed proposed here  (1 + e) λe  λ  TABLE I  AGE SCALING COMPARISON FOR DIFFERENT GOSSIPING SCHEMES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  In this paper, we propose two new gossiping schemes, one  semi-distributed and the other fully-distributed, that both yield  O(1) performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' These schemes are able to circumvent  the previously mentioned downsides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' In the semi-distributed  scheme, each time a node gets updated by the source, it  transmits a pilot signal to the neighboring nodes and starts  gossiping with the maximum capacity until it receives a signal  from some other node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' In the fully-distributed scheme, each  time a node gets updated by the source, it gossips for a  fixed duration with the maximum capacity and stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' The  age scaling comparison of these schemes is shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  Further, we prove that the semi-distributed gossiping scheme  yields the best age performance among all possible symmetric  gossiping schemes with an upper bound on the instantaneous  maximum gossip rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' For our analysis, we use stochastic  hybrid system (SHS) formulation [27], similar to [1], [5], to  calculate of mean steady-state version age of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' SYSTEM MODEL  We consider a gossip network consisting of a source labeled  node 0, and a set of nodes labeled N = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=', n}, as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' The source updates its information with  Poisson arrivals of rate λe, and it sends Poisson updates to  the network with a total rate of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' For simplicity, we consider  a symmetric network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=', each of the nodes receives updates  from the source with a rate λ  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' In the timely gossiping papers  in the literature [5], [20]–[26], it is assumed that each node  of the network gossips with a rate of λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' thus, on a fully  connected network where each node is connected to (n − 1)  other nodes, each node i gossips with a node j with a rate  of  λ  n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Therefore, the total update capacity of the network is  B = nλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' As in [1], in this paper, we consider allocating this  total update rate B to users dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Once a gossip rate is  assigned to a node, it gossips with its (n − 1) neighbors with  equal rates in the fully connected network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  Thus, the network has an upper bound of B on the instan-  taneous gossiping rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' If at any time, multiple nodes transmit  and the total instantaneous gossip rate exceeds B, there will be  interference, and the gossiped data is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Hence, for effective  gossiping, at any time instant, the total instantaneous gossip  rate has to be less than or equal to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' The goal of our work  is to improve the timeliness of such a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' To measure  the timeliness of the ith node, we use version age, denoted  as ∆i(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' This measure counts how many versions the data at  the ith node is lagging, compared to the data available at the  source at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Mathematically, we write  ∆i(t) = Ns(t) − Ni(t),  (1)  where Ns(t) and Ni(t) are the versions of the data available  at the source and at the ith node, respectively, at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  We denote all the ages of nodes at time t as the age vector  ∆(t) = [∆1(t), ∆2(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' , ∆n(t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' When the source updates  itself, all the ages of the nodes increase by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' If the source  sends an update to a node, its age becomes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' When node  i sends a gossip update to node j, it stores the data with  the freshest version, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=', the age of the jth node becomes  ˆ∆j(t) = ∆{i,j}(t) = min{∆i(t), ∆j(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' SEMI-DISTRIBUTED GOSSIPING  In this section, we introduce the semi-distributed gossiping  scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' The motivation for this is to allow the freshest node  of the network to gossip with maximum capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Suppose  we denote the kth source-to-ith node update as t(i)  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' In this  scheme, at time t(i)  k , i transmits a small pilot signal to all the  other nodes in the network and starts gossiping with rate B to  the other nodes with equal rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' While gossiping, if i receives  a pilot signal from any other node, it will stop gossiping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' We  define the gossiping node at any given time t as M(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Since,  the probability of two simultaneous Poisson arrivals is 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=',  P(|M(t)| ≥ 2) = 0, here we do not face the problem of  overlapping pilot signals like ASUMAN [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  0  1  2  3  4  λe  λ  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Source 0 updates itself with rate λe and sends updates to the nodes  N = {1, 2, 3, 4, 5} uniformly with total rate λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=', with rate λ/5 to each  of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' The nodes gossip with each other with total update rate B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  We investigate the mean steady-state age of an individual  node, denoted as,  ai = lim  t→∞ ai(t) = lim  t→∞ E[∆i(t)],  (2)  in particular, how network size n affects ai, in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  Theorem 1 If B = nλ, the average version age of a node ai  in a semi-distributed gossip network scales as O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  Proof: We use SHS formulation of [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Note that, for any  time t, the gossiping node is the minimum age node in the  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Let us denote this minimum age as ∆min(t) =  min{∆1(t), ∆2(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' , ∆n(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' From [5], we know that  limt→∞ E[∆min(t)] =  λe  λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Since for any given t, only the  node with the minimum age is gossiping, we can express  the state transition of the system as an SHS with only one  type of transition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=', Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' We choose the test function  ψi :  Rn × [0, ∞) →  R, where i ∈ N, as  ψi(∆(t), t) = ∆i(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  (3)  Now, following [27, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 1], we evaluate the extended gen-  erator function as  E[(Lψi)(∆(t), t)] =  �  (j,ℓ)∈L  λj,ℓ(∆(t), t)E  �  ψi(φj,ℓ(∆(t), t))  − ψi(∆(t), t)  �  ,  (4)  where L denotes all possible state transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' We define the  reset maps φj,ℓ(∆(t), t) = ˆ∆(t) = [ ˆ∆1(t), ˆ∆2(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' , ˆ∆n(t)]  as follows  ˆ∆i(t) =  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  ∆i(t) + 1,  if j = 0, ℓ = 0  0,  if j = 0, ℓ = i  min(∆j(t), ∆ℓ(t)),  if j ∈ N, ℓ = i  ∆i(t),  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  (5)  The update rates λj,ℓ are given as  λj,ℓ(∆(t), t) =  \uf8f1  \uf8f2  \uf8f3  ��e,  if j = 0, ℓ = 0  λ  n,  if j = 0, ℓ = i  B  n−1  1{j = M(t)},  otherwise,  (6)  where  1{·} denotes the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Now, we can  rewrite (4) as  E[(Lψi)(∆(t), t)]  = E  �  λe(∆i(t) + 1 − ∆i(t)) + λ  n(0 − ∆i(t))  +  �  j∈N  B  n − 1  1{j = M(t)}  �  ∆{j,i}(t) − ∆i(t)  � �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' (7)  Since the gossiping node is always the minimum age node,  we can write  E[(Lψi)(∆(t), t)]  = λe − λ  nai(t) + E  �  �  j=M(t)  B  n − 1(∆min(t) − ∆i(t))  �  = λe − λ  nai(t) +  B  n − 1(amin(t) − ai(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  (8)  Now, since the version age is a piece-wise constant function  of time, we obtain  dE[ψi(∆(t), t)]  dt  = dE[∆i(t)]  dt  = 0,  (9)  for any continuity point t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Hence, the expected value in (8) is  0, by Dynkin’s formula, as given in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Thus, (8) becomes  0 = λe − λ  nai(t) +  B  n − 1(amin(t) − ai(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  (10)  Hence, the mean age of an individual node is expressed as  ai(t) =  λe +  B  n−1amin(t)  λ  n +  B  n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  (11)  To evaluate the steady-state mean age, we take t → ∞ in (11)  which gives  ai =  λe +  B  n−1  λe  λ  λ  n +  B  n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  (12)  Finally, to calculate the scaling of the average age, we use  B = nλ, which yields  lim  n→∞ ai = lim  n→∞  λe  λ  �  1 +  n  n−1  1  n +  n  n−1  �  = 2λe  λ ,  (13)  concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' ■  Next, we show that this semi-distributed scheme gives  the best version age performance for any possible gossip-  ing scheme with a constraint on the instantaneous gossiping  scheme, in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  Theorem 2 For any symmetric network with maximum in-  stantaneous gossip rate of B, the semi-distributed gossiping  scheme yields the minimum average age for the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  Proof: Suppose we use any arbitrary gossiping policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Since  the total gossip rate is upper bounded by B, we have  �  j,i∈N,j̸=i  λj,i(∆(t), t) ≤ B,  ∀t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  (14)  From the symmetry of the network, we can write  E  \uf8ee  \uf8f0  �  j∈N,j̸=i  λj,i(∆(t), t)  \uf8f9  \uf8fb ≤  B  n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  (15)  Note that the sum in (14) is over all i, j whereas the sum  in (15) is over j only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Now, equating the extended generator  function to 0, yields  λ  nai(t) + E  \uf8ee  \uf8f0  �  j∈N,j̸=i  λj,i(∆(t), t)∆i(t)  \uf8f9  \uf8fb  = λe + E  \uf8ee  \uf8f0  �  j∈N,j̸=i  λj,i(∆(t), t)∆{j,i}(t)  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  (16)  Using the inequality in (15) and by definition the fact that  ∆{j,i}(t) ≥ ∆min(t), we can rewrite (16) as  λ  nai(t) +  B  n − 1ai(t) ≥ λe +  B  n − 1amin(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  (17)  Taking t → ∞ in (17) and using the expression of amin(t),  we obtain  ai ≥  λe +  B  n−1  λe  λ  λ  n +  B  n−1  ,  (18)  where the right-hand side of the inequality is the average  age of a node with the proposed semi-distributed policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' This  concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' ■  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' FULLY-DISTRIBUTED GOSSIPING  In this section, we introduce a gossiping policy which is  fully-distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' In ASUMAN [1], the nodes need to be age-  aware and in the semi-distributed scheme, the nodes need to  implement a pilot-signal based communication in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  We improve upon them and formulate a gossiping policy that  does not require age-awareness or pilot-signal transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  In this scheme, whenever node i receives an update from the  source at time t(i)  k , it starts gossiping to all the other nodes  with rate B for a fixed time duration δ, and then it stops, as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' We investigate the age performance of this  scheme in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  Theorem 3 If B = nλ, the average version age of a node in  a fully-distributed gossip network scales as O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  Proof: From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 2, we observe that at any given time, if  there is any effective gossiping, only the minimum age node  is responsible for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' This is because, effective gossiping is  possible only if a single node is gossiping and in that case, the  node has to be a minimum age node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Whereas, when multiple  nodes are gossiping with rate B, there will be no effective  gossiping due to interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Additionally, each update from  the source is a Poisson arrival with rate λ, and gossiping starts  immediately for a time duration of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Hence, we can model  this process as an M/D/∞ queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Now, from [28], [29], we  version age  t  t(1)  1  t(2)  1  t(1)  2  t(2)  2  t(1)  3  t(2)  3  t(1)  4  t(2)  4  t(1)  5  t(1)  6  t(2)  5  t(2)  6  t(1)  7  δ  δ  number of entries in M/D/∞ queue  effective gossiping  interference  1  2  1  2  t  ∆1(t)  ∆2(t)  ∆min(t)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Distributed gossiping in a 2 node network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' At each t(i)  k , ∆i(t) becomes zero and node i starts gossiping for a δ duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' The corresponding M/D/∞  queue indicates the number of nodes gossiping simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Effective gossiping only happens when only one node is gossiping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Presence of multiple  gossiping nodes creates interference, resulting in no net gossip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  know that the stationary distribution for any general M/G/∞  queue follows the Poisson distribution,  πk = (λ/µ)ke−λ/µ  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  ,  k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  (19)  For this M/D/∞ queue, µ = 1  δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Since effective gossip happens  only when there is one entry in the queue, the effective gossip  rate becomes  ˜B = π1B = λδe−λδB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  (20)  The rest of the analysis is the same as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Therefore,  we can directly substitute ˜B instead of B in (12) to obtain the  mean age of the ith node as  ai =  λe +  ˜  B  n−1  λe  λ  λ  n +  ˜  B  n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  (21)  Using B = nλ and taking n → ∞ in (21), we get the age  scaling as  lim  n→∞ ai = lim  n→∞  λe + λδe−λδnλ  n−1  λe  λ  λ  n + λδe−λδnλ  n−1  (22)  = λe  λ  �  1 +  1  λδe−λδ  �  ,  (23)  which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' ■  Finally, we note that the age expression in (23) for the fully-  distributed gossiping scheme depends on the chosen gossiping  duration δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Thus, we can improve the age expression in (23)  by choosing an optimal δ that minimizes the mean age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Since  λδe−λδ ≤ 1  e, the maxima being at δ∗ = 1  λ, the lower bound  of mean age of distributed gossiping is λe  λ  �  1 +  1  e−1  �  = (1 +  e) λe  λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' This result matches our intuition, because if δ is too  small, it will not allow sufficient time to gossip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' On the other  hand, if δ is too large, there will not be effective gossiping  due to interference from simultaneous gossiping nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' The  minimum age is achieved when the effective gossiping rate ˜B  is maximized, which is ˜B|δ∗ = B  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' NUMERICAL RESULTS  In this section, we present simulation results for the two  proposed gossiping schemes, and compare them with the  theoretically derived age expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' We also show the results  for ASUMAN [1] as a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 3, we present the numerical results for λe  λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='4,  λe  λ  = 1 and  λe  λ  = 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 3(a), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 3(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 3(c),  respectively, with λ = 1 in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' From the figures,  it is evident that all the gossiping schemes result in O(1)  performance and the semi-distributed gossiping scheme yields  the best performance among all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 3(a), where λe  λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='4 <  1  e−1, ASUMAN gives the  worst age performance among the three schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' However,  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 3(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 3(c), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=', for  λe  λ  >  1  e−1, ASUMAN  performs worse than the semi-distributed scheme, but is better  than the fully-distributed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' This matches our intuition  because, in ASUMAN, we use the information about source  self-updates to allocate gossip rate more efficiently, while in  the fully-distributed scheme, multiple nodes gossiping together  causes interference to lose some portion of the total gossip  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' This effect of interference becomes more prominent when  the source to network update rate λ is high as compared  to source self-update rate λe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' We have chosen δ =  1  λ = 1  for the simulation to get the minimum average age for fully-  distributed gossiping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  For  ASUMAN,  the  asymptotic  age  scales  as  limn→∞ λe  λ  �  1+  n  n−1 (1+ λ  λe )  1  n +  n  n−1  �  = 2 λe  λ + 1, while the other two  0  100  200  300  400  500  600  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='8  2  Theorem 1 formula (12)  semi-distributed gossip  Theorem 3 formula (21)  fully-distributed gossip  ASUMAN  (a) λe  λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='4  0  100  200  300  400  500  600  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='5  4  Theorem 1 formula (12)  semi-distributed gossip  Theorem 3 formula (21)  fully-distributed gossip  ASUMAN  (b) λe  λ = 1  0  100  200  300  400  500  600  2  3  4  5  6  7  8  Theorem 1 formula (12)  semi-distributed gossip  Theorem 3 formula (21)  fully-distributed gossip  ASUMAN  (c) λe  λ = 2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  Average version age of a single node versus the total number of  nodes in the network n for semi-distributed, fully-distributed and ASUMAN  schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  schemes obtain 2 λe  λ  and (1 + e) λe  λ , as shown in (13) and  (23) (with optimized δ), respectively, and as listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  The numerical simulation results exactly match the derived  formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' With an increase in the ratio  λe  λ , the average  age increases due to source being updated more frequently  compared to the network for all schemes, as we observe  going from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 3(a) to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 3(b) to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' CONCLUSION  We proposed a semi-distributed and a fully-distributed  gossiping scheme for a fully-connected network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' The semi-  distributed scheme allows the freshest node to communicate  in the network through pilot signals and to gossip with full  capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' This scheme archives the lowest possible average  age for any symmetric network, with a constraint on the  instantaneous gossip rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' On the other hand, in the fully-  distributed scheme, the freshest node gossips for a fixed time  duration with full capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' The effective gossip happens only  a fraction of the total time, when there is no interference  from multiple nodes gossiping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Both of the proposed schemes  yield O(1) age performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Compared to our previous work  ASUMAN, which also gives O(1) age scaling, this work is an  improvement because here we do not require the nodes to be  age-aware or to transmit pilot signals for channel reservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
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+page_content=' Kadota, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Talak, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
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+page_content=' Bolch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Greiner, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' de Meer, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Trivedi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  Queueing  Networks and Markov Chains: Modeling and Performance Evaluation  with Computer Science Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' John Wiley & Sons, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content='  [29] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' Newell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' The M/G/∞ queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}
+page_content=' SIAM Journal on Applied Mathe-  matics, 14(1):86–88, January 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'}

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+Learning Modulo Theories
+Matt Fredrikson 1 Kaiji Lu 1 Saranya Vijayakumar 1 Somesh Jha 2 Vijay Ganesh 3 Zifan Wang 1
+Abstract
+Recent techniques that integrate solver layers
+into Deep Neural Networks (DNNs) have shown
+promise in bridging a long-standing gap between
+inductive learning and symbolic reasoning tech-
+niques. In this paper we present a set of tech-
+niques for integrating Satisfiability Modulo Theo-
+ries (SMT) solvers into the forward and backward
+passes of a deep network layer, called SMTLayer.
+Using this approach, one can encode rich domain
+knowledge into the network in the form of mathe-
+matical formulas. In the forward pass, the solver
+uses symbols produced by prior layers, along with
+these formulas, to construct inferences; in the
+backward pass, the solver informs updates to the
+network, driving it towards representations that
+are compatible with the solver’s theory. Notably,
+the solver need not be differentiable. We imple-
+ment SMTLayer as a Pytorch module, and our
+empirical results show that it leads to models that
+1) require fewer training samples than conven-
+tional models, 2) that are robust to certain types
+of covariate shift, and 3) that ultimately learn
+representations that are consistent with symbolic
+knowledge, and thus naturally interpretable.
+1. Introduction
+A recent class of techniques aims at integrating solver
+layer(s) within deep neural networks (DNNs) (Wang et al.,
+2019a; Pogancic et al., 2020; Huang et al., 2021; Manhaeve
+et al., 2018), both during training and inference. A class
+of problems which can benefit from such an integration is
+one that has both a perceptual and a symbolic sub-problem,
+such as “visual” Sudoku (Wang et al., 2019a), or the prob-
+lem of determining the shortest path from a picture of a
+map (Pogancic et al., 2020).
+The most straightforward way to incorporate a solver layer
+1Carnegie Mellon University, Pittsburgh, PA, USA 2University
+of Wisconsin-Madison, Madison, WI, USA 3University of Water-
+loo, Waterloo, ON, Canada. Correspondence to: Matt Fredrikson
+<[email protected]>.
+into an ML model is to learn models with representations
+that are compatible with symbols used by the solver. For
+example, if one wanted to leverage symbolic domain knowl-
+edge to classify images of birds, or diagnose ailments from
+CT scans, then one could train a model in a fashion similar
+to “concept bottlenecking” (Koh et al., 2020). This requires
+detailed labels for supervision, which may be prohibitively
+expensive to obtain and keep consistent with a potentially
+evolving domain theory.
+We present a set of techniques for incorporating a Satisfia-
+bility Modulo Theories (SMT) solver into a DNN layer so
+that symbolic knowledge can be leveraged to learn such a
+compatible representation, without requiring label super-
+vision. Our approach is general, and can handle a broad
+range of domain knowledge encoded as SMT constraints,
+provided that they interface with the surrounding neural net-
+work layers over propositional variables. Unlike the most
+closely related prior work (Wang et al., 2019a), our approach
+does not approximate the solver’s behavior by formulating
+a differentiable relaxation. Rather, we extract information
+from the solver as it works on a set of constraints, that is
+geared towards checking the correctness of the output of the
+model that precedes the solver, and use that information to
+construct updates to the model during training (Section 4.2).
+We present two different approaches for this, one based
+on unsatisfiable cores, and another based on weighted
+MaxSMT (Section 4.2). There are several advantages to
+this approach. Aside from the mild interface constraints
+mentioned earlier (i.e., solver and neural layers interface
+with each other via boolean variables), our approach does
+not place any restrictions on the theory solver embedded in
+the layer, such as linearity (Pogancic et al., 2020) or even
+decidability—if the solver is capable of efficiently discharg-
+ing the relevant constraints, then the layer can operate as
+intended. Because there is no need to provide a differen-
+tiable relaxation for each theory or solver technique that one
+may want to incorporate, we can leverage the continuous
+and unabated progress being made in solver technology.
+We implement our approach as a PyTorch (Paszke et al.,
+2019) layer, using the Z3 (De Moura & Bjørner, 2008) SMT
+solver as the solver layer to solve SMT and MaxSMT con-
+straints. On three applications involving vision and natural
+language: visual arithmetic, algebraic equation solving, and
+arXiv:2301.11435v1  [cs.LG]  26 Jan 2023
+
+Learning Modulo Theories
+a so-called natural language “liar’s puzzle,” we demonstrate
+that our implementation can be incorporated into DNN ar-
+chitectures to solve problems more effectively than conven-
+tional DNNs (Section 5). In particular, our results show that
+the data needed to train a DNN with symbolic knowledge
+may be much simpler than may be necessary otherwise, and
+that while doing so is more expensive computationally, of-
+ten times the more efficient (i.e., not involving MaxSMT)
+algorithms perform well in practice.
+Our contributions are as follows:
+1. We present SMTLayer, a framework for incorporating
+an SMT solver into a DNN, as a layer that leverages
+symbolic knowledge during training and inference.
+2. We prototype our approach in Pytorch1, and show that
+it can be applied to solve a range of problems that
+incorporate symbolic knowledge.
+3. Our empirical evaluation, over four diverse applica-
+tions, shows that models using SMTLayer require sig-
+nificantly less training data, can be trained more ef-
+ficiently, and are more robust than those based on
+closely-related prior work (Wang et al., 2019a; Huang
+et al., 2021).
+Section 3 provides background on ERM and the first-
+order theories used in our framework. Section 4 describes
+SMTLayer, Section 5 gives our empirical evaluation, and
+Section 6 concludes the paper.
+2. Related Work
+Combining logical solvers and deep models can be diffi-
+cult because logic has discrete structure while the most
+successful way to construct neural networks today requires
+differentiability (Riegel et al., 2020).
+Combinatorial Solver Layers.
+Vlastelica et al. (2019) in-
+tegrate a blackbox, non-differentiable combinatorial solver
+on top of a deep network. To propagate the gradient through
+the solver on the backward pass, they linearly interpolate
+the loss w.r.t the solver’s input and define the gradient of the
+solver as the slopes of the line segments. CSL solves a set
+of problems where the solver’s objective must be linear w.r.t
+its input, e.g. finding the shortest path and travel salesman
+problem (TSP). Further, the authors assume that the only
+labels available are the outputs of solvers, e.g. the minimum
+cost in TSP, and hence their tool has to discover the label for
+the output of the network itself. These requirements limit
+the choices one has for the solver layer.
+1We plan to release our implementation as an open source
+library upon publication of this paper
+Neural Logic Programming.
+While SATNet integrates
+a logic-based solver on top of a network, DeepProbLog
+takes the opposite approach, extending the capability of a
+probabilistic logic programming language with neural pred-
+icates Manhaeve et al. (2018). In the context of our work,
+the logic program can be viewed as a “solver layer” that
+explicitly encodes symbolic knowledge. Scallop (Huang
+et al., 2021) extends DeepProbLog to scale without sacri-
+ficing accuracy compared to DeepProbLog. Similarly to
+DeepProbLog, each possible result of the sum of two digits
+in MNIST is given a probability, in the form of a weighted
+Boolean formula. They prune unlikely clauses of the for-
+mula, represented by proofs, only keeping the top-k most
+likely. Likelihood is computed using weighted model count-
+ing (Huang et al., 2021; Chavira & Darwiche, 2008). These
+techniques are well-suited to problems that benefit from
+probabilistic Datalog, but have inherent limitations: they
+cannot handle quantifiers, general negation, and the range
+of supported first-order theories is more restrictive.
+SATNet.
+Wang et al. (2019b) present SATNet, a network
+architecture with a differentiable approximate MAXSAT
+solver layer. Their approximation is based on a coordinate
+descent approach to solving the semidefinite program (SDP)
+relaxation of the MAXSAT problem. SATNet does not as-
+sume that the logical structure of the problem is given, and
+instead attempts to learn it. By placing the MAXSAT solver
+layer on top of a convolution network to learn represen-
+tations from images, SATNet directly solve problems like
+Visual Sudoku, for which neural networks alone are not well
+suited (Wang et al., 2019b).
+Differentiable Logic.
+Another recent direction has ex-
+plored differentiable logics (Fischer et al., 2019; Varnai &
+Dimarogonas, 2020; van Krieken et al., 2022). These ap-
+proaches provide ways of integrating symbolic knowledge
+into training, by making logical formulas differentiable, and
+therefore amenable to optimization when included in a loss
+function. This line of work does not explicitly aim to make
+use of symbolic information during inference. In contrast,
+the information that our approach extracts from the solver
+during training is used to condition the model towards a rep-
+resentation that will allow it to communicate effectively with
+the solver during inference. Additionally, we do not require
+the logical formulas, or the solver, to be differentiable.
+3. Background
+Let X denote a domain of features, Y a domain of labels,
+and D a distribution over X × Y. Formally, D is a proba-
+bility measure on a space given by a σ-algebra over subsets
+of ℘(X × Y). The goal of a learning algorithm A is to
+find a function h : X → Y that, for (x, y) ∼ D, can be
+used to predict y when given x. To do this, A is given a set
+
+Learning Modulo Theories
+of training examples S = (x1, y1), . . . , (xm, ym) sampled
+i.i.d. from D, and uses some criterion to select h from a
+hypothesis class H of functions. We refer to h as the model
+learned by A on S. When the learning algorithm A is clear
+from the context, we will write hS to denote the model pro-
+duced from the given sample. Throughout this paper, we
+will generally assume that the loss is either the 0-1 loss ℓ01
+or binary cross-entropy ℓbce.
+A theory T consists of a signature Σ of constant, predicate,
+and function symbols, as well as a set of axioms over Σ.
+Formulas in a theory are composed of elements of Σ, vari-
+ables, and logical symbols such as quantifiers and Boolean
+operations. We use the term decision procedure to refer to
+an algorithm that is given an open T-formula, and returns
+true if it is satisfiable, and false otherwise. Additionally, it
+may return an assignment to all of the variables that demon-
+strates satisfiability, or if the formula is not satisfiable, then
+it may return an unsatisfiable core, which is a subset of
+clauses taken from the formula’s representation in conjunc-
+tive normal form that remains unsatisfiable. Loosely, we
+also refer to such an algorithm as a “solver”, but this term is
+more general, and could also refer to an algorithm that iden-
+tifies the maximal set of clauses, possibly weighted by some
+user-defined values, that are satisfiable when conjoined.
+4. Constructing SMTLayer
+In this section, we present SMTLayer, a set of algorithms
+for computing the forward and backward passes of a layer
+whose behavior is defined by a set of user-defined SMT
+constraints. SMTLayer does not have trainable parame-
+ters, and its functionality is wholly defined by a set of
+SMT constraints φ that are provided by the model de-
+signer. SMTLayer can be used in modern deep-learning
+frameworks as a drop-in replacement for more conven-
+tional neural network layers, e.g., dense, convolutional, and
+LSTM (Hochreiter & Schmidhuber, 1997) are prominent
+examples of widely-used layers.
+Section 4.1 provides a high-level overview of our approach,
+Section 4.2 describes them in detail, and Section 4.3 begins
+an analysis of this setting that we hope future work will
+continue developing.
+4.1. Overview
+We envision SMTLayer being used primarily at the top of
+a DNN taking inputs from a stack of conventional DNN
+layers that convert raw input features into ground terms for
+the constraints φ(z0, . . . , zp−1, y0, . . . , yq−1) embedded in
+SMTLayer, and producing outputs that are consistent with
+φ and the given ground terms. Figure 1 shows an illustrative
+example, with the previously-studied problem of MNIST
+addition (Manhaeve et al., 2018; Huang et al., 2021).
+Algorithm 1 Fφ
+max(z)
+MaxSMT-based forward pass of SMTLayer
+Inputs: z ∈ Rp layer input
+φ(z0, . . . , zp−1, y0, . . . , yq−1) T-formula
+Output: y ∈ Rq
+1 begin
+2
+zb ← [z[i] > 0 : i = 0 . . . p − 1]
+3
+C ← �
+i∈I softmax(|z|)[i]
+4
+y ← arg maxyb maxI C·1(φ∧�
+i∈I yi = yb[i]∧zi = zb[i])
+5
+return y
+6 end
+During the forward pass the outputs of the previous layer
+are mapped to designated free variables z0, . . . , zp−1. The
+layer then checks the satisfiability of φ, a formula in an
+appropriate combination of first-order theories, after substi-
+tuting these ground terms for the zi, and the output of the
+layer consists of the solver’s model for y0, . . . , yq−1. These
+outputs are converted from Boolean to floating-point values
+by mapping false to -1 and true to 1. At the moment, the
+only restriction on φ that our layer requires is that z and y
+be vectors of Booleans, so that they can be appropriately
+mapped to continuous values; any other symbols appear-
+ing in φ can come from arbitrary domains (e.g. strings)
+supported by the underlying SMT solver.
+In the backward pass, the layer receives the gradient of
+its output with respect to the function whose derivative
+is being computed, which we will assume is the binary
+cross-entropy loss ℓ(y, y⋆). Unless stated otherwise, we
+will assume this loss for the remainder of the section. This
+gradient is used, along with the inputs and outputs of the
+corresponding forward pass, to first compute an amended
+output ˆy which corresponds to an output that would have
+yielded a smaller loss. Because the outputs are Boolean,
+it is always possible to determine the ground truth output
+y⋆ from this information. Using ˆy, the layer determines
+which of components of its inputs are inconsistent with
+φ and ˆy, and provides the corresponding gradients to the
+previous layer. Section 4.2 details the manner in which these
+gradients are computed.
+4.2. SMTLayer, forward and backward
+We now present the details of the forward and backward
+passes of SMTLayer. There are two algorithms for each
+pass, Fφ
+max and Fφ
+smt are forward passes, and Bφ
+max, Bφ
+core
+are backward passes. Fφ
+max and Bφ
+max both make use of
+MaxSMT solvers, whereas Fφ
+smt and Bφ
+core rely on satisfia-
+bility solvers (SMT). Despite the symmetry in which type
+of solver each algorithm uses, they are all compatible with
+each other. That is, Fφ
+smt can be used with either Bφ
+max or
+Bφ
+core, and the same for Fφ
+max.
+
+Learning Modulo Theories
+Features X
+Symbolic Domain Z
+0010000111
+Neural
+Network
+φ(
+z1∥ . . . ∥z10,
+y
+) ≡
+a = 1z1>0∥ . . . ∥1z5>0 ∧ b = 1z5>0∥ . . . ∥1z10>0 ∧
+a + b = y
+Prediction Logic φ
+Labels Y
+{01011}
+Satisfying
+Assignments
+Figure 1. MNIST Addition example.
+Algorithm 2 Fφ
+smt(z)
+SMT-based forward pass of SMTLayer
+Inputs: z ∈ Rp layer input
+φ(z0, . . . , zp−1, y0, . . . , yq−1) T-formula
+Output: y ∈ Rq
+1 begin
+2
+zb ← [z[i] > 0 : i = 0 . . . p − 1]
+3
+ˆφ ← φ(zb[0], . . . , zb[p − 1])
+4
+if ˆφ is satisfiable then
+5
+yb[0], . . . , yb[q − 1] ← solve(ˆφ, y0, . . . , yq−1)
+6
+y ← [yb[i] > 0
+: i = 0 . . . q − 1]
+7
+else
+8
+y ← 0
+9
+end
+10
+return y
+11 end
+Forward pass.
+Algorithms 1 and 2 illustrate Fφ
+max and
+Fφ
+smt, the methods for computing the forward pass based
+on weighted MaxSMT and SMT, respectively. Both of the
+algorithms are parameterized by a user-provided first-order
+formula φ, and take a single vector-valued input consisting
+of unscaled floating-point values (logits). These values are
+cast to Boolean constants by taking their sign on line 2
+of both algorithms, so that they can be equated with the
+corresponding free variables z0, . . . , zp−1.
+The key difference between Fφ
+max and Fφ
+smt is the way in
+which they handle inputs that are inconsistent with φ when
+interpreted as Booleans. Fφ
+smt addresses this by provid-
+ing an output that is also inconsistent with φ, i.e. a vector
+of zeroes, effectively signaling that the network below it
+did not provide consistent inputs. Alternatively, we can
+interpret the values provided by the network as Booleans
+enriched with “confidence” values. Although we expect
+inputs to SMTLayer to be unscaled floating-point values,
+Algorithm 1 scales them to a formal probability distribution
+via the softmax function (line 3) for use as weights to find
+the weighted MaxSMT solution of φ. With this approach,
+Algorithm 3 Bφ
+max(z, y, ∂yℓ(y, y⋆))
+MaxSMT-based backward pass of SMTLayer
+Inputs: z ∈ Rp input of forward pass
+y ∈ Rq output of forward pass
+∂yℓ(y, y⋆) gradient with respect to output
+φ(z0, . . . , zp−1, y0, . . . , yq−1) a T-formula
+Output: ∂zℓ(y, y⋆) ∈ Rp approximate gradient of ℓ
+1 begin
+2
+Gz ← ∂zℓ(z, sign(z))
+3
+ˆy = sign(y) − 2 · sign (∂yℓ(y, y⋆))
+4
+if sign(y) ̸= sign(ˆy) then
+5
+zb ← [z[i] > 0 : i = 0 . . . p − 1]
+6
+ˆyb ← [ˆy[i] > 0 : i = 0 . . . q − 1]
+7
+φy ← �
+0≤i<q yi = ˆyb[i]
+8
+C ← �
+i∈I softmax(|z|)[i]
+9
+I ← arg maxI⊆[0,p) 1(φ ∧ φy ∧ �
+i∈I zi = zb[i]) · C
+10
+foreach i ∈ ¯I do
+11
+Gz[i] ← ∂z[i]ℓ(z[i], 1 − sign(z[i]))
+12
+end
+13
+end
+14
+return Gz
+15 end
+SMTLayer will always provide a valid, although not neces-
+sarily correct, output that is consistent wrt φ with the inputs
+of which the network below is most “confident.” (line 4).
+Backward pass.
+The backward pass is responsible for
+computing the gradient of the loss with respect to the layer
+inputs. It is given the gradient of the loss with respect to the
+layer outputs, and is assumed to have memoized the inputs
+that it received previously on the forward pass, as well as the
+outputs that they produced. The gradients returned by this
+pass are then used by the backward pass of previous layers,
+and ultimately to derive updates to trainable parameters that
+will yield smaller loss.
+The key issue in designing a backward pass for SMTLayer
+is the geometry of the functions computed by either for-
+ward pass. For any vector v ∈ {−1, 0, 1}p and x, x′ with
+sign(x) = sign(x′) = v, then Bφ
+· (x) = Bφ
+· (x′), so these
+
+47Learning Modulo Theories
+Algorithm 4 Bφ
+core(z, y, ∂yℓ(y, y⋆))
+unsat core-based backward pass of SMTLayer
+Inputs: z ∈ Rp input of forward pass
+y ∈ Rq output of forward pass
+∂yℓ(y, y⋆) gradient with respect to output
+φ(z0, . . . , zp−1, y0, . . . , yq−1) a T-formula
+Output: ∂zℓ(y, y⋆) ∈ Rp approximate gradient of ℓ
+1 begin
+2
+Gz ← ∂zℓ(z, sign(z))
+3
+ˆy = sign(y) − 2 · sign (∂yℓ(y, y⋆))
+4
+if sign(y) ̸= sign(ˆy) then
+5
+zb ← [z[i] > 0 : i = 0 . . . p − 1]
+6
+ˆyb ← [ˆy[i] > 0 : i = 0 . . . q − 1]
+7
+φz, φy ← �
+0≤i<p zi = zb[i], �
+0≤i<q yi = ˆyb[i]
+8
+I ← arg minI⊆[0,p) 1(¬φ∨¬φy ∨�
+i∈I zi ̸= zb[i])·|I|
+9
+foreach i ∈ I do
+10
+Gz[i] ← ∂z[i]ℓ(z[i], 1 − sign(z[i]))
+11
+end
+12
+foreach i ∈ ¯I do
+13
+Gz[i] ← 0
+14
+end
+15
+end
+16
+return Gz
+17 end
+functions are piece-wise constant step functions ranging
+over the corners of the Rq unit hypercube. Thus, while
+they are differentiable almost everywhere, the gradient is
+not helpful for training because it is always zero. Prior
+work on integrating such step functions into deep networks
+primarily addresses this problem by relaxing the function
+computed by the forward pass, so that its gradients are no
+longer constant and hopefully more informative.
+In contrast, Bφ
+max (Algorithm 3) and Bφ
+core (Algorithm 4)
+do not attempt to provide gradients for a relaxation of the
+forward pass. Instead, they use information provided by
+the solver in its computation of the forward pass to identify
+which components of the input may have contributed to
+higher loss. The gradient is then computed by constructing
+a counterfactual variant of the input provided to the for-
+ward pass, which differs on the identified components, and
+returning the gradient of the binary cross-entropy loss of
+the original input on this counterfactual variant. The two
+algorithms differ in the information that they extract from
+the solver, i.e., either solutions to a MaxSMT instance or an
+unsatisfiable core.
+Both algorithms begin by initializing the gradient to be the
+loss between the logit inputs, and their hard labels (line 2).
+Recall that we assume the loss ℓ is binary cross-entropy, so
+the result will not be zero. The purpose of this initialization
+is to emulate the dynamics of training with cross-entropy
+loss with a conventional layer; when the rounded output
+matches the target, the loss is not zero, and training will
+continue to move the parameters in a direction that makes
+them agree “more” with the hard target. One line 3, they
+then use the provided gradient from the subsequent layer
+together with the memoized output from the forward pass
+to construct ˆy, a “corrected” output that satisfies ℓ(ˆy, y⋆) ≤
+ℓ(y, y⋆). If the sign of ˆy is the same as that of y, then both
+algorithms return the initialized gradient. Otherwise, they
+extract information from the solver using the inputs to the
+forward pass z and ˆy.
+Bφ
+max constructs a set of clauses φy that constrain the free
+y0, . . . , yq−1 to take the values of ˆyb, the Boolean conver-
+sion of ˆy. It then computes the softmax values of the abso-
+lute incoming logits |z|, and uses them to find the maximally-
+weighted set of clauses (softmax(|z|)[i], zi = zb[i]) that are
+consistent with φ ∧ φy. Intuitively, these are the inputs that
+the previous layer is most confident in that can be made
+consistent with the corrected label ˆy by changing some of
+the less confident inputs. Bφ
+max then updates the initialized
+gradient at each index for which the solution to this instance
+does not match the sign of the original input.
+Bφ
+core also constructs φy, but instead identifies a set of con-
+straints zi = zb[i] that are inconsistent with φ ∧ φy. Note
+that line 7 specifies a minimal unsatisfiable core, but this is
+not necessary. All that is needed is that none of the clauses
+in the core be superfluous, i.e., deleting any singleton clause
+from I will cause it to be satisfiable. If a superfluous clause
+remains in the core, then the gradient returned for the corre-
+sponding input will have the incorrect sign, which may lead
+to issues with training. Bφ
+core then updates the gradient at
+each input identified in the core using the loss of z[i] with
+respect to 1 − sign(z[i]), which will lead to updates in a
+direction that would have modified the input such that i was
+not in the unsat core. The indices not in the unsat core have
+their gradients set to zero, as their absence in the core is not
+evidence that these inputs were correct or incorrect.
+4.3. Analysis
+To understand the settings where SMTLayer will provide op-
+timal results, we introduce a class of decomposable learning
+problems (Definition 1).
+Definition 1 (Decomposable problem). Let T be a first-
+order theory with constants in Z. An ERM problem D, H
+is decomposable by T if there exists a function f : X → Z,
+companion hypothesis class Hf ⊆ X → Z, and T-formula
+φ such that:
+1. For any h ∈ H, there exists hf ∈ Hf and h′ such that
+h = h′ ◦ hf.
+2. There exists a random function g : ℘(Y) → Y such
+that for any n > 0 and ∀S in the support of Dn,
+Pr
+(x,y)∼D[(x, y) ∈ S] =
+Pr
+(x,·)∼D[(x, g(⟨x⟩f,φ)) ∈ S]
+where ⟨x⟩f,φ = {y : φ(f(x), y) is satisfied}.
+
+Learning Modulo Theories
+In (2), f is called the grounding function and φ is called the
+prediction logic.
+Intuitively, a learning problem defined in terms of a distribu-
+tion D and hypothesis class H is decomposable if members
+of H can be decomposed into functions that are responsible
+for grounding and prediction, and D can be expressed in
+terms of a grounding function and a first-order formula φ.
+There are a few important things to note. First, there is no re-
+quirement that the grounding function f be a member of Hf.
+While this may be realized at times, we should not assume
+that the data is actually generated, or otherwise described
+with perfect fidelity, by a function in the class that one learns
+over. In fact, we do not assume that f is efficiently com-
+putable, as it may correspond to a natural process, or an
+aspect of data generation that is not understood well enough
+to make such computational claims.
+Second, for a given x, there may be more than one satis-
+fying assignment for y to φ(f(x), y). The function g in
+(2) accounts for this, requiring only that when solutions
+to φ(f(x), y) are sampled by g, the result is distributed
+identically to D. This paper will focus on the case where
+satisfying assignments for y are unique, as these are more in
+line with ”classic” ERM classification problems. We leave
+exploration of the more general setting to future work.
+We note that if the grounding function is known, can be
+computed efficiently, and φ is efficiently solvable, then the
+learning problem effectively has a closed-form solution.
+Rather, we assume that only φ and perhaps g are known,
+and a sample of D is given. The remaining challenge is
+to identify a grounding hypothesis hf ∈ Hf for which the
+construction in (2) is an effective solution to the end-to-end
+learning problem posed by D, H. This stands in contrast to
+traditional ERM, in which a good solution h ∈ H must ei-
+ther solve both grounding and prediction, or find a “shortcut”
+that manages to predict D as well as the decomposition.
+Convergence.
+Regarding the backward passes, Theo-
+rem 2 below demonstrates that when φ satisfies certain
+conditions, and the companion hypothesis class Hf satisfies
+conditions that are sufficient to guarantee convergence with
+SGD, then training with Fφ
+smt and Bφ
+max will converge to
+the optimal solution in the number of iterations. The proof
+of this theorem is based on the observation that when the
+conditions on φ are met, then training with Bφ
+max obtains the
+same solution that would be obtained if the labels of φ were
+available for supervised learning. Thus, the conditions on
+Hf are sufficient to ensure the stated convergence, as stated
+in a well-known result outlined in Chapter 14 of (Shalev-
+Shwartz & Ben-David, 2014).
+It is also worth noting that Theorem 2 does not necessar-
+ily hold if Bφ
+core is used instead of Bφ
+max. The reason is
+that there may be many unsatisfiable cores that are locally
+minimal in cardinality, and gradients are set only for in-
+puts that appear in the computed core. These gradients will
+not match those of the loss on a grounding sample, so the
+training dynamics are likely to be different. We believe that
+training with Bφ
+core may have more in common with block
+coordinate descent than gradient descent, and save a more
+detailed exploration of the topic for future work.
+Theorem 2. Let D, H be a T-decomposable problem with
+grounding function f and prediction logic φ where:
+1. Z and Y are Cartesian products of Booleans.
+2. For any (x, y) ∼ D and y′ ̸= y, φ(f(x), y′) is T-
+equivalent to false and there is exactly one z such that
+φ(z, y) is T-equivalent to true.
+3. Hf is a convex set and for all hf ∈ Hf, ∥hf∥ ≤ B,
+and the loss ℓ(hf(·), z) is M-Lipschitz and convex in
+x for any fixed z.
+Then for any ϵ > 0, selecting hf by minimizing either
+LS(Fφ
+smt(hf(·))) with τ ≥ M 2B2/ϵ2 iterations of stochastic
+gradient descent, with gradients provided by Bφ
+max, and
+learning rate η =
+�
+B2/M 2τ yields a grounding hypothesis
+ˆhf ∈ Hf that satisfies: E[LD( ˆhf)] ≤ minhf ∈Hf LD(hf)+
+ϵ. The randomness in this expectation is taken over the
+choices of the SGD algorithm.
+5. Experimental Evaluation
+In this section we present an empirical evaluation of
+SMTLayer on four learning problems that can be decom-
+posed into perceptual and symbolic subtasks. Our results
+demonstrate the following primary findings. 1) SMTLayer
+is effective: on every benchmark, it provides superior re-
+sults over “conventional” learning that takes place without
+encoded symbolic knowledge. 2) SMTLayer has distinct
+advantages over prior approaches. Compared with SAT-
+Net (Wang et al., 2019a), it requires significantly less train-
+ing data to converge, and in all cases yields a more accurate
+model; compared with Scallop (Huang et al., 2021), it is
+less computationally expensive, requires less training data,
+and it is more expressive in terms of the knowledge that
+it can encode. 3) Models trained with SMTLayer may be
+more robust to certain types of covariate shift that occur
+relative to the symbolic component of the problem; when
+SMTLayer succeeds at learning a compatible representation,
+then it will continue to produce correct inferences provided
+the perceptual component remains stationary.
+5.1. Datasets
+Additional details on the datasets and corresponding ar-
+chitectures used in our evaluation can be found in Ap-
+
+Learning Modulo Theories
+pendix A.2, and specific hyperparameters used when train-
+ing on each dataset are in Appendix A.3.
+MNIST Addition.
+The MNIST addition problem is illus-
+trated in Figure 1, and is similar to the benchmark described
+by Huang et al. (2021). For training, we use “MNIST +p%”
+to denote a training set of size 60,000 that contains p% of
+the possible pairs of digits. So p = 100 indicates all pos-
+sible pairs of digits are used, and for p = 10, we only use
+pairs of the same digit. We use p = 10, 25, 50, 75 and 100
+in our experiments. In all cases, we use the same test set
+consisting of instances from all possible pairs of digits.
+Visual Algebra.
+The task is to solve for the variable x in
+a graphical depiction of the equation ax + b = c, where
+a, b and c are randomly-chosen numbers, and each symbol
+is depicted visually using EMNIST (Cohen et al., 2017a)
+and HASY graphics (Thoma, 2017b). Similar to MNIST
+addition, the training sample selects a and b uniformly from
+pairs of the same digit, and x uniformly from the odd num-
+bers between 0 and 9. The test sample was generated by
+sampling a, b uniformly from all pairs of digits, and x from
+all numbers 0 to 9.
+Liar’s Puzzle.
+The liar’s puzzle is comprised of three
+sentences spoken by three distinct agents: Alice, Bob, and
+Charlie. One of the agents is “guilty” of an unspecified
+offense, and in each sentence, the corresponding agent either
+states that one of the other parties is either guilty or innocent.
+For example, “Alice says that Bob is innocent.” It is assumed
+that two of the agents are honest, and the guilty party is not.
+The solution to the problem is an identification of the guilty
+party. A formal characterization of the underlying logic
+is given in Appendix A.2. We note that the logic has non-
+stratified occurrences of negation, so it cannot be encoded
+with Scallop. We select a training sample that does not
+fully specify the logic, so conventional training should be
+insufficient to identify a good model.
+Visual Sudoku.
+This task is to complete a 9 × 9 Sudoku
+board where each entry is an MNIST digit. We use the
+dataset from the SATNet evaluation (Wang et al., 2019a),
+and examine three configurations obtained by sampling 10%,
+50%, and 100% of the original training set. Although there
+are examples of Sudoku solvers implemented as logic pro-
+grams, we were not able to implement one in Scallop with-
+out violating stratified negation. When calculating accuracy,
+we check that the entire Sudoku board is correct.
+5.2. Setup
+We implemented a prototype of our approach using Py-
+torch (Paszke et al., 2019) and Z3 (De Moura & Bjørner,
+2008), which will be made available in open-source when
+this paper is published.
+When training models with
+SMTLayer, we use SGD with Nesterov momentum at rate
+0.9 and gradient clipping rate 0.1. Before training a model
+with SMTLayer (or a comparison technique, unless stated
+otherwise), we first pre-train the neural network by replac-
+ing SMTLayer with a dense network containing one hidden
+layer of 512 neurons. This can potentially limit the num-
+ber of training updates needed at lower layers, but will not
+result in a model with a representation that is compatible
+with symbolic knowledge, so further training is needed. The
+models labeled “conventional” in our evaluation have the
+same architecture as the one used for pre-training. Results
+were averaged over five runs of training.
+Our evaluation was performed on a machine with an Intel
+i9 1050K CPU, 64GB memory, and a GeForce RTX 3080
+accelerator running Ubuntu 20.04.4, with CUDA 11.1.0 and
+cuDNN 8.0.4. We developed and tested our prototype with
+Pytorch version 1.7.0a0+7036e91 and Z3 4.8.14, and the
+results in our evaluation use these versions as well.
+5.3. Results
+Overall performance.
+In terms of accuracy, Table 1
+shows that SMTLayer outperforms both the conventional
+network and prior work in terms of accuracy, training
+time, or both, on all configurations. While training with
+SMTLayer (or any of the above approaches) is more ex-
+pensive than conventional, SMTLayer is consistently faster
+than Scallop (nearly 4× in the case of visual algebra). The
+per-epoch time to train the SATNet models is less expensive
+than SMTLayer, but this is not always conclusive. In the
+case of visual sudoku, the 10% SMTLayer model achieved
+superior error rates in 15 epochs, compared with 100 epochs
+for the 100% SATNet model; this means that the SMTLayer
+model took less than one-tenth the amount of time to train.
+It is also worth noting that although Theorem 2 suggests
+that Algorithm 3 might have learning advantages over Algo-
+rithm 4, we found this not to be the case on these datasets.
+All of the results in Table 1 were trained with Algorithm 4,
+and test inference was done using Algorithm 1.
+Training sample size.
+Because SMTLayer encodes ex-
+plicit knowledge that is essential to correct inference on
+these datasets, our approach is able to perform well in data-
+impoverished settings where the training sample is insuffi-
+cient to fully specify the symbolic component of the learning
+task. This is readily apparent across the results in Table 1: in
+the MNIST addition and first visual algebra configuration,
+SMTLayer yields a model that performs nearly perfectly
+despite not being given a sufficient sample in most cases.
+Because SATNet must learn the symbolic component, it is
+at a disadvantage, and in these settings performs similarly
+to a conventional model. In theory, Scallop should be able
+
+Learning Modulo Theories
+Conventional
+w/ SMTLayer
+w/ SATNet
+w/ Scallop
+configuration
+test
+epoch
+test
+epoch
+test
+epoch
+test
+epoch
+acc. (%)
+time (sec.)
+acc.(%)
+time (sec.)
+acc.(%)
+time (sec.)
+acc. (%)
+time (sec.)
+MNIST+ 10%
+10.0
+7.1
+98.1
+75.4
+10.0
+31.0
+33.7
+96.3
+MNIST+ 25%
+32.5
+7.1
+98.3
+74.8
+34.2
+30.9
+65.8
+96.4
+MNIST+ 50%
+51.5
+7.0
+98.6
+75.8
+54.8
+32.8
+98.4
+96.5
+MNIST+ 75%
+76.1
+7.0
+98.5
+75.0
+78.4
+31.9
+93.5
+96.4
+MNIST+ 100%
+98.3
+7.1
+98.5
+75.8
+96.7
+33.5
+98.6
+96.6
+Vis. Alg. #1
+24.1
+13.2
+98.2
+168.2
+19.6
+80.1
+18.7
+602.8
+Vis. Alg. #2
+25.4
+11.2
+25.4
+127.2
+18.6
+52.5
+21.3
+636.1
+Liar’s Puzzle
+54.2
+3.1
+86.1
+28.7
+84.6
+3.0
+—
+—
+Vis. Sudoku 10%
+0.0
+6.3
+66.0
+135.7
+0.0
+9.9
+—
+—
+Vis. Sudoku 50%
+0.0
+28.3
+73.1
+608.1
+0.0
+45.4
+—
+—
+Vis. Sudoku 100%
+0.0
+26.7
+79.1
+1199.0
+63.2
+86.5
+—
+—
+Table 1. Results after training and inference with SMTLayer versus a conventional architecture. We use the publicly-available imple-
+mentations of SATNet (Wang et al., 2019a) and Scallop (Huang et al., 2021), with hyperparameters matching those in their code. All
+SMTLayer test accuracies were measured with the MaxSMT forward pass. Epoch times are averaged over all epochs on which the model
+was trained. Cells marked — denote that the problem is not compatible with the approach.
+to perform as well as SMTLayer, as it also encodes explicit
+knowledge. However, it is unable to learn a useful model
+for either visual algebra configuration, and does not learn
+the correct representation for MNIST addition until it is
+exposed to half of the possible digit pairs during training.
+SMTLayer does particularly well on the visual Sudoku
+dataset introduced by Wang et al.. When trained on just
+10% of the original sample, it learns a function that exceeds
+the performance of the SATNet model by a healthy margin,
+which continues to grow as it is exposed to more training
+data. On the other hand, we found that SATNet failed to
+converge with less than the full original training sample.
+Robustness
+&
+interpretability.
+The
+reason
+that
+SMTLayer is able to perform well, and often near the
+optimum, in configurations that other approaches perform
+poorly on, is that it learns a representation that is consistent
+with the symbolic knowledge encoded in the SMTLayer.
+For example, the constraints that we use for MNIST
+addition, visual algebra, and visual sudoku all encode digits
+as bitvectors. In order to make a correct inference, the
+neural network must learn to encode MNIST digits in their
+correct bitvector representation. If learning succeeds at
+this, then there are two positive outcomes that follow. First,
+the model’s representation will be inherently interpretable,
+because it will coincide with the provided symbolic domain
+knowledge, which is also (presumably) interpretable.
+Second, the resulting model is naturally robust to covariate
+shift that does not affect the distribution of perceptual data
+that the network translates into theory symbols, but that
+does affect the statistics of their composition.
+This type of shift is on display in the MNIST 10% and visual
+algebra experiments, where at training time, the model only
+sees pairs of same-numbered digits, and at test time it is
+exposed to a substantially different distribution of digit pairs
+or formulas. We verified this by examining the representa-
+tions learned by SMTLayer and Scallop on MNIST Addi-
+tion 10%; it is unreasonable to expect SATNet to learn an
+interpretable representation, as it is not provided with an in-
+terpretable theory during training. As expected, SMTLayer
+produces the correct representation at the rate of accuracy of
+a typical MNIST model (≈ 99%), whereas Scallop’s digit
+representation was correct roughly 50% of the time. How-
+ever, architecture plays a role in this robustness, as shown in
+the SMTLayer results for the second visual algebra configu-
+ration. Because the network is shown the full instance, and
+not the individual digits, it learns the training bias. Despite
+having access to the symbolic formulas in SMTLayer, it
+cannot disentangle the perceptual symbols from their covari-
+ance. Understanding this issue is an important direction for
+future work.
+6. Conclusion
+Our approach for integrating logical theories into deep learn-
+ing, SMTLayer, provides a pragmatic solution to the prob-
+lem of incorporating symbolic knowledge into learning for
+training and inference, which we demonstrate on several
+problems involving both perceptual tasks—vision and natu-
+ral language—and logical reasoning. Notably, we show that
+models which incorporate symbolic knowledge during train-
+ing and inference can outperform conventional models as
+well as prior work in this area, especially in settings where
+training data is limited. Continued progress on automated
+reasoning techniques has played a pivotal role in the devel-
+opment of several fields over the past decades, and our hope
+is that the contributions in this paper will aid in progress
+towards realizing their potential in challenges that surpass
+the capabilities of existing learning techniques.
+
+Learning Modulo Theories
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+A. Appendix
+A.1. Proofs
+Theorem 2. Let D, H be a T-decomposable problem with grounding function f and prediction logic φ where:
+1. Z and Y are Cartesian products of Booleans.
+2. For any (x, y) ∼ D and y′ ̸= y, φ(f(x), y′) is T-equivalent to false and there is exactly one z such that φ(z, y) is
+T-equivalent to true.
+3. Hf is a convex set and for all hf ∈ Hf, ∥hf∥ ≤ B, and the loss ℓ(hf(·), z) is M-Lipschitz and convex in x for any
+fixed z.
+Then for any ϵ > 0, selecting hf by minimizing either LS(Fφ
+smt(hf(·))) with τ ≥ M 2B2/ϵ2 iterations of stochastic gradient
+descent, with gradients provided by Bφ
+max, and learning rate η =
+�
+B2/M 2τ yields a grounding hypothesis ˆhf ∈ Hf that
+satisfies: E[LD( ˆhf)] ≤ minhf ∈Hf LD(hf) + ϵ. The randomness in this expectation is taken over the choices of the SGD
+algorithm.
+Proof. To prove this result, we introduce the notion of a grounding sample.
+Definition 4 (Grounding sample). Let D, H be a T-decomposable problem with grounding function f. The grounding
+sample Sf for S ∼ D is given by [(xi, f(xi)) : (xi, yi) ∈ S], i.e., tuples that consist of the first element of each instance in
+S and its image under f.
+Now observe that the conditions stated in assumption (3) are sufficient to yield the result if instead of optimizing
+LS(Fφ
+smt(hf(·))), we were given the grounding sample Sf and minimized LSf (hf) (see (Shalev-Shwartz & Ben-David,
+2014), Theorem 14.8). The result follows as stated then because of assumptions (1) and (2), which imply that the update
+vectors provided by Bφ
+max are the gradients of LSf (hf).
+To understand why, observe that the sign of ˆy computed on line 3 of both algorithms must be equal to that of y⋆. This
+follows from two facts:
+1. At any coordinate i where y[i] ̸= y⋆[i], sign(∂yℓ(y, y⋆))[i] = sign(y)[i].
+2. At any coordinate i where y[i] = y⋆[i], sign(∂yℓ(y, y⋆))[i] = −1 · sign(y)[i].
+Now there are two cases to consider.
+Case 1: sign(y) = sign(ˆy).
+In this case the algorithm returns ∂zℓ(z, sign(z)). Because solutions to φ(·, y) are unique,
+sign(z) is the correct grounding, i.e., z = f(x) for the original features x.
+Case 2: sign(y) ̸= sign(ˆy).
+Because of assumption (2), the set of indices computed on line 8 will contain all of the
+coordinates at which z matches the correct value z⋆ = f(x). Note that at these coordinates, the vector returned by the
+algorithm matches the gradient of ℓ(z, z⋆), which is ∂z[i]ℓ(z[i], sign(z[i])). In the remaining coordinates, the vector will
+contain ∂z[i]ℓ(z[i], 1 − sign(z[i])), which also matches the gradient of ℓ(z, z⋆). The result follows.
+A.2. Dataset details
+Two of the three problems that we examine are based on the MNIST handwritten digit dataset (Deng, 2012), which consists of
+60,000 28x28 gray-scale images of handwritten numerals for training and 10,000 instances for testing. The digits on the left
+of Figure 1 are examples of instances from this data. To generate data for the visual algebra problem, we additionally drew
+from EMNIST (Cohen et al., 2017b), which extends MNIST with handwritten letters, and HASY (Thoma, 2017a), which
+contains handwritten symbols with similar characteristics to MNIST. For the liar’s puzzle, inspired by examples (Yurichev,
+2020) which formulate similar examples as SMT constraints, we constructed examples using a set of common phrases that
+we devised ourselves, and did not otherwise draw from publicly-available data.
+
+Learning Modulo Theories
+Below, we describe the way in which we used these data sources to construct training and test samples, and the neural
+network architectures that we used with SMTLayer to solve them.
+MNIST Addition.
+The MNIST addition problem is described in Example 1. In each instance, two MNIST digits are
+presented as features, and the task of the model is to provide their sum represented as a bitvector. The architecture that
+we use consists of four convolutional layers with kernels of size 3, depths in the order 64, 64, 128, 128, and a stride of
+width 2 on the first layer, and two dense layers of width 256 and 4. This network is applied to each digit, and the results are
+concatenated to obtain a vector of size 8 that is passed to an instance of SMTLayer with SMT constraints from Example 1,
+which ultimately produces a vector of width 5 that represents the bitvector sum of the digits.
+We generated five training samples starting with one containing only pairs of the same digit, i.e. (0, 0), (1, 1), . . .. We then
+added progressively more from the full set of possible pairs, using 25%, 50%, and 100%, and trained on batches of 128
+across all datasets. Note that although we change the number of digit pairs that appear between samples, we always map
+these pairs to random MNIST images to obtain 60,000 training instances. This is to ensure that the training sample contains
+a sufficient sample of MNIST images to be able to perform well on the test data. In all cases, we use the same test set
+consisting of instances from all possible pairs of digits. The purpose of this is to demonstrate that the conventional network
+will not generalize until it has seen the full distribution, whereas the model with SMTLayer should be able to generalize
+after seeing many fewer examples.
+Visual Algebra.
+The visual algebra problem is described in Example 2 and Example 3. Recall that features depict
+handwritten depictions of linear equations of the form ax + b = c. Values for a, x and b are randomly drawn from the
+range 1-9 to ensure that solutions are unique. Then the corresponding value of c is decomposed into c = 10 · c1 + c2,
+and MNIST digits are selected at random to represent a, b, c1, c2. A random EMNIST alphabet character is drawn for the
+variable, and random multiplication, addition, and equality symbols are drawn from HASY. A minor note is that HASY does
+not contain the standard equality symbol “=”, so we instead use “ .=”. These images are then concatenated horizontally in
+the appropriate order.
+We evaluate two architectures for this problem. The first uses the same neural network that was used for MNIST addition,
+and an instance of SMTLayer with the constraints given in Example 2. It assumes that the four numeric digits in the problem
+have already been extracted, e.g. by a separate vision routine that can recognize digits from letters and arithmetic symbols,
+and are provided directly to the model. The four inputs are given separately to the neural network, which produces four
+4-bit bitvectors that are concatenated and passed to SMTLayer, which produces a 4-bit bitvector result. We refer to this as
+configuration #1 in our results.
+The second uses an architecture which takes the entire image containing the problem as a whole, and produces a 16-bit
+bitvector that is passed directly to SMTLayer. This architecture uses a similar stack of convolutional layers, but has a larger
+initial dense layer containing 26,112 neurons, as it is given a larger image. The difference in the convolutional stack is at the
+second layer, which also has a stride of width 2, to reduce the size of the feature map and mitigate the need for an even larger
+dense connection. The difference between these architectures relates to one of the challenges of this problem. Much of the
+information contained in the features is irrelevant to the solution, e.g., it is irrelevant which letter is chosen for the variable,
+or what the arithmetic operators look like, so this architecture must also learn to disregard these parts of the instance. We
+refer to this as configuration #2 in our results and in Example 3.
+We generated a training sample by selecting a and b uniformly from pairs of the same digit, i.e. (0, 0), (1, 1), . . ., and
+sampling x uniformly from the odd numbers between 0 and 9. The test sample was generated by sampling a, b uniformly
+from all pairs of digits, and x from all numbers 0 to 9. We then map these values to random MNIST, EMNIST, and HASY
+images to obtain 60,000 samples. The intention is to study a problem wherein the model is not shown all possible problems
+(modulo representation as digits), or all of the solutions. This is more challenging than MNIST addition for two reasons:
+for a given solution, there are many more compatible ground terms, and the model does not see examples of some of
+the solutions it must provide for the test set. Thus, in order for SMTLayer to succeed, it must use the provided symbolic
+knowledge to approximate the correct grounding function, despite these deficiencies in the data.
+Liar’s Puzzle.
+The liar’s puzzle is comprised of three sentences spoken by three distinct agents: Alice, Bob, and Charlie.
+One of the agents is “guilty” of an unspecified offense, and in each sentence, the corresponding agent either states that one
+of the other parties is either guilty or innocent. It is assumed that two of the agents are honest, and the guilty party is not.
+The solution to the problem is an identification of the guilty party. An example is described in Example 4
+
+Learning Modulo Theories
+Features
+Symbolic Domain Z
+0010000111
+Grounding
+Function f
+φ(
+z1∥ . . . ∥z10,
+y
+) ≡
+a = 1z1>0∥ . . . ∥1z5>0 ∧ b = 1z5>0∥ . . . ∥1z10>0 ∧
+c = 1z5>0∥ . . . ∥1z10>0
+∧
+ax + b = c
+Prediction Logic φ
+Labels Y
+{00101}
+Satisfying
+Assignments
+Figure 2. Visual Algebra configuration 1 example.
+We synthesized a dataset for the liar’s puzzle based on a limited set of utterances about who speaks in each sentence,
+which agent is the subject, and whether the subject is guilty or innocent. There are five ways of denoting the speaker: “*
+says”, “* says that”, “* said”, “* said that”, and a colon “* :” separating the speaker’s name from the rest of the sentence.
+There are five ways of uttering either innocence or guilt: “* did it/did not do it”, “* is guilty/innocent”, “* is definitely
+guilty/innocent”, and “* definitely did it/did not do it”, “* is the criminal/is a good person”. We generated all of the
+combinations of subject, speaker, and proclaimed innocence or guilt, and took the product with all possible combinations of
+these utterances. The result is a dataset of 375,000 instances, each containing three natural language sentences.
+The
+prediction
+logic
+for
+this
+problem
+assumes
+a
+set
+of
+ground
+predicates
+speaker(agent, sentence),
+subject(agent, sentence), accuses(sentence), and guilty(agent).
+For example, if the first sentence was “Alice
+says that Bob is innocent”, the ground predicates would be speaker(alice, 1), subject(bob, 1), and ¬guilty(bob). Then the
+prediction logic is shown in Equation 1.
+|{a : guilty(a)}| = 1
+∧
+∀a.|{s : speaker(a, s)}| = 1
+∧
+∀s.|{a : subject(a, s)}| = 1
+∧
+∀s, a1, a2.speaker(a1, s) ∧ subject(a2, s) →
+(¬guilty(a1) ↔ accuses(s) ↔ guilty(a2))
+(1)
+In our implementation, agents and sentence identifiers are encoded unary as 3-bit bitvectors. The quantifiers are removed
+by substituting for each ground term or sentence identifier, and the cardinality constraints are expanded into propositional
+formulas. The architecture that we adopt is a two-layer bidirectional long short-term model (LSTM) with 512 dimensions at
+each layer, and a 300-dimension trainable embedding layer initialized from GloVe-6B (Pennington et al., 2014). The hidden
+units of the last LSTM layer were connected to 2-layer dense network containing 128 followed by 7 neurons. This network
+is applied to each sentence in the input, and the concatenated results are passed to the SMTLayer, which solves the formula
+in Equation 1 to produce a unary encoding of the guilty party.
+To evaluate solver layers on this problem, we selected training and test samples by first subsampling half of the full 375,000
+available instances. We then selected half of the speaker, subject, accuses predicate configurations for all three sentences
+appearing in this subsample to appear in the training sample, and the other half to appear in the test sample. To further limit
+the amount of information in the training sample, we randomly selected one ordering for each predicate configuration to
+remain for training. There were 9,400 resulting training instances, and 28,062 test instances. Restricting the training set as
+described ensures that the model is trained on a limited subset of possible sentence configurations, and one that is logically
+disjoint from those that appear in the test sample. Because there is not enough information in the training sample to learn
+Equation 1, we expect only the model with SMTLayer to succeed, but to do so it must approximate the grounding function
+well from a limited sample.
+A.3. Hyperparameters
+The four problems that our evaluation studies vary considerably in size and complexity, and the models used to train them
+require different considerations. This section details these differences. Table 2 relates the optimizers and epochs for each
+
+4X+Learning Modulo Theories
+Features
+Symbolic Domain Z
+0100010000100100
+Grounding
+Function f
+φ(
+z1∥ . . . ∥z10,
+y
+) ≡
+a = 1z1>0∥ . . . ∥1z5>0 ∧ b = 1z5>0∥ . . . ∥1z10>0 ∧
+ax + b = c
+Prediction Logic φ
+Labels Y
+{00101}
+Satisfying
+Assignments
+Figure 3. Visual Algebra configuration 2 example.
+Input
+Alice said that Bob
+did not do it.
+Bob: Alice is
+definitely innocent
+Charlie: Alice did it
+Symbolic Domain Z
+100010001010100100110
+Grounding
+Function f
+|{a : guilty(a)}| = 1
+∧
+∀a.|{s : speaker(a, s)}| = 1
+∧
+∀s.|{a : subject(a, s)}| = 1
+∧
+∀s, a1, a2.speaker(a1, s) ∧ subject(a2, s) →
+(¬guilty(a1) ↔ accuses(s) ↔ guilty(a2))
+Prediction Logic φ
+Labels (Liar) Y
+{2}
+Satisfying
+Assignments
+Figure 4. Liar’s Puzzle example.
+dataset and solver layer configuration. For SATNet and Scallop, we use the optimization settings described in their respective
+papers, and found in their public implementations. For SATNet models, the MaxSAT clause parameters were trained at a
+rate of 2e-3, and the convnet at rate 1e-5. When SGD(1.0) is stated, we used a warmup period spanning the first epoch, and
+cosine annealing for the remainder of training.
+MNIST Addition.
+The MNIST addition problem is the easiest of the problems that we study, at least in its full (100%)
+configuration. We find that for the 100% configuration, all of the solver layers converge to a nearly optimal solution with
+three epochs of supervised pre-training, and five epochs of subsequent training with the solver layer attached. For the
+subsample configurations, all of the solver layers converge to a stable, although in many cases suboptimal, solution within
+these parameters as well. For SMTLayer, we clipped gradients for all parameters at 0.1, and did not clip gradients for the
+other solver layer models. For all configurations, we used batches of size 128.
+Visual Algebra.
+Although visual algebra is a more difficult learning problem than MNIST addition, as evidenced by the
+results in Table 1, we find that the same parameters allow all of the configurations studied in our evaluation to converge. After
+3/5 epochs of training, the models stabilize, and in some cases, further training yields an overfit model. For SMTLayer, we
+clipped gradients for all parameters at 0.1, and did not clip gradients for the other solver layer models. For all configurations,
+we used batches of size 64.
+Liar’s Puzzle.
+The Liar’s Puzzle is the only problem to use a recurrent model, and we found that it required more epochs
+of pre-training to reduce the variance of the final model with the solver layer. Additionally, the SGD optimizer used
+with SMTLayer on other datasets caused the model to converge at local minima. We found that pre-training at a higher
+learning rate, and using Adam with a default learning rate, let to the best results. Additionally, we did not clip gradients
+for the SMTLayer model. We used the same parameters, but the normal SATNet optimizer, for the SATNet model. For all
+configurations, we used batches of size 32.
+
+4x+4=ayLearning Modulo Theories
+Conventional
+w/ SMTLayer
+w/ SATNet
+w/ Scallop
+optimizer
+epochs
+optimizer
+epochs
+optimizer
+epochs
+optimizer
+epochs
+MNIST+ 10%
+SGD(1.0)
+0/5
+SGD(1.0)
+3/5
+Adam(2e-3, 1e-5)
+3/5
+Adam(1e-3)
+3/5
+MNIST+ 25%
+SGD(1.0)
+0/5
+SGD(1.0)
+3/5
+Adam(2e-3, 1e-5)
+3/5
+Adam(1e-3)
+3/5
+MNIST+ 50%
+SGD(1.0)
+0/5
+SGD(1.0)
+3/5
+Adam(2e-3, 1e-5)
+3/5
+Adam(1e-3)
+3/5
+MNIST+ 75%
+SGD(1.0)
+0/5
+SGD(1.0)
+3/5
+Adam(2e-3, 1e-5)
+3/5
+Adam(1e-3)
+3/5
+MNIST+ 100%
+SGD(1.0)
+0/5
+SGD(1.0)
+3/5
+Adam(2e-3, 1e-5)
+3/5
+Adam(1e-3)
+3/5
+Vis. Alg. #1
+SGD(1.0)
+0/5
+SGD(1.0)
+3/5
+Adam(2e-3, 1e-5)
+3/5
+Adam(1e-3)
+3/5
+Vis. Alg. #2
+SGD(1.0)
+0/5
+SGD(1.0)
+3/5
+Adam(2e-3, 1e-5)
+3/5
+Adam(1e-3)
+3/5
+Liar’s Puzzle
+Adam(2e-3)
+0/15
+Adam(1e-3)
+15/5
+Adam(2e-3, 1e-5)
+15/5
+—
+—
+Vis. Sudoku 10%
+SGD(1.0)
+0/100
+SGD(1.0)
+30/15
+Adam(2e-3, 1e-5)
+30/100
+—
+—
+Vis. Sudoku 50%
+SGD(1.0)
+0/100
+SGD(1.0)
+30/5
+Adam(2e-3, 1e-5)
+30/100
+—
+—
+Vis. Sudoku 100%
+SGD(1.0)
+0/100
+SGD(1.0)
+30/5
+Adam(2e-3, 1e-5)
+0/100
+—
+—
+Table 2. Training hyperparameters. Numbers in parentheses after the optimizer denote the learning rate; when there are multiple numbers,
+different learning rates were applied to different parameter groups, as detailed in the text. Each epoch pair corresponds to the pre-training
+and training phases, i.e., 3/5 denotes three epochs of pre-training and five subsequent training epochs with the solver layer.
+Visual Sudoku.
+Visual Sudoku is the most challenging problem that we studied, for all solver layers as well as the
+conventional model. We did not use supervised pre-training, as the supervision in this problem leaks the correct labels
+directly to the model, bypassing the solver layer and the need for its updates (Chang et al., 2020). We instead used the
+unsupervised pre-training method described in (Topan et al., 2021), and found that 30 epochs of unsupervised pre-training
+was sufficient to yield consistent and quick convergence with the solver layer. For the most data-scarce configuration (10%),
+15 epochs of training with the solver layer were needed to converge, and for the others five epochs were sufficient. For the
+SMTLayer model, we used batches of size 1 after pretraining (batch size 64 during pre-training), primarily due to the fact
+that SMTLayer returns only the indices masked as non-hint elements on the Sudoku board. Because each instance has a
+different number of hint elements, this would lead to ragged tensors during training, which Pytorch does not support.
+When assessing SATNet on Visual Sudoku, we were unable to converge to a useful model on any except the full (100%)
+configuration, as discussed in Section 5.2. Using the authors’ public implementation and training script, we attempted the
+10% and 50% configurations with and without unsupervised pre-training, with and without the measures taken in (Chang
+et al., 2020) to prevent label leakage, and with batches of size 40 (as used in the original paper) as well as 1, to no avail. For
+the 100% configuration, we were able to reproduce a useful model; we use the accuracy reported in the original paper in
+Table 1 for consistency, as the average that we obtained did not differ significantly from this.
+

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+What is a Mathematical Structure
+of Conscious Experience?
+Johannes Kleiner1,2,3 and Tim Ludwig4
+1Munich Center for Mathematical Philosophy, LMU Munich
+2Munich Graduate School of Systemic Neurosciences, LMU Munich
+3Association for Mathematical Consciousness Science
+4Institute for Theoretical Physics, Utrecht University
+Princetonplein 5, 3584 CC Utrecht, The Netherlands
+Abstract. In consciousness science, several promising approaches have been
+developed for how to represent conscious experience in terms of mathematical
+spaces and structures. What is missing, however, is an explicit definition of
+what a ‘mathematical structure of conscious experience’ is. Here, we propose
+such a definition. This definition provides a link between the abstract formal
+entities of mathematics and the concreta of conscious experience; it comple-
+ments recent approaches that study quality spaces, qualia spaces or phenome-
+nal spaces; it provides a general method to identify and investigate structures
+of conscious experience; and it may serve as a framework to unify the various
+approaches from different fields. We hope that ultimately this work provides
+a basis for developing a common formal language to study consciousness.
+Attempts to represent conscious experiences mathematically go back at least to 1860 [15],
+and a large number of approaches have been developed since. They span psycho-physics [35,
+36, 39, 51, 73], philosophy [9, 10, 12, 17, 18, 42, 43, 55, 56, 57], phenomenology [50, 72], neu-
+roscience [64, 74], theories of consciousness [21, 22, 46, 49] and mathematical consciousness
+science [19, 32, 52, 63, 66, 67], and are known under various different names, including qual-
+ity spaces [9], qualia spaces [63], experience spaces [33, 34], Q-spaces [8], Q-structure [45],
+Φ-structures [65], perceptual spaces [74], phenomenal spaces [16], spaces of subjective experi-
+ence [64], and spaces of states of conscious experiences [31].1 The mathematical structures and
+spaces, which these approaches introduced, have enabled important results in their respective
+fields. Yet, they remain largely fragmented. What is missing, from our perspective, is a defi-
+nition of what the term ‘mathematical structure of conscious experience’ refers to in the first
+place.
+1There is a vast literature of articles that either make use of these constructions or which offer important
+insights that concern these structures; for examples, see [1, 2, 6, 7, 13, 14, 20, 23, 28, 30, 29, 38, 40, 41, 43, 47,
+54, 60, 61, 62, 67, 68, 69].
+1
+arXiv:2301.11812v1  [q-bio.NC]  26 Jan 2023
+
+2
+In this article, we propose a definition of mathematical structures of conscious experience.
+Our main desideratum is that for a mathematical structure to be of conscious experience,
+there must be something in conscious experience that corresponds to that structure. We call
+this “something” a structural aspect of conscious experience.
+Our key idea is to use variations to identify and investigate structural aspects of conscious
+experience. That is because the concept of variation can serve as a binding link between con-
+scious experiences and mathematical structures: on the one hand, variations relate to conscious
+experiences, because they change some of their aspects (like qualia, qualities, or phenomenal
+properties); on the other hand, variations relate to mathematical structures, because they may
+or may not preserve them.
+In defining what it means for a mathematical structure to be of conscious experience, our
+proposal does not answer the question of what this mathematical structure is.
+Rather, it
+provides an analysandum for future work on spaces and structures of conscious experience.
+Furthermore, by providing a general way to identify and investigate such structures, our pro-
+posal provides a framework to unify the various approaches from different fields.
+Before presenting the details of our proposal, we discuss how recent approaches relate math-
+ematical structures to conscious experience in Section 1; we identify three key problems. In
+Section 2, we present our proposal together with the necessary background information. In
+Sections 3, 4, and 5, we consider three important examples; namely relative similarity, metric
+spaces, and topological spaces. In Section 6, we show how our proposal resolves the three
+problems identified in Section 1.
+1. The Status Quo
+So where do things stand? Most of the early work that has attributed mathematical struc-
+ture to conscious experience was grounded in intuition. That is, whether or not a specific
+mathematical structure pertains to consciousness was not assessed systematically; instead, it
+was assessed based on an intuitive insight of appropriateness. More recent approaches have
+realized the need for a more systematic method, for example [18, 42, 44, 50, 52, 56, 57]. In
+this section, we analyze what we take to be the condition that underlies these approaches:
+a condition on what justifies prescribing a mathematical structure to conscious experience.
+As we will see, this condition is quite natural. But, as we will show, when understood as a
+sufficient condition, it is problematic.
+In a nutshell, a mathematical structure consists of two building blocks; for a detailed intro-
+duction, see Section 2.2. The first building block brings in one or more sets called the domains
+of the structure. The second building block are relations or functions which are defined on the
+domains. For reasons explained below, we will denote them as structures in the narrow sense
+of the term. A metric space, for example, is a mathematical structure that is defined on the
+two domains: a set of points and the real numbers. Furthermore, it comprises a function—the
+so-called metric function—which maps two points to a real number. A topological space, to
+give another example, is a mathematical structure that is defined on a single domain: a set
+
+3
+of points. Furthermore, it comprises a collection of unary relations, which are subsets of the
+domain.2
+Usually, a mathematical structure also comes with axioms. The axioms establish conditions
+that the functions or relations have to satisfy. In the case of a metric structure, the axioms
+require the metric function to satisfy three conditions, called positive definiteness, symmetry,
+and triangle inequality. In the case of a topological structure, the axioms ensure the collection
+includes the empty set and the whole domain, that it is closed under finite intersections, and
+that it is closed under arbitrary unions.
+When put in these terms, recent proposals that go beyond intuitive assessments, make use,
+either directly or indirectly, of the following condition to justify that a specific mathematical
+structure pertains to consciousness.
+A mathematical structure describes conscious experience (MDC) if and only if
+the following two conditions are satisfied:
+(D1) The domains of the structure are sets whose elements correspond to aspects of
+conscious experiences.
+(D2) The axioms of the structure are satisfied.
+Here, we use the term aspect as a placeholder for qualia, qualities, (instantiated) phenomenal
+properties or similar concepts.
+In the case of the metric structure introduced in [9], for example, (D1) is satisfied because
+the set of points corresponds to qualities of conscious experience. The real numbers might
+have a phenomenal interpretation as describing degrees of similarity, as for example in [42]; for
+details see Section 4. Condition (D2) requires positive definiteness, symmetry, and the triangle
+inequality to hold. This includes, for example, the condition that “points should have distance
+zero just in case the qualities represented by those points are phenomenally identical” [42,
+p. 14]. In the case of the topological structure introduced in [63], to give another example,
+(D1) is satisfied because the domain of the structure refers to qualia. Condition (D2) would
+require, then, that the chosen collection of subsets satisfies the axioms of a topological space.
+Prima facie, (MDC) could be taken to define what a mathematical structure of conscious
+experience is. However, if understood as sufficient condition, three problems arise. This implies,
+in particular, that (MDC) cannot be used to justify that a mathematical structure pertains to
+consciousness.
+Problem 1: Incompatible Structures. A first reason why (MDC) cannot be a sufficient
+condition to asses whether a mathematical structure pertains to consciousness is that it would
+allow for incompatible structures to pertain to consciousness.
+Consider, as an example, the case of topology. A basic question in topology is whether a
+target domain is discrete or not. A target domain is discrete if and only if its topology contains
+all subsets of the domain [26]. Otherwise, the target domain is not discrete. These two cases
+are exclusive, meaning that discrete and non-discrete topological structures are incompatible.
+2A unary relation on a domain, in the mathematical sense, is a subset of the domain; see Section 5.
+
+4
+According to (MDC), conscious experience has a discrete structure. That is because any set
+whatsoever can be equipped with the discrete topology. Therefore, picking a set X of aspects
+(qualia, qualities, phenomenal properties, etc.) and choosing its discrete topology provides a
+mathematical structure that satisfies both conditions (D1) and (D2). But, according to (MDC),
+consciousness also has a non-discrete structure. That is because any set can also be equipped
+with a non-discrete topology. We can, for example, take an arbitrary decomposition of the set
+X into two subsets A and A⊥, where A⊥ is the complement of A, and consider the topology
+{∅, A, A⊥, X}.
+This choice satisfies all axioms of a topology, and therefore satisfies (D2).
+Furthermore, it is built on the same set X as the discrete topology above, which implies that
+it also satisfies (D1). Therefore, the discrete and the non-discrete topological structures are
+both structures of conscious experience, according to (MDC).
+This example shows that, if understood as sufficient condition, (MDC) implies that two
+incompatible structures pertain to conscious experience and that they do so with respect to the
+exact same domain of aspects. The condition fails to determine which of the two incompatible
+structures is the right one, or to remain silent on the issue.
+Problem 2: Arbitrary Re-Definitions. A second reason why (MDC) cannot be a suf-
+ficient condition is that it allows for arbitrary re-definitions: if one structure is given that
+satisfies (MDC), then any arbitrary definition of a new structure in terms of the given struc-
+ture also satisfies (MDC), so long as the domains of the structure remain unchanged. If the
+former pertains to consciousness, so does the latter.
+A simple and well-behaved example of this is given by rescaling a metric function. Let us
+suppose that (M, d) is a metric structure which pertains to consciousness according to (MDC),
+where M is a set of aspects and d is the metric function, which provides a real number
+d(a, b) for every two aspects a and b. Since (M, d) satisfies (MDC), so does every structure
+(M, C · d), where C · d is the multiplication of the function d by a positive real number C.
+Here, the number C can be chosen arbitrarily. If one metric structure pertains to consciousness
+according (MDC), so does an uncountably infinite number of metric structures.
+What is more, when re-defining structures, one is free to change the axioms as one pleases.
+For example, we could pick any function f that maps M to the positive real numbers and
+define a new distance function by (f(a)+f(b))·d(a, b). This is not a metric structure anymore,
+because the triangle inequality axiom does not hold. But it still satisfies positive definiteness
+and symmetry, and therefore satisfies (MDC), with a new set of axioms. One could even break
+asymmetry to get a distance function like the one applied by IIT [34]. More severe cases appear
+with more complicated structures.
+This is a problem, not only because of the unlimited number of structures that appear, but
+also because there is an arbitrariness in the definition of a new structure, specifically concerning
+the axioms. It seems strange that the axioms can be redefined at will, so as to always satisfy
+Condition (D2). Something is missing that restricts this arbitrariness in (MDC).
+Problem 3: Indifference to Consciousness. The third reason, which speaks against the
+sufficiency of (MDC), is that the proposed condition seems somewhat indifferent to details of
+conscious experience.
+
+5
+To illustrate this indifference, let us consider again the discrete and non-discrete topological
+structures from above. As we have shown, these structures pertain to conscious experience
+according to (MDC). Yet, nothing more than a few lines needed to be said to establish this
+fact. In particular, we did not need to use any noteworthy input related to consciousness other
+than picking some set of aspects; and it didn’t matter which aspects we picked.
+It is a red flag if so short an analysis, which does not depend on consciousness in a meaningful
+way, establishes facts about the mathematical structure of conscious experience. This is another
+hint that condition (MDC) is missing some important piece, if used as sufficient condition.
+The Way Forward. To resolve the three problems, our task is to find the missing piece and
+to propose a definition for a mathematical structure of conscious experience that makes sense
+as a necessary and sufficient condition. Two desiderata guide our search. First, as is the case
+with (MDC), the definition should be about conscious experience in the sense that it describes
+aspects of conscious experience. Second, there should be something in conscious experience
+that relates to a mathematical structure if that structure is a mathematical structure of con-
+scious experience. This “something” should make sure that the definition is not indifferent
+to conscious experience (Problem 3) and that it relates to the mathematical structure in a
+meaningful way, so as to stop arbitrary re-definitions (Problem 2). The proposal which we
+present in the next section is the result of our search.
+Looking back at Condition (MDC) after our analysis, we think that (MDC) is best under-
+stood as an expression of what it takes for a mathematical structure to describe conscious
+experience. Because of the problems with sufficiency, a structure that satisfies this condition
+might not pertain to consciousness; but it might still be a valuable descriptive tool that is dis-
+tinguished from other structures by its relation to aspects of conscious experience. This is why,
+retrospectively, we have chosen the term ‘describes conscious experiences’ when specifying the
+condition. Our first desideratum will lead us to develop a new condition that contains (MDC)
+as necessary part; this is aligned with the intuition that any mathematical structure of con-
+scious experience also describes conscious experience.
+2. Mathematical Structures of Conscious Experience
+For a mathematical structure to be of conscious experience, rather than just a descriptive
+tool for conscious experience, there should be something in conscious experience that relates
+to that structure. Denoting a mathematical structure by S, we call this structural aspect of
+consciousness an S-aspect.
+To make sense of what an S-aspect is, we need to understand how aspects (like qualia, qual-
+ities or phenomenal properties) relate to mathematical structures. While aspects may have
+an arity (meaning they may be instantiated relative to other aspects), they are not experi-
+enced as having a mathematical structure per se.3 Therefore, relating aspects to mathematical
+structures requires a tool that applies both; concreta and abstract formal entities. Variations
+provide such a tool.
+3With the exception of experiences of mathematical structures themselves, of course.
+
+6
+In general, a variation is a change of something into something else; in our case, it is a
+change of one experience into another experience. Such variations may be induced by external
+stimuli or interventions, come about naturally, or be subjected to imagination (‘imaginary
+variations’ [24]). Variations are intimately related to aspects of conscious experiences because
+they may or may not change them. And they are intimately related to mathematical structures,
+because they may or may not preserve them. An S-aspect, then, is an aspect that is changed
+by a variation if and only if the variation does not preserve the structure S. To explain this in
+detail is the purpose of the remainder of this section.
+2.1. Terminology and Notation. Here, we introduce the key terms we use to define math-
+ematical structures of conscious experience. These terms are conscious experiences, aspects
+of conscious experiences, and variations of conscious experiences. The introduction proceeds
+axiomatically, so that our construction does not rely on a specific choice of these concepts.
+Rather, any choice that is compatible with what we say here can be the basis of an application
+of our definition.
+Our construction is based on a set E of conscious experiences of an experiencing subject. We
+denote individual conscious experiences in that set by symbols like e and e′; formally e, e′ ∈ E.
+From a theoretical or philosophical perspective, one may think of the set E as comprising all
+conscious experiences which one experiencing subject can have, i.e. all nomologically possible
+experiences of that subject. From an experimental or phenomenological perspective, one may
+think of this set as comprising all conscious experiences that can be induced in the lab or
+in introspection. Different such choices may lead to different mathematical structures being
+accessible.
+We use the term aspect as a placeholder for concepts such as qualia [70], qualities [55],
+or (instantiated) phenomenal properties.4 For every experience e ∈ E, we denote the set of
+aspects instantiated in this experience by A(e). The set of all aspects of the experiences in E,
+denoted by A, is the union of all A(e); formally A = �
+e∈E A(e). Individual aspects, that is
+members of A, will be denoted by small letters such as a, b, c. When explaining examples, we
+will often use the abbreviation ‘a is the experience of ...’ as a shorthand for saying ‘a is a ...
+aspect of an experience’. For example, ‘a is the experience of red color’ means ‘a is a red color
+aspect of an experience’.
+Some aspects may require other aspects for their instantiation. For example, it is usually the
+case that an experience of relative similarity is an experience of relative similarity of something,
+for example two color aspects relative to a third color aspect. If an aspect a requires other
+aspects for its instantiation, we will say that the aspect a is instantiated relative to aspects
+b1, ..., bm, or simply that a is relative to b1, ..., bm. Aspects which are instantiated relative to
+other aspects are the building blocks for the structure of conscious experience.
+4Many other concepts work as well.
+For example, if one works with an atomistic conception of states of
+consciousness, where the total phenomenal state of a subject—what it is like to be that subject at a particular
+time—is built up from individual atomic states of consciousness, one can take e to denote the total phenomenal
+state and aspects to be the states of consciousness in that total state. Another example would be to take
+aspects to denote phenomenal distinctions as used in Integrated Information Theory [65]. What matters for
+our definition to be applicable is only that according to one’s chosen concept of conscious experience, every
+conscious experience exhibits a set of aspects.
+
+7
+A variation of a conscious experience e changes e into another experience e′.
+Because
+experiences have structure, there may be various different ways to go from e to e′.5 Therefore,
+in addition to specifying e and e′, a variation is a partial mapping
+v : A(e) → A(e′) .
+This mapping describes how aspects are replaced or reshuffled by the variation. A mapping
+which is not surjective, meaning that it does not map to all aspects in A(e′), makes room for
+appearance of new aspects. A mapping which is partial, meaning that it does not specify a
+target for every aspect in A(e), makes room for aspects to disappear.
+2.2. What is a Mathematical Structure? To find a rigorous definition of the mathematical
+structure of conscious experience, we need to work with a rigorous definition of mathematical
+structure. But, what is a mathematical structure? Fortunately, mathematical logic provides
+us with an answer to just that question.
+A mathematical structure S consists of two things: domains, on the one hand, and func-
+tions or relations, on the other hand. We now introduce these concepts based on two simple
+examples.
+The domains of a structure S are the sets on which the structure is built. We denote them
+by Ai, where i is some index in a parameter range I. In the case of a metric structure, for
+example, the domains would be A1 = M and A2 = R, where M is a set of points and R
+denotes the real numbers, understood as a set. In the case of a strict partial order, there is
+just one domain A, which contains the elements that are to be ordered.
+The second ingredient are functions and/or relations. Functions f map some of the domains
+to other domains. In the case of a metric structure, the function would be a metric function
+d : M × M → R, which maps from A1 × A1 to A2. A relation R, in the mathematical sense,
+is a subset of the m-fold product Ai × ... × Ai. Here, Ai is the domain on which the relation
+is defined, and m is the arity of the relation, which expresses how many relata the relation
+relates. The product is usually just written as Am
+i . In the case of a strict partial order, the
+relation is binary, which means that R is a subset of A2. For binary relations, one usually uses
+notation like a < b instead of writing (a, b) ∈ R; still, it is important to keep in mind that
+relations are subsets in that sense.
+In almost all cases, mathematical structures also come with axioms, which establish condi-
+tions that the functions or relations have to satisfy. They are useful because they constrain
+and classify the structure at hand. For S to be a metric structure, for example, the function d
+has to satisfy the axioms of positive definiteness, symmetry, and triangle inequality [59]. For S
+to be a strict partial order, the relation R has to be irrefelxive, asymmetric, and transitive [27].
+5To illustrate this point, consider, for example, the following two mappings v and v′ which map the numbers 1,
+2, and 3 to the numbers 2, 4, and 6. The mapping v is the multiplication of every number by 2, meaning that
+we have v(1) = 2, v(2) = 4, v(3) = 6. The mapping v′, on the other hand, is defined by v(1) = 6, v(2) = 2,
+v(3) = 4. If we only cared about the sets of elements that these mappings connect, the mappings would be
+equivalent: there is no difference between the set {2, 4, 6}, which is the image of v, and {6, 2, 4}, which is the
+image of v′. If, however, we care about the ordering of the elements of the sets, which we usually do in the
+case of numbers, then there is a difference. While 2 ≤ 4 ≤ 6, it is not the case that 6 ≤ 2 ≤ 4. Because we care
+about the order of the elements, we need to say which element goes where.
+
+8
+To have a nice and compact notation, we will use one symbol Sj to denote both functions
+relations. That is because, in any concrete proposal, it is always clear whether Sj is a function
+or a relation.6 The index j takes values in some parameter range J that specifies how many
+functions or relations there are. In summary, the desired rigorous definition is:
+A mathematical structure S is a tuple
+S =
+�
+(Ai)i∈I, (Sj)j∈J
+�
+of domains Ai and functions or relations Sj.
+For given domains Ai, the mathematical structure S is fully determined by the Sj. Thus,
+we can also refer to Sj as ‘structures’, if the domains are clear from context. For simplicity,
+we can drop the index j and simply write S whenever we consider just one such structure.
+As a final step in this section, we introduce the relata of a structure S.
+This will be
+helpful to write things concisely below. The term relata designates those elements that are
+related by a structure. In the case where S is a relation R on a domain A and has arity m,
+these are the elements of the m-tuples (b1, ..., bm) ∈ R. In the case where S is a function
+f : A1 × ... × Am−1 → Am, the relata are the elements of the m-tuples (b1, ..., bm−1, bm) where
+bm = f(b1, ..., bm−1), and where the other bi range over their whole domains. For notational
+simplicity, we write b1, ..., bm instead of (b1, ..., bm) when designating relata below.
+2.3. What is a Mathematical Structure of Conscious Experience? Finally, to the
+heart of the matter! We recall that we have so far identified two desiderata for a mathematical
+structure S to be a mathematical structure of conscious experience. First, it should be about
+conscious experiences in the sense that it describes aspects of conscious experiences. Second,
+there should be aspects in conscious experience that relate to the structure S. The following
+definition satisfies these two desiderata.
+A mathematical structure S is a mathematical structure of conscious experi-
+ence (MSC) if and only if the following two conditions hold:
+(S1) The domains Ai of S are subsets of A.
+(S2) For every Sj, there is a Sj-aspect in A.
+Here, A denotes the set of all aspects of the experiences in E; formally A = �
+e∈E A(e),
+the Ai denote the domains of the structure S, and the Sj-aspects are defined below.
+Condition (S1) guarantees that the first desideratum is satisfied. Condition (S2) guarantees
+that the second desideratum is satisfied. Furthermore, whenever a certain type of structure
+(metric, topological, partial order, manifold, etc.) is claimed to be a structure of conscious
+experience, the axioms that constrain and classify that type have to hold. Therefore, any
+mathematical structure of conscious experience (MSC) is also a mathematical structure that
+describes conscious experience according to (MDC). The requirement that has been applied in
+previous proposals remains a necessary condition.
+6In mathematical logic, mathematical structures are denoted as triples of domains, relations, and functions.
+However, in our case, using just one symbol for functions and relations improves readability substantially.
+
+9
+The remaining task of this section, then, is to explain what an Sj-aspect is. For notational
+simplicity, we use the symbol S to denote Sj. As we have emphasized before, variations are key
+to understand the structure of conscious experience, because they link aspects and structure.
+Therefore, to be able to precisely define what an S-aspect is, we need to understand how
+variations relate to aspects, on the one hand, and structures, on the other hand. Our strategy
+is to first discuss how variations relate to aspects. This amounts to specifying what precisely
+it means for a variation to change an aspect. Second, we focus on how variations relate to
+mathematical structure. This amounts to explaining what it means for a variation to preserve
+a structure. Finally, combing these two steps allows us to understand S-aspects and provide a
+useful definition.
+What does it mean for a variation v : A(e) → A(e′) to change aspects? The underlying idea
+is simply that an aspect is present in the source of the variation, A(e), but not present any
+more in the target of the variation, A(e′). We need to take into account, though, that aspects
+are often instantiated relative to other aspects (see Section 2.1). This can be done as follows.
+A variation v : A(e) → A(e′) changes an aspect a ∈ A(e) relative to b1, ..., bm ∈ A(e) if
+and only if a is instantiated relative to b1, ..., bm in A(e), but a is not instantiated relative
+to v(b1), ..., v(bm) in A(e′).
+In the case where a ∈ A(e) is not instantiated relative to other aspects, the definition indeed
+reduces to the simple condition that a ∈ A(e) but a ̸∈ A(e′). The negation of the definition is
+also as intuitively expected: the aspect is present both in the source and in the target.7
+For applications it is important to understand that this definition can fail to apply in
+two ways. First, it can fail because there is no a in A(e′) which is instantiated relative to
+v(b1), ..., v(bm). This, in turn, can be the case either because there is not a in A(e′) at all, or
+because there is an a in A(e′) but it is instantiated relative to other aspects. Second, it can fail
+because one or more of the v(b1), ..., v(bm) do not exist. The second case is possible because v
+is a partial mapping, which means aspects can disappear.
+What does it mean for a variation to preserve a mathematical structure? The underlying
+idea is that a variation preserves the structure if and only if the structure is satisfied before the
+variation and remains to be satisfied after the variation. By its very nature, this is a mathemat-
+ical condition, namely the condition of being a homomorphism, as specified by mathematical
+logic [48]. The definition of a homomorphism, though, always applies to all elements of a
+domain at once. For our case, it is best to refine this definition to a single set of relata.8
+A variation v : A(e) → A(e′) preserves a structure S with respect to relata b1, ..., bm ∈
+A(e) if and only if we have
+7Because the definiendum already includes the first part of the condition, the negation is as follows:
+A variation v : A(e) → A(e′) does not change an aspect a ∈ A(e) relative to b1, ..., bm ∈ A(e) if and only if a
+is instantiated relative to b1, ..., bm in A(e) and a is also instantiated relative to v(b1), ..., v(bm) in A(e′).
+We felt that is the best way of writing things to optimize clarity.
+8For notational simplicity,
+we write R
+�
+b1, ..., bm
+�
+=
+R
+�
+v(b1), ..., v(bm)
+�
+instead of R
+�
+b1, ..., bm
+�
+⇔
+R
+�
+v(b1), ..., v(bm)
+�
+.
+
+10
+(P1) R
+�
+b1, ..., bm
+�
+= R
+�
+v(b1), ..., v(bm)
+�
+if S is a relation R, or
+(P2) v
+�
+f(b1, ..., bm−1)
+�
+= f
+�
+v(b1), ..., v(bm−1)
+�
+if S is a function f.
+As in the previous case, the negation of this definition is exactly what is intuitively expected:
+a variation does not preserve the structure if and only if the structure is satisfied before the
+variation, but not satisfied after the variation.9
+For applications it is again important to see that the definition can fail for two reasons. First,
+it could be the case that one or more of the v(bi) do not exist in A(e′), if the corresponding
+aspect disappears. Second, the identities may fail to hold.
+We finally have the keys to understand S-aspects and provide a useful definition.
+The
+underlying idea is that an S-aspect is an aspect that, under any variation, behaves exactly as
+the structure S does. Whenever S is preserved, the S-aspect does not change. Whenever the
+S-aspect changes, the structure S is not preserved. That is, it needs to satisfy the following
+definition.
+An aspect a ∈ A is a S-aspect if and only if the following condition holds:
+A variation does not preserve S with respect to relata b1, ..., bm if and only if the variation
+changes a relative to b1, ..., bm.
+Here, the condition needs to hold true for all variations and all relata. This means that it
+needs to hold true for all variations of all experiences e in the set E that instantiate relata of
+the structure S.
+This concludes our proposal for the definition of the mathematical structure of conscious
+experience. It is a structure whose domains correspond to sets of aspects, and which contains
+an S-aspect for every relation or function of the structure. In the next three sections, we apply
+this definition to three examples. On the one hand, these examples illustrate the definition. On
+the other hand, they provide new insights to structures that have been featured prominently
+in previous approaches.
+3. Relative Similarity
+Our first example concerns relative similarity, which plays an important role, for example,
+in the construction of quality spaces by Austen Clark [9, 10]. In this example, we use natural
+language to pick out experiences and aspects. As we will see, this works fine to a large extend,
+but at one point we will have to show a bit of good faith when it comes to the precision of
+natural language.
+9A variation v : A(e) → A(e′) does not preserve a structure S with respect to relata b1, ..., bm ∈ A(e) if and only
+if we have R
+�
+b1, ..., bm
+�
+̸= R
+�
+v(b1), ..., v(bm)
+�
+if S is a relation R, or v
+�
+f(b1, ..., bm−1
+�
+̸= f
+�
+v(b1), ..., v(bm−1)
+�
+if S is a function f.
+This negation agrees with the intuition because the definiendum already states part of the condition that
+follows, namely that b1, ..., bm are relata of the structure S in A(e), which implies that (b1, ..., bm) ∈ R if S is a
+relation and that f(b1, ..., bm−1) exists in A(e) if S is a function, meaning that the structure is satisfied before
+the variation.
+
+11
+A first step in applying our definition is to choose a set E. Here we take E to comprise
+experiences of three color chips, as indicated in Figure 1A, where one of the chip (the reference)
+has a fixed color coating and the others vary in a range of color coatings Λ.10
+The second step is to specify the set of aspects A(e) for every experience e ∈ E. Here,
+we take A(e) to comprise:11 (a) the color qualities in e, that is, the experienced colors of the
+individual chips; (b) positional qualities of the color experiences, that is, which chip has which
+color; and (c) the experience of relative similarity. Relative similarity is an experience of one
+pair of aspects to be more, less, or equally similar to each other than another pair of aspects;
+here, the two pairs have to have one aspect—the reference—in common. In Figure 1A, for
+example, the color of the top left chip will, for many readers, be less similar to the reference
+chip than the color of the top right chip.
+To pick out relative similarity more precisely, we let b0, b1 and b2 denote the color aspects of
+the three chips in an experience e, where b0 is the color aspect of the reference; see Figure 1B.
+For some experience e, it might be the case that the colors b1 and b0 are experienced as less
+similar to each other than the colors b2 and b0. In this case, the experience e has a relative
+similarity aspect in the above sense; we denote this “less-similar” relative similarity aspect
+by a. So, a is an aspect of e, and it is instantiated relative to b1 and b2.12
+Variations change one experience e into another experience e′. An example for a variation
+would be a swap of the coatings of the two non-reference chips, as in Figure 1C. Another
+example for a variation would be to change the coatings of both non-reference chips to some
+other coating in Λ, as in Figure 1D. Formally, variations are represented by mappings v :
+A(e) → A(e′). In the first example, Figure 1C, the mapping is of the form v(b1) = b2 and
+v(b2) = b1, and v(c) = c for all other aspects c, except for the relative similarity aspect a,
+which is discussed in detail below. In the second example, Figure 1D, the mapping is as in the
+first example but with v(b1) = b3 and v(b2) = b4.
+The key question of this example is: Is there a mathematical structure of conscious experi-
+ence which corresponds to relative similarity? To answer this question, we propose a structure
+and check whether (MSC) applies.
+The words “less similar than” in the description of relative similarity already indicate that
+some order, in the mathematical sense of the word, might be involved. For reasons that will
+become clear below, we propose a strict partial order as mathematical structure. A strict
+partial order (C, <), consists of a set C, which is the domain of the structure, and a binary
+relation ‘<’ on C. For all x, y, z ∈ C, this binary relation has to satisfy the following axioms:
+▶ Irreflexivity, meaning that there is no x ∈ C with x < x.
+▶ Asymmetry, meaning that if x < y, then it is not the case that y < x.
+▶ Transitivity, meaning that if x < y and y < z, then also x < z.
+10The “color coating” here denotes the physical stimuli. Alternative choices would be to speak of wavelength
+mixtures, presentation, etc.
+11The experience e may contain many other aspects. However, in A(e) we only include those which are relevant
+for our investigation.
+12To be precise, a is also relative to b0. But since b0 does not vary in E we can leave this implicit.
+
+12
+Figure 1. To help explain the example of relative similarity, this figure il-
+lustrates experiences with color qualities and variations thereof. Subfigure A
+illustrates an experience of three color chips as well as the concept of relative
+similarity: many readers will experience the color of the top-left color chip to
+be less similar to the reference chip than the color of the top-right color chip.
+Subfigure B illustrates our notation for the color aspects corresponding to the
+color chips. Subfigures C and D illustrate variations v of experiences: a swap
+of two color aspects in C; and a replacement of two color aspects in D.
+In order to propose a strict partial order structure of conscious experiences, we need to
+specify how the set C and the relation < relate to (aspects of) conscious experience. For the
+set C we choose the color qualities of the experiences in E, meaning that C now comprises the
+color qualities evoked by the coatings Λ of the chips we consider. For example, it contains
+what we have labelled b0, b1, b2, b3 and b4 in Figure 1. For the relation, we define bi < bj if
+and only if bi is experienced as less similar to b0 than bj is to b0.13
+For this proposal to make sense, we first need to check whether the axioms are satisfied.
+If they were not satisfied, the proposal could still be a structure of conscious experience; but
+it wouldn’t be a strict partial order.
+That’s why the axioms are not explicitly mentioned
+in (MSC). Irreflexivity is satisfied because no color quality is less similar to the reference than
+itself. Asymmetry is satisfied because if a bi is less similar to the reference than bj, then bj is
+not less similar to the reference than bi.
+Transitivity is a more interesting case. The use of terms like ‘less similar to’ in natural
+language suggests that transitivity is also satisfied; it suggests that, if bi is less similar to the
+reference than bj and bj is less similar to the reference than bk, then bi should be less similar
+to the reference than bk. But it might very well be the case that natural language is not
+precise enough to describe its target domain. The use of natural language may be justified
+in simple cases, or even in a majority of cases, but whether or not transitivity holds for all
+bi, bj, bk ∈ C is, ultimately, an empirical question. Already in such a simple example one can see
+how the mathematical-structure approach might be used to identify subtle details of conscious
+experiences for which natural language alone is not precise enough. Still, for the purpose of this
+13Since relative similarity, as defined above, depends on the choice of reference b0, it would be more precise to
+write <b0 instead of <. However, to simplify the notation, we keep the reference implicit.
+
+bo
+60
+62
+b
+bo
+6013
+example, we’re going to show the above-mentioned bit of good faith and assume transitivity
+to hold as well.
+Having checked that the axioms hold—that is, that the proposal is indeed a strict partial
+order—we can proceed to check whether the structure is a mathematical structure of conscious
+experience according to (MSC). Concerning Condition (S1), there is one domain C and it
+consists of color qualities, so this condition is satisfied. Therefore, only Condition (S2) remains
+to be checked.
+We claim that the relative similarity aspect a, as defined above, is in fact a <-aspect. To
+see that this is true, we have to show that a variation does not preserve < with respect to
+relata b1 and b2 if and only if the variation changes a relative to b1 and b2.
+Consider any variation v : A(e) → A(e′) that does not preserve < with respect to relata
+b1, b2 ∈ A(e). Two aspects b1 and b2 are relata of < if either b1 < b2 or b2 < b1. We focus
+on the first case as the other one follows from the first by renaming b2 and b1 in what follows.
+By definition of the < relation, b1 < b2 means that b1 is experienced as less similar to the
+reference than b2. Therefore, there is also a relative similarity aspect a ∈ A(e) as defined
+above. As explained in Section 2.3, there can be two ways in which the variation v might not
+preserve <. Either v(b1) or v(b2) are not defined, or, if they are defined, it is not the case that
+v(b1) < v(b2). In the former case, there cannot be an a in A(e′) relative to v(b1) or v(b2),
+simply because the latter do not both exist. In the latter case, it follows from the definition
+of < that v(b1) is not experienced as less similar to the reference than v(b2). So, there is no
+a ∈ A(e′) relative to v(b1) and v(b2). Hence, we may conclude that v changes a relative to b1
+and b2.
+For the opposite case, let v : A(e) → A(e′) be a variation which preserves < with respect
+to relata b1 and b2. As before, this implies that a is in A(e) relative to b1 and b2. Because v
+preserves <, v(b1) and v(b2) both exist and we also have v(b1) < v(b2). Applying the definition
+of < then implies that a is also in A(e′) relative to v(b1) and v(b2). Hence v does not change a
+relative to b1 and b2.
+Because in both of these cases, v was arbitrary, it follows that a is indeed a <-aspect.
+Therefore, Conditions (S1) and (S2) of (MSC) are both satisfied, and the strict partial order
+(C, <) is indeed a mathematical structure of conscious experience; it is the mathematical
+structure of relative similarity of color experiences with respect to b0.
+4. Metric Structure
+Our next example is a metric structure. Metric structures feature prominently in various
+different approaches, for example [9, 10, 42, 55, 56, 57]. In some cases, such as [9], their intro-
+duction is motivated by mathematical convenience. In others, such as [56], their introduction
+is closely tied to laboratory features, such as the topology of physical stimuli. Therefore, it is
+not so clear whether the prominent role metric structures play is due to an intimate relation
+to consciousness or just due to them being a very handy mathematical tool. A principled
+investigation, we take it, is highly imperative.
+
+14
+A metric structure (M, d) consists of a set M of ‘points’ together with a function d :
+M × M → R. Therefore, the domains of the structure are M and R. For the function to be a
+metric function, three axioms have to be satisfied for all points x, y, z ∈ M:
+▶ Positive definiteness, which requires that d(x, y) ≥ 0 and that d(x, y) = 0 if and only
+if x = y.
+▶ Symmetry, which requires that d(x, y) = d(y, x).
+▶ The triangle inequality, which requires that d(x, y) ≤ d(x, z) + d(z, y).
+We will turn to previous work on metric structure momentarily, but before doing so, we
+can ask whether this type of structure could possibly be a structure of conscious experience,
+according to either (MDC) or (MSC)?
+To answer this question, we look at one of the two domains of the metric structure, namely
+the real numbers R. What is not well-known outside of mathematics is that the real numbers
+are not simply given, as the natural numbers or rationals might be, but that they have to be
+constructed in a comparably involved procedure. There are a small number of such procedures
+which yield equivalent results; the most common procedure is to construct the real numbers
+as equivalence classes of Cauchy sequences of rational numbers.14
+If real numbers are equivalence classes of sequences of rational numbers, it is hard to see
+how one might reasonably claim that these correspond to aspects of conscious experiences, as
+required by both (MDC) and (MSC). Independently of whether aspects are taken to denote
+qualities, qualia or phenomenal properties, there do not seem to be aspects that are equivalence
+classes of sequences of rational numbers. Phenomenal similarity, relative similarity, or similar
+aspects which have been associated to metric structures in previous work (see for example [9,
+42, 55]), do not have anything to do, in our eyes, with Cauchy sequences of rational numbers
+or any of the other real number construction schemes, such as Dedekind cuts.
+Therefore, when understood in the precise sense of the term, a metric structure can neither
+be a structure of conscious experience (MSC), nor a structure to describe conscious experience
+in the sense of (MDC); it can only be an auxiliary tool.
+The way to move forward, if one wants to consider something like a metric structure as
+a structure of conscious experience, is to restrict the target domain of the metric function d
+to something that could reasonably be “in” conscious experience; that is, something closer to
+degrees of phenomenal similarity or a similar concept, for example, natural or rational numbers.
+We will now turn to this option. But it is important to note that most theorems about metric
+spaces cease to hold true if the real numbers are replaced by something else. That’s because
+the convergence properties of sequences of real numbers (called completeness [58]) are crucial
+for the definition of metric spaces.
+The received wisdom on how to link metric structures to conscious experience is summarized
+concisely by Lee when discussing the “standard framework” for modeling mental qualities [42]:
+“There are three main desiderata when constructing a model in the standard
+framework. First, there should be one-to-one correspondence between points in
+14A Cauchy sequence is, roughly speaking, a sequence of rational numbers that get arbitrarily close to each
+other. Two Cauchy sequences are defined to be equivalent if the difference between their elements tends to
+zero.
+
+15
+the model and qualities in the targeted domain. Second, points that are more
+distant in the model should represent qualities that are less phenomenally
+similar to each other. Third, points should have distance zero just in case the
+qualities represented by those points are phenomenally identical.” [42, p. 14]
+The first desideratum states that the metric structure contains a set of points M that corre-
+sponds to the targeted domain; for example, qualities of a certain type. The third desideratum
+alludes to one of the three axioms of the metric structure, namely positive definiteness; it
+is implicitly understood that the other axioms should also hold. But what does the second
+desideratum mean? And how does it relate to a metric function that takes values in some
+number range?
+To interpret the second desideratum, we have to understand how phenomenal similarity
+may relate to numbers; this will lead us to “degrees” of phenomenal similarity. To have a
+foundation on which to build this understanding, we will assume that phenomenal similarity
+can be described by a strict partial order (M, <) as introduced in Section 3. This assumption
+is in line with Clark’s proposal in [9] and [10], where a metric structure is based on relative
+similarity.
+Given a strict partial order (M, <) that describes phenomenal similarity (meaning that
+x < y if and only if x is less similar to some reference than y), a metric-like function d on M
+can be given by defining
+d(x, y) = length<(x, y) ,
+where length<(x, y) denotes the number of elements of the shortest path that connects x and
+y.15 This definition of d(x, y) means that the metric function counts degrees of phenomenal
+similarity in-between x and y, so that “points that are more distant in the model [...] represent
+qualities that are less phenomenally similar to each other” (ibid.), as required by the second
+desideratum.
+As the reader can check, this definition of d(x, y) indeed satisfies the three axioms of a
+metric function.
+Furthermore, it takes values in the natural numbers N, so that the real-
+number problem is avoided. So, with d : M × M → N defined as above, is (M, d) a structure
+of conscious experience?
+By assumption, M consists of aspects of conscious experiences: qualities of the targeted
+domain. Furthermore, there is no obvious reason why a bounded subset of the natural numbers
+should not be aspects of conscious experiences of a suitable set E. Therefore, Condition (S1)
+might well be satisfied.
+So, whether or not (M, d) is a structure of conscious experience
+(MSC), as compared to just a structure to describe conscious experiences (MDC), boils down
+to whether there is a d-aspect for d as defined above. Let us check if there is one.
+15Formally, it is defined as
+length<(x, y) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+if x = y
+min |{x → y}|
+if x < y
+min |{y → x}|
+if y < x
+∞
+otherwise ,
+where x → y denotes a path along the strict partial order from x to y, |{x → y}| denotes the number of
+elements in the path (with x counted but y not counted), and where the minimum selects the shortest path.
+
+16
+An aspect a is a d-aspect if and only if every variation v which does not preserve d with
+respect to relata b1, ..., bm changes a relative to b1, ..., bm, and vice versa. The definition of
+relata in case of the function d means that m = 3 and b3 = d(b1, b2). Therefore, a variation
+v doesn’t preserve d with respect to b1, b2, b3 if either the v(bi) do not all exist or if v(b3) ̸=
+d(v(b1), v(b2)), which means that the variation changes the experience of the number b3 =
+d(b1, b2) in such a way that it does not agree any more with the experienced distance of v(b1)
+and v(b2). A variation v preserves d with respect to relata b1, b2, b3, on the other hand, if
+v(b3) = d(v(b1), v(b2)). Therefore, what we’re looking for is an aspect a of some experience e
+in which the identity b3 = d(b1, b2) holds, that changes relative to b1, b2, b3 if the above identity
+breaks, and that does not change relative to b1, b2, b3 if the above identity holds true. This
+must be true of all relata b1, b2, b3 of d, that is, of all experiences which exhibit aspects b1, b2
+and b3 such that b3 = d(b1, b2).
+Taking into account that b3 is the experience of a number, the only aspect which can satisfy
+these two conditions is the experience of b1 and b2 having distance b3, or put differently: the
+aspect in question would have to be an experience of ‘having distance’, which is instantiated
+relative to b1, b2 and the number b3. We do not think there is such an aspect. Aspects might
+be experienced as more or less similar, but we doubt that they are experienced as being a
+specific number apart, be that number natural or rational. Therefore, even if watered down
+to avoid the issues with real numbers, a metric structure does not seem to be a structure of
+conscious experience.
+So, in summary, there is one immediate and one deep reason why conscious experience does
+not have a metric structure. The immediate reason is that the real numbers with their very
+involved construction scheme do not seem to have anything to do with aspects of conscious
+experience. The deep reason is that we simply do not experience qualities or other aspects as
+being a particular number of degrees of similarity apart. This might explain why in approaches
+such as that of Clark [9, 10], the details of the introduction of a metric structure point outside
+the realm of conscious experience.
+5. Phenomenal Unity and Topological Structure
+Our final example concerns topological structure. Interestingly, this is intimately tied to
+phenomenal unity, the thesis that phenomenal states of a subject at a given time are unified [5].
+Recall that we have introduced the set A(e) to denote aspects of the conscious experience e,
+where we have used the term ‘aspect’ as a placeholder for concepts like qualia, qualities, or
+(instantiated) phenomenal properties.
+Most examples of these concepts are “independent”
+from the experience in which they occur; they could be experienced together with a largely
+different set of aspects in a different experience. Yet, experiences seem unified; their aspects are
+experienced as tied together in some essential way. This raises the question of what underlies
+this experience of the unity of a conscious experience? As we will see, somewhat surprisingly,
+the answer is: a topological structure of conscious experience.
+Much has been written about the question of phenomenal unity in the literature, for exam-
+ple [4, 5, 11, 47, 53, 71], and in order to make use of some of the results, we assume that the
+
+17
+term ‘aspect’ denotes an instantiated phenomenal property or quale. The set of aspects A(e),
+then, comprises the phenomenal properties or qualia which are instantiated in the experience e,
+also called the phenomenal states of the experience e.16 Our question, then, is what it means
+that “any set of phenomenal states of a subject at a time is phenomenally unified” [5, p. 12].
+There are various answers one might give to this question.
+A promising answer is the
+so-called subsumptive unity thesis, developed in [5]:
+“For any set of phenomenal states of a subject at a time, the subject has a
+phenomenal state that subsumes each of the states in that set.” [5, p. 20]
+According to this thesis, what underlies the experience of the unity of a conscious experience
+is that for any set X of phenomenal states in the conscious experience, there is a further
+phenomenal state that subsumes each of the states in X. This phenomenal state characterizes
+what it is like to be in all of the states of X at once [5, p. 20].
+Put in terms of aspects, the subsumptive unity thesis says that for any set X ⊂ A(e) of
+aspects of an experience, there is an additional aspect in A(e) that subsumes the aspects in X.
+This aspect is the experience of what it is like to experience the aspects in X as part of one
+experience e together, the experience that they are unified, as we will say. Let us call this
+aspect the phenomenal unity aspect of X and denote it by aX. It is instantiated relative to
+the elements of X.
+Phenomenal unity gives rise to a mathematical structure of conscious experience. To see
+how, let us use the symbol T to denote a collection of subsets of A(e), to be specified in
+more detail below. Every subset of A(e) is a unary relation on A(e),17 and hence also on the
+set A that comprises all aspects of the experiences in E. Therefore, (A, T ) is a mathematical
+structure; it has domain A and its structures are the unary relations in T . The next paragraph
+shows that because of the subsumptive unity thesis, the mathematical structure (A, T ) is a
+mathematical structure of conscious experience according to (MSC).
+Because A is the set of all aspects of E, Condition (S1) of (MSC) is satisfied. Therefore,
+only Condition (S2) remains to be checked. This condition is satisfied because for every set
+X ∈ T , the phenomenal unity aspect aX is an X-aspect. To show that this is the case, we need
+to check that a variation does not preserve X with respect to relata b1, ..., bm if and only if it
+changes aX relative to b1, ..., bm. Let v : A(e) → A(e′) be a variation that does not preserve X
+with respect to relata b1, ..., bm . The relata of the subset X are the elements of that subset.
+Therefore, we have b1, ..., bm ∈ A(e), so that the subsumptive unity thesis implies that there
+is a phenomenal unity aspect aX relative to the b1, ..., bm in A(e). The condition that v does
+not preserve X furthermore implies that either not all of the v(bi) exist or that at least one
+of them is not in the set X. Therefore, there is no phenomenal unity aspect aX relative to
+v(b1), ..., v(bm) in A(e′). Hence, the variation v changes aX relative to b1, ..., bm ∈ X. Vice
+versa, let v : A(e) → A(e′) be a variation which preserves X with respect to relata b1, ..., bm.
+16A phenomenal state is an instantiation of a phenomenal property, or quale, by a subject at a given time. This
+instantiation constitutes part of the experience of the subject at the time. An experience e, in our terminology,
+is an experience of a subject at a given time. Hence, a phenomenal state is an instantiation of a phenomenal
+property, or quale, in an experience e.
+17An m-ary relation on a set X is a subset R of Xm. Hence, a unary relation, where m = 1, is a subset of X.
+
+18
+This implies that aX is instantiated relative to b1, ..., bm in A(e). The condition that v preserves
+X furthermore implies that v(b1), ..., v(bm) exist, and that they are elements of X. Therefore,
+aX is also instantiated relative to v(b1), ..., v(bm) in A(e′). This shows that the variation does
+not change aX relative to b1, ..., bm. Thus, aX is indeed an X-aspect. And because that is true
+for any X ∈ T , (A, T ) indeed satisfies Condition (S2) and hence (MSC).
+The previous paragraph proves that, if the subsumptive unity thesis holds true for all sets
+X in T , then (A, T ) is indeed a mathematical structure of conscious experience. As we will
+explain next, this structure is intimately tied to a topological structure.
+A topological structure (M, T ) consists of a set M and a collection T of subsets of M. The
+collection has to satisfy three axioms, and there are a few different ways of formulating these
+axioms. Here, we choose the formulation that corresponds to what is usually called ‘closed
+sets’. The axioms are:
+▶ The empty set ∅ and the whole set M are both in T .
+▶ The intersection of any collection of sets of T is also in T .
+▶ The union of any finite number of sets of T is also in T .
+Having specified what a topological structure is, we return to the structure (A, T ), which is
+induced by phenomenal unity, and ask what the collection T of subsets is?
+First, it is important to note that the subsumptive unity thesis does not provide a phenom-
+enal unity aspect aX for every subset of A; it can only provide such an aspect for a set of
+aspects that are actually experienced together, that is, for a subset X of A(e). Therefore, T
+is not the discrete topology introduced in Section 1. Second, it also cannot be the case that it
+provides a phenomenal unity aspect for every subset of A(e). That’s because then there would
+be an infinite regress: for every subset X of A(e) there would be a new aspect aX in A(e),
+giving a new subset X ∪ {aX} that would give a new phenomenal unity aspect aX∪{aX}, and
+so forth. This problem is well-known in the literature [3, 71]. Rather, we take it, the quantifier
+‘any set’ in the subsumptive unity thesis must be understood as ‘any set of aspects that are
+experienced as being unified’. While it is arguably the case that the whole set of aspects A(e)
+of an experience is always experienced as unified, introspection suggests that we consciously
+experience only a select group of aspects as unified at a time.18
+So, which sets of aspects do we experience as unified? While it might be difficult to give a
+general answer to this question, there is a special case where a sufficiently detailed specification
+can be given: the case of regions in visual experience. Here, ‘regions’ are sets of positions of
+the space that visually perceived objects occupy.19 The positions in a region are experienced
+as unified. Therefore, the regions of visual experience are members of the collection T which
+is induced by phenomenal unity. Furthermore, they appear to satisfy the axioms of a topology
+as stated above: the whole set of positions in a visual experience is a region; it seems to be the
+case that intersections of regions in visual experience are also regions in visual experience; and
+18This solves the infinite regress problem because, arguably, we do not always experience the phenomenal unity
+aspects as unified with the sets they correspond to. So, there is not always a phenomenal unity aspect aX∪{aX}
+for the set that consists of aX and X.
+19It is also plausible to think that visual experiences do not contain positions as aspects, but only regions.
+However, assessing whether or not this is the case goes beyond the scope of this paper. Here, we assume that
+positions are aspects of visual experiences.
+
+19
+it seems to be the case that the union of any two regions in visual experience is also a region
+in visual experience. For the empty set, no S-aspect of consciousness is required (there are
+no relata of the corresponding unary relation), so we can take the empty set to be a member
+of T . Thus, all axioms of a topology are satisfied.
+Therefore, if we take M to denote the position aspects of visual experiences, and choose T to
+comprise the regions of visual experience, then (M, T ) is indeed a topological structure. And,
+as shown above, it is a structure of conscious experience as defined in (MSC). We thus find that,
+because of the subsumptive unity thesis, this topological structure is indeed a mathematical
+structure of conscious experience; much like conjectured in [64], it is a topology of the visual
+content of subjective experience.
+6. The Three Problems Revisited
+In this section, we discuss how the new approach (MSC), which we have developed in
+Section 2.2, resolves the three problems discovered in Section 1.
+Problem 1: Incompatible Structures. The first problem was that the condition (MDC),
+which has been applied in previous approaches, admits incompatible structures to conscious
+experience. Is this also true of (MSC)?
+If two structures are incompatible, then there exists at least one automorphism of one struc-
+ture that is not an automorphism of the other structure.20 As we explain below, this condition
+implies that two incompatible structures cannot have an S-aspect in common. Therefore, it
+is not possible for two incompatible structures to pertain to conscious experience in the exact
+same way; so, (MSC) indeed resolves the problem of incompatible structures.
+Let S and S′ denote two incompatible structures with the same domains. Then, there is at
+least one automorphism of one structure that is not an automorphism of the other structure.
+Let us denote such an automorphism by v and assume that it is an automorphism of S but
+not of S′. Because v is not an automorphism of S′, it follows that there is at least one set of
+relata b1, ..., bm of S′ in some A(e), such that the variation v : A(e) → A(e) induced by the
+automorphism does not preserve S′ with respect to these relata. On the other hand, because
+v is an automorphism of S, it follows that this variation preserves S with respect to b1, ..., bm.
+If an aspect a is an S′-aspect, then, applying the definition of S′-aspects, we find that the
+variation v needs to change it. In contrast, if an aspect a is an S-aspect, then, applying the
+definition of S-aspects, we find that the variation v needs to not change it; either because the
+b1, ..., bm do not constitute relata of S, or because the variation v preserves S with respect
+to relata b1, ..., bm. So, because an aspect cannot be both changed and not changed under a
+single variation, there cannot be an aspect a that is both an S-aspect and an S′-aspect.
+20Automorphisms are structure-preserving mappings from a structure to itself. Put in terms of the terminology
+we have introduced in Section 2.2, automorphisms are mappings v that map the domains of a structure to
+themselves. These mappings have to be bijective, and they have to preserve the structure, meaning that they
+have to satisfy (P1) for all elements of the domain in case of relations, and (P2) for elements of the domains in
+the case of functions.
+
+20
+Problem 2: Arbitrary Re-Definitions. The definition (MSC) also resolves the problem of
+arbitrary re-definitions. That’s because any re-definition changes the relata of the respective
+structure, and therefore generates an own, independent condition for something to be an S-
+aspect of the redefined structure. Whether or not this new S-aspect is a part of conscious
+experience is a substantive question that depends on the actual experiences of the subject
+under consideration; it is not automatically the case.
+Consider, as examples, the cases of rescaling a metric, which we have introduced in Sec-
+tion 1.
+If, per assumption, (M, d) were a structure of conscious experience, then for any
+relata (b1, b2, d(b1, b2)), the condition for d-aspects would have to be satisfied.
+Rescaling
+this to (M, C · d) generates a new condition because now, the relata to be considered are
+(b1, b2, C · d(b1, b2)).
+These are different relata, and correspondingly, different experiences
+and different variations will enter the definition of a C · d-aspect. The same is true for an
+(f(a) + f(b)) · d(a, b)-aspect. Whether or not these structures satisfy (MSC) depends on the
+details of the conscious experiences under consideration; but they do not automatically sat-
+isfy (MSC) just because (M, d) does.
+Problem 3: Indifference to Consciousness. The third problem is resolved, finally, because
+of the introduction of S-aspects, which are a counterpart “in” conscious experience to the
+proposed mathematical structure. Because S-aspects are part of the definition (MSC), any
+application of (MSC) requires one to engage with details of the conscious experiences of the
+subject under consideration; (MSC) is not indifferent to conscious experience.
+Consider, for example, the two topological structures of Section 1.
+While (MDC) only
+required us to check whether the structures are about aspects and satisfy the axioms, (MSC)
+also requires us to check whether there is an S-aspect in conscious experience that corresponds
+to the structures. As we have seen in Section 5, this involves a careful investigation of conscious
+experience, for example, concerning the phenomenal unity.
+7. Conclusion
+In this article, we investigated mathematical structures of conscious experience. Our main
+result is a definition of what mathematical structures of conscious experience are. This defi-
+nition provides a general method to identify and study structures of conscious experience; it
+is grounded in a foundational understanding of mathematical structures as laid out by math-
+ematical logic; and it provides a link between the abstract formal entities of mathematics,
+on the one hand, and the concreta of conscious experience, on the other hand, see Section 2.
+Our definition also resolves three problems that interfere with recent approaches that relate
+mathematical structures to conscious experience, see Section 1.
+What we consider noteworthy about our definition is that it is conceptually neutral, meaning
+that it does not rely on any specific conception of conscious experience or aspects. Rather, it
+is applicable to any conception of ‘conscious experience’ and ‘aspects’ in which every conscious
+experience comes with a set of aspects. This includes common conceptions built on qualities,
+qualia, or phenomenal properties, but also less common ideas built on atomistic conceptions of
+
+21
+states of consciousness or phenomenal distinctions. Furthermore, our definition is methodolog-
+ically neutral, meaning that it can be combined with many methods, practices, and procedures
+that are used to investigate conscious experience, spanning empirical, analytical, and phe-
+nomenological research. That is because the definition rests on the concept of variations, and
+variations can be induced in three major ways: introspectively (for example, as in Husserl’s
+imaginary variations [24]); in a laboratory by change of stimuli; or theoretically based on a
+proposed theory of consciousness.
+As first applications of our definition, we considered relative similarity, metric spaces, and
+topological spaces. We found that relative similarity, which plays an important role in several
+constructions of quality spaces, is indeed a mathematical structure of conscious experience, see
+Section 3. Topological spaces are too, but for a surprising reason: they are intimately related
+to phenomenal unity, see Section 5. Metric spaces, however, are not structures of conscious
+experience; see Section 4. One of the two reasons for that is that metric spaces are built on
+the real numbers, which have to be constructed with involved procedures using, for example,
+equivalence classes of Cauchy sequences that do not seem to correspond to aspects of conscious
+experience, however conceived.
+We view the result presented here as one further step in a long journey to investigate
+conscious experience mathematically. This step raises new questions and creates new oppor-
+tunities, both of which can only be explored in an interdisciplinary manner. A new question,
+for example, is whether our result on mathematical structures might open new perspectives
+on measurements of consciousness [25], as arguably promised by the Representational Theory
+of Measurement [37] whenever an axiomatic structure on a target domain is available. A new
+opportunity, in our eyes, is the unification of the various approaches to represent consciousness
+mathematically that have emerged in different fields. Because our result is conceptually and
+methodologically neutral, it can provide a framework for this unification; but to develop a
+unification that is both useful and valuable in practice requires experts from the respective
+fields. We hope that, ultimately, our result provides a basis for developing a common formal
+language to study consciousness.
+Acknowledgments. We would like to thank the participants of the 2022 Modelling Con-
+sciousness Workshop and of the Models of Consciousness 3 conference, both organized under
+the umbrella of the Association for Mathematical Consciousness Science, as well as members of
+the Munich Center for Mathematical Philosophy for fruitful discussions and helpful comments,
+and in particular Jonathan Mason for valuable feedback on the manuscript. This research was
+supported by grant number FQXi-RFP-CPW-2018 from the Foundational Questions Institute
+and Fetzer Franklin Fund, a donor advised fund of the Silicon Valley Community Foundation.
+We would like to thank the Dutch Research Council (NWO) for (partly) financing TL’s work
+on project number 182.069 of the research programme Fluid Spintronics, and the Mathematical
+Institute of the University of Oxford for hosting JK while working on this project.
+
+22
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+arXiv:2301.00343v1  [hep-th]  1 Jan 2023
+On the A∞-Category of a Holomorphic Moment Map
+Ahsan Z. Khan∗
+School of Natural Sciences
+Institute for Advanced Study
+Einstein Drive, Princeton NJ 08540
+January 3, 2023
+Abstract
+Let M be a hyperK¨ahler manifold equipped with a U(1) hyperK¨ahler isometry, and
+let I be a complex structure on M. In this note, we study the A∞-category of A-branes
+for the Landau-Ginzburg model with target space (M, I), and superpotential being the
+I-holomorphic moment map. We show that if I is a generic complex structure, the
+A∞-category is semi-simple. For exceptional complex structures, though typically not
+semi-simple, the category still has no instanton corrections.
+We illustrate the A∞-
+category at both generic and exceptional loci when M is the cotangent bundle of the
+projective line.
+The Morse-Smale-Witten (MSW) complex is a cochain complex associated to a real func-
+tion1 h on a Riemannian manifold (M, g). From a physics perspective, the MSW complex
+is the space of perturbative ground states of an N = 2 supersymmetric quantum mechan-
+ics (SQM) system [Wit82]. The differential on this complex is constructed from instanton
+effects.
+There is a special situation in which the MSW complex simplifies rather dramatically.
+This is when (M, g) admits a g-compatible complex structure I (so that (M, g, I) is a K¨ahler
+manifold), and h is the real part of an I-holomorphic function W on M. If this is the case,
+it is well-known that all critical points of h have the same Morse index, equal to the complex
+dimension of M. The MSW complex is thus simply a vector space concentrated in a single
+degree, with one basis vector for each critical point.
+This immediately implies that the
+differential must vanish. In terms of physics, there is also something special that happens
+∗[email protected]
+1Throughout this paper we assume that all critical points of h are isolated and non-degenerate
+1
+
+in this situation. The N = 2 supersymmetry of the SQM with target space (M, g) and
+superpotential h enhances to N = 4 precisely when M is K¨ahler and h = Re W. Systems
+with twice the supersymmetry often have simpler properties for their supersymmetric ground
+states.
+Remark
+Another case where the supersymmetry enhances to N = 4 is when h is the
+moment map of a continuous K¨ahler isometry of M. We will use a complex version of this
+statement later in the note.
+We conclude that the MSW complex of Re W is not a very interesting invariant of the
+pair
+�
+(M, g, I), W
+�
+. There is, however, a richer invariant of the same pair which, in a sense
+that can be made precise, categorifies the MSW complex of Re W. The categorification of
+the MSW complex of Re W is known (to mathematicians) as the Fukaya-Seidel A∞-category
+of (M, W). In terms of physics what’s responsible for the existence of this categorification
+is the fact that there is an uplift of the N = 4 SQM to a two-dimensional N = (2, 2)
+Landau-Ginzburg (LG) model such that the original SQM supercharge lifts to a topological
+supercharge2 of the LG model. The Fukaya-Seidel A∞-category is then simply the category
+of boundary conditions of the Landau-Ginzburg model that preserve this topological super-
+charge. These boundary conditions are known as A-branes. The MSW complex of Re W
+can be recovered upon taking the Hochschild homology of the Fukaya-Seidel category.
+To be more precise, there are a few different versions of the A∞-category associated to
+(M, W) (all conjectured to be A∞-equivalent). Usually the term “Fukaya-Seidel category”
+refers to a formulation by Seidel based on vanishing cycles and their intersections [Seid].
+We will be working mostly with a formulation developed by Gaiotto, Moore and Witten,
+which is based on particular kinds of solutions to the ζ-soliton and ζ-instanton equations3 4
+[GMW15].
+It is natural to wonder if there is a class of K¨ahler manifolds and superpotentials for
+which the associated A∞-category simplifies. A possible criteria for such pairs could be that
+the supersymmetry of the Landau-Ginzburg model of (M, W) enhances. There is indeed a
+special situation where this is the case. Suppose that the target space (M, g, I) admits an
+I-holomorphic two-form Ω satisfying
+Ωg−1Ω + g = 0,
+∇Ω = 0,
+(1)
+2A supercharge Q is called topological if all translations are in its image.
+3A similar (but not identical) formulation was also developed by Haydys [Hay10].
+4See also [KKS14] for a reformulation of the formalism of [GMW15] that generalizes to higher dimensions.
+2
+
+so that the space (M, g, I, Ω) is a hyperK¨ahler manifold5. Moreover, suppose (M, g, I, Ω)
+admits a hyperK¨ahler isometry generated by a vector field V , so that
+LV g = LV I = LV Ω = 0,
+(2)
+and W is the corresponding holomorphic moment map
+dW = ιV Ω.
+(3)
+If this is the case, the Landau-Ginzburg model for the pair
+�
+(M, g, I), W
+�
+has an additional
+symmetry6 coming from rotating the fermions of the theory by the endomorphism
+J = g−1Re
+�
+Ω
+�
+(4)
+of the tangent bundle. The commutator of this symmetry transformation with the N =
+(2, 2) supersymmetries generates four additional fermionic symmetries resulting in a total of
+eight supercharges. The supersymmetry of the Landau-Ginzburg model of (M, W) therefore
+enhances from N = (2, 2) to N = (4, 4).
+The above example in fact gives us an infinite family of pairs of K¨ahler manifolds and
+holomorphic functions for which the supersymmetry enhances.
+In addition to I we can
+define the integrable complex structures J = g−1Re(Ω) and K = g−1Im(Ω) that moreover
+satisfy the quaternion relations
+I2 = J2 = K2 = IJK = −1.
+(5)
+This brings us to the well-known fact that a hyperK¨ahler manifold M has a two-sphere’s
+worth of complex structures, since for any v = (v1, v2, v3) ∈ S2, the endomorphism
+I(v) = v1I + v2J + v3K
+(6)
+is an integrable complex structure on M. Moreover, given a hyperK¨ahler isometry generated
+by a vector field V , there is a holomorphic moment map
+W (v) : M → C
+5Recall that one definition of a hyperK¨ahler is as a K¨ahler manifold with a compatible parallel holomor-
+phic symplectic form Ω.
+6The axial R-symmetry FA, the symmetry coming from rotating by J, and their commutator, generate
+an so(3) R-symmetry algebra. Another way of seeing both the supersymmetry enhancement and this SO(3)
+R-symmetry is to note that the two-dimensional theory comes from reducing a six dimensional hyperK¨ahler
+sigma model, where the hyperK¨ahler isometry is gauged by a six-dimensional U(1) vector multiplet, to two
+dimensions. When doing the reduction, we set the gauge fields in the four internal directions to a non-zero
+vector in R4, while setting all other vector multiplet fields to vanish. The choice of a non-zero vector breaks
+the SO(4) R-symmetry coming from the internal directions to SO(3).
+3
+
+in every complex structure I(v). Explicitly, W (v) can be obtained as follows. Let µI be the
+moment map corresponding to the real symplectic form, ωI = gI and let
+W = µJ + iµK,
+Ω = ωJ + iωK,
+(7)
+so that the triple (µI, µJ, µK) satisfies
+dµI = ιV ωI,
+(8)
+dµJ = ιV ωJ,
+(9)
+dµK = ιV ωK.
+(10)
+We can obtain the complex structure I(v) from I by a hyperK¨ahler rotation: there is a
+quaternion
+q = q0 + q1i + q2j + q3k ∈ H
+(11)
+such that
+¯q i q = v1i + v2j + v3k.
+(12)
+The quaternion q is unique up to a redefinition by a phase q → eiθq. Organize the hy-
+perK¨ahler moment map in terms of the imaginary quaternions
+⃗µ = µIi + µJj + µKk.
+(13)
+With respect to the complex structure I, the decomposition of the hyperK¨ahler moment
+map into real and complex parts is given by
+⃗µ = h i + Wj,
+(14)
+so that h = µI is the real moment map and W = µJ +iµK is the complex moment map. The
+real and holomorphic moment maps with respect to I(v) are then given by doing a rotation
+of µ with respect to q:
+q ⃗µ q = h(v)i + W (v)j.
+(15)
+Redefinition of q by a phase,
+q → eiθq
+(16)
+leaves h(v) invariant whereas it transforms
+W (v) → e2iθW (v).
+(17)
+Since redefinition of W (v) by a phase does not change the associated Landau-Ginzburg model,
+we get an LG model ((M, g, I(v)), W (v)) for every point on the unit two-sphere. Because the
+supersymmetry enhances for all such models, it is natural to expect that the A∞-category
+of each such pair
+�
+(M, g, I(v)), W (v)�
+simplifies.
+4
+
+The purpose of this note is to show that this is indeed the case7.
+For the rest of the note we assume that the hyperK¨ahler moment map has isolated and
+non-degenerate critical points, and v ∈ S2 is such that the critical values of W (v) are in
+general position on the complex plane.
+The first basic simplification in the A∞-category of a moment map is the lack of instan-
+ton corrections.
+This can be seen as follows.
+Recall that in the Gaiotto-Moore-Witten
+formulation of the A∞-category of W, all instanton corrections come from solutions of the
+ζ-instanton equation. The ζ-instanton equation for a map φ : C → M is
+∂φi
+∂¯z = ζgi¯j ∂W
+∂ ¯φ¯j ,
+(18)
+where ζ = eiθ is a phase we are required to choose in order to define the category, (φi, φ
+¯i) are
+local complex coordinates on M and z is the standard complex coordinate on C. The par-
+ticular solutions that contribute to the A∞-structure are solutions with “fan-like” boundary
+conditions at infinity, that are rigid. Rigidity means that a solution has no moduli other
+than overall translations of the Euclidean spacetime complex plane. That there cannot be
+any such solution in the hyperK¨ahler case can be easily seen from the fact that the operator
+obtained from linearizing the ζ-instanton equation has a zero-mode
+δφi = V i
+(19)
+in addition to the translational zero-mode. There are therefore no non-trivial rigid instan-
+tons.
+Next we prove that at a generic point v on the two-sphere of complex structures, a stronger
+statement holds: the A∞-category of the pair
+�
+(M, g, I(v)), W (v)�
+is semi-simple. What we
+mean by this is the following. Recall that A∞-category of a superpotential W has a generating
+set of “thimble” objects {Ti} that are in one-to-one correspondence with critical points of
+W. The A∞-category is said to be semi-simple if the morphism spaces between the thimble
+objects satisfy
+Hom(Ti, Tj) = δijC.
+(20)
+This property follows from an elementary Lemma.
+Lemma
+Let (M, g, I, J, K) be a hyperK¨ahler manifold,
+⃗µ : M → R3
+7For previous work that is related to the setting of the present note see [SV18] and [Jin21]
+5
+
+be a hyperK¨ahler moment map, and ⃗n = (n1, n2, n3) ∈ S2 be a point on the unit sphere.
+Suppose φ : R → M is a (non-constant) solution to the gradient flow equation
+dφA
+dy = gAB ∂(⃗n · ⃗µ)
+∂φB
+(21)
+where
+⃗n · ⃗µ = n1µI + n2µJ + n3µK.
+(22)
+Then the composition ⃗µ ◦ φ : R → R3 is an embedding with image a straight line in the
+⃗n-direction.
+Proof. We show the claim for ⃗n = (0, 1, 0), so that we study the gradient flow equation for
+µJ. Since µJ is the real part of the I-holomorphic function
+WI = µJ + iµK,
+(23)
+the Cauchy-Riemann equation
+dµJ = ItdµK
+(24)
+implies that the the gradient vector field of µJ is equivalent to the Hamiltonian vector field
+of µK with respect to the symplectic form ωI
+g−1dµJ = ω−1
+I dµK.
+(25)
+Therefore µK is constant along a flow line. But µJ is also the real part of the K-holomorphic
+function
+WK = µJ − iµI
+and so the Cauchy-Riemann equation for WK implies that the gradient flow of µJ is also
+equivalent to the Hamiltonian flow for −µI with respect to the symplectic form ωK. Therefore
+µI is also constant. Since µJ ◦ φ is monotonic, the image of the ⃗µ ◦ φ is a line parallel to the
+J-axis. The case when ⃗n is arbitrary can be obtained by a hyperK¨ahler rotation.
+Consider now the Landau-Ginzburg model with target space (M, g, I(v)) and superpoten-
+tial given by the I(v)-holomorphic function W (v). A fundamental role in Landau-Ginzburg
+models is played by BPS solitons. Let φi and φj be critical points of W (v). Recall that an
+ij-BPS soliton is a solution of the gradient flow equation for Re(e−iθijW (v)) where
+eiθij = W (v)(φi) − W (v)(φj)
+|W (v)(φi) − W (v)(φj)|.
+(26)
+Let h(v) be the real moment map in the complex structure I(v), explicitly
+h(v) = v1µI + v2µJ + v3µK = ⃗v · ⃗µ.
+(27)
+6
+
+According to the Lemma, the real moment map h(v) is conserved along the gradient flow of
+Re
+�
+eiθW (v)�
+(as is Im
+�
+eiθW (v)�
+) for any value of θ. An ij soliton can therefore only exist if
+h(v)(φi) = h(v)(φj),
+(28)
+Letting ⃗µi ∈ R3 be the critical value of the hyperK¨ahler moment map in the ith vacuum,
+this condition is simply saying that
+⃗v · (⃗µi − ⃗µj) = 0,
+(29)
+so that ⃗v is perpendicular to the vector
+⃗Zij = ⃗µi − ⃗µj
+(30)
+pointing from the jth critical value to the ith critical value. If this does not happen for any
+pair of distinct critical points, the soliton spectrum of the Landau-Ginzburg theory is empty.
+In the notation of [GMW15], the soliton space Rij is trivial
+Rij = {0},
+(31)
+for each pair of distinct critical points (φi, φj). Since, in the Gaiotto-Moore-Witten formal-
+ism, morphism spaces between different thimble objects are given as direct sums of tensor
+products of soliton spaces, we obtain the following.
+Corollary
+Suppose ⃗v ∈ S2 is such that ⃗v · (⃗µi − ⃗µj) ̸= 0 for any pair of critical points
+(φi, φj). Then the A∞-category of
+�
+(M, g, I(v)), W (v)�
+is semi-simple:
+Hom(Ti, Ti)
+=
+C, for all i,
+(32)
+Hom(Ti, Tj)
+=
+0
+for i ̸= j.
+(33)
+One can also rephrase the discussion in terms Lefschetz thimbles and their intersections.
+The Lefschetz thimble Li(θ; v) for a critical point φi, and a given angle θ, is the space of
+all gradient flow trajectories for the function Re(e−iθW (v)) that go to φi in the far past. We
+are simply saying that if v is away from the exceptional locus, then the Lefschetz thimbles
+Li(θ; v) and Lj(θ; v) for a given angle θ, even when slightly rotated away from θ so that their
+images in the W (v)-plane intersect, will have an empty intersection
+Li(θ ± ǫ; s) ∩ Lj(θ ∓ ǫ; s) = ∅
+(34)
+in M.
+It is interesting to consider the exceptional loci, namely the points on the sphere of complex
+structures where ⃗v · ⃗Zij = 0 for some pair (i, j) of vacua. The exceptional locus is typically a
+set of great circles on the sphere, one such great circle Sij for each pair (i, j) of vacua. Along
+a point on the great circle Sij there can be an ij-soliton, and so a morphism space between
+distinct objects can be non-trivial.
+7
+
+To get a better feeling for what happens at these exceptional points, we consider an
+example. Let M be the cotangent bundle of the projective line M = T ∗P1. Consider the
+complex structure induced from the complex structure on P1, and consider the holomorphic
+coordinates (p, q) in one patch. We call this complex structure I. The real and holomorphic
+symplectic forms in this complex structure are given by
+ω
+=
+2i dq ∧ d¯q
+(1 + |q|2)2 − 2i(1 + |q|2)2dp ∧ d¯p,
+(35)
+Ω
+=
+dp ∧ dq.
+(36)
+The real and complex moment maps corresponding to the hyperK¨ahler isometry
+(p, q) → (e−iθp, eiθq),
+(37)
+are given by
+hI
+=
+1 − |q|2
+1 + |q|2 − (1 + |q|2)2|p|2
+(38)
+WI
+=
+ipq.
+(39)
+There are two critical points, φ1 and φ2 given by (p, q) = (0, 0) and (p, q) = (0, ∞) respec-
+tively, with corresponding critical values
+(hI, WI)|(0,0)
+=
+(1, 0),
+(40)
+(hI, WI)|(0,∞)
+=
+(−1, 0).
+(41)
+In the complex structure I, the critical values of the real moment map are distinct, so that it
+satisfies the criteria of the Corollary. The Landau-Ginzburg superpotential in this complex
+structure,
+WI = ipq
+(42)
+has no flows between the distinct critical points, since the value of WI on both critical points
+is the same, being equal to zero. Therefore the A∞-category of the pair
+�
+(T ∗P1, g, I), WI
+�
+is
+indeed semi-simple. On the other hand, consider the point on the sphere corresponding to
+the complex structure K. The holomorphic moment map in this complex structure is
+WK = hI + iRe(WI),
+(43)
+and the real moment map is
+hK = Im(WI).
+(44)
+For the complex structure K we indeed find that the real moment map hK has the same
+(vanishing) critical value for both critical points. The critical values of WK on the other
+hand are
+WK(φ1) = 1,
+WK(φ2) = −1,
+(45)
+8
+
+so the phase eiθ12 of a BPS soliton interpolating between the critical points φ1 and φ2 must
+be real. We therefore study the gradient flow equation for Re(WK) = hI, or equivalently, the
+Hamiltonian flow equation for Re(WI) with respect to ωK = Im(Ω). This is the equation
+dq
+dy
+=
+−i∂WI
+∂p ,
+(46)
+dp
+dy
+=
+i∂WI
+∂q ,
+(47)
+which simply becomes
+dq
+dy = q,
+dp
+dy = −p.
+(48)
+There is a family of solutions: for each q0 ∈ C\{0} we have
+q(y)
+=
+q0ey,
+(49)
+p(y)
+=
+0,
+(50)
+is a soliton interpolating between the critical points. Writing q0 = ey0+iα we find that y0
+corresponds to an overall translation of the center, and α is the internal collective coordinate
+corresponding to the U(1) isometry. Quantizing the latter collective coordinate, we conclude
+that the soliton space R12 is isomorphic to the deRham cohomology of a circle.
+To give a little more detail on the latter point, note that the space of one-particle BPS
+states M12, namely the subspace of the Hilbert space H12 annihilated by the A-type su-
+percharge (with ζ = 1) and its adjoint is four-dimensional, since there are four fermion
+zero-modes: the superpartners b, b to the translational mode, and the superpartners c, ¯c to
+the U(1) global symmetry. We can also see these as coming from the fact that the BPS
+soliton equation is half-BPS in an eight-supercharge theory, giving us four broken super-
+symmetries which we identify as the fermion zero modes. These fermion zero modes act on
+the vacuum in H12 to generate an irreducible representation of a Clifford algebra with two
+creation and two annihilation operators
+{b, b} = {c, c} = 1,
+(51)
+thus giving us the four-dimensional vector space M12. To obtain R12, as done in [GMW15],
+we factor out the Clifford module corresponding to the translational mode b, so that
+M12 =
+�
+C ⊕ C[1]�
+⊗ R12
+(52)
+leaving us with
+R12 = C ⊕ C[1].
+(53)
+9
+
+In a theory with two vacua 1 and 2, and ζ chosen so that T1 < T2, the morphism space
+�R12 := Hom(T2, T1),
+coincides with the soliton space R12. We therefore conclude that at the exceptional locus,
+the thimble objects T1, T2 satisfy
+�R12 = C ⊕ C[1].
+(54)
+The A∞-structure on the category with objects T1 and T2 is the obvious one: the only
+non-trivial compositions come from the units in Hom(T1, T1) and Hom(T2, T2).
+What happens at the complex structure K, and indeed for any I(0,a,b) = aJ + bK where
+a2 + b2 = 1, is that we are considering an A-model for a real symplectic form
+ω(0,a,b) = Im
+�
+(a + ib)Ω
+�
+,
+(55)
+that is exact (since Ω is an exact form on T ∗P1). On the other hand, if we work with a complex
+structure I(c,a,b) with c ̸= 0 (such as the complex structure I), then the real symplectic form
+ω(c,a,b) is non-exact (since there’s a non-zero contribution from the non-exact symplectic form
+ω). The A∞-categories for exact and non-exact symplectic forms indeed behave differently.
+On the exceptional circle the symplectic manifold (M, ω(0,a,b)) is symplectomorphic to the
+real cotangent bundle T ∗S2, in which the non-trivial morphism space Hom(T2, T1) was first
+worked out in [Seid04].
+Another noteworthy feature of the exact locus is the following. Consider the zero-section
+of T ∗P1, which we denote as C. Note that C is an I-holomorphic submanifold of T ∗P1 such
+that the holomorphic symplectic form Ω vanishes when restricted to it. C is therefore (the
+support of) what is known as a (B, A, A) brane. In particular it is Lagrangian with respect
+to the symplectic structure ωK. We therefore expect it to be an object in the A∞-category
+of the pair
+�
+(T ∗P1, g, K), WK
+�
+. Recall that a general object or brane in the A∞-category
+generated by thimbles is given by a twisted complex. A twisted complex B is specified by
+a choice of graded vector space Vi for each thimble object Ti, along with a Maurer-Cartan
+element (also known as a boundary amplitude): an element
+γB ∈ ⊕i,jVi ⊗ �Rij ⊗ V ∨
+j
+(56)
+of degree +1 that solves the Maurer-Cartan equation. We write such a brane as
+B = [⊕iVi ⊗ Ti, γB].
+(57)
+We claim that the Lagrangian brane with support C is equivalent to the following twisted
+complex:
+BC = [T [1]
+1 ⊕ T2, γC]
+(58)
+10
+
+where T [1] denotes a choice of a multiplicity space of T being a one-dimensional vector space
+in degree +1
+T [1] = C[1] ⊗ T,
+(59)
+and γC is a Maurer-Cartan element
+γC ∈ Hom(T [1]
+1
+⊕ T2, T [1]
+1 ⊕ T2).
+(60)
+The degree one part of this space is one-dimensional and we let γC be a spanning vector.
+Since
+m2(γC, γC) = 0,
+(61)
+it is a solution to the Maurer-Cartan equation. An elementary computation then shows that
+the cohomology of the endomorphism space of this object is
+H∗�
+Hom(BC, BC)
+�
+= C ⊕ C[2].
+(62)
+Moreover as an A∞-algebra this is nothing but the ring C[x]/x2 where x is an element
+carrying homological degree +2.
+Thus the cohomology of Hom(BC, BC) is the deRham
+cohomology ring of C, as we expect. We see that at an exceptional point the A∞-category
+admits a Lagrangian sphere as an object.
+In summary, for M = T ∗P1, the exceptional locus is a great circle on the twistor sphere
+corresponding to when the real symplectic form is exact. If we are away from this locus, the
+A∞-category of the holomorphic moment map is semi-simple, whereas on the exceptional
+locus there is a non-trivial morphism space isomorphic to the cohomology of a circle. It is
+also precisely at the exceptional locus that the A∞-category admits a Lagrangian sphere as
+an object.
+Going back to the general case, at the exceptional locus, we see that the A∞-category de-
+pends on the spectrum of solitons for which in general there is no universal answer. However,
+there is one feature which we comment on: A BPS soliton in the hyperK¨ahler moment map
+setting is the critical point of a holomorphic functional. Indeed, the gradient flow equation
+for µI is equivalent to both the ωJ-flow for µK and the ωK-flow for −µJ. We can obtain the
+latter as the critical locus of the following holomorphic functional on the space of maps from
+R to M:
+W[ϕ] =
+� �
+ϕ∗(Λc) + iµc dy
+�
+,
+(63)
+where ϕ : R → M, Λc is the Liouville form for Ωc = ωJ + iωK and µc = µJ + iµK. The
+functional W is indeed holomorphic in the complex structure on Map(R, M) induced from I,
+and a critical point of Re W obeys the ωJ-flow equation for µK. Thus we find that a soliton
+is now the critical point of a holomorphic functional. This gives another argment for why
+there are no instanton corrections even when there are non-trivial solitons.
+11
+
+To conclude, this note studies the A∞-category of a holomorphic moment map in some
+complex structure of a hyperK¨ahler manifold. Away from exceptional loci on the two-sphere
+of complex structures, the A∞-category is semi-simple. At the exceptional loci, while the
+category is not semi-simple in general, there are still no instanton corrections. We therefore
+find that the A∞-category of a holomorphic moment map is not very interesting.
+Just like the A∞-category of a holomorphic function categorifies the MSW complex of
+its real part, it is natural to wonder if there is a categorification of the A∞-category of a
+holomorphic moment map that is richer, and more intrinsic to hyperK¨ahler moment maps.
+Such a proposal is supported by the fact that there is an uplift of the N = (4, 4) theory to
+a three-dimensional theory with N = 4 supersymmetry, where the A-type supercharge lifts
+to a topological supercharge. This would suggest that the natural invariant associated to a
+hyperK¨ahler moment map is a suitable version of a 2-category.
+The conjectural 2-category associated to (M, ⃗µ) is currently under investigation.
+Acknowledgements
+I thank Justin Hilburn, Mikhail Kapranov, Semon Rezchikov, Paul Seidel, and Edward
+Witten for stimulating discussions. This work is supported by the Institute for Advanced
+Study and the National Science Foundation under Grant No. PHY-2207584.
+References
+[GMW15] D. Gaiotto, G. W. Moore and E. Witten, “Algebra of the Infrared:
+String
+Field Theoretic Structures in Massive N = (2, 2) Field Theory In Two Dimensions,”
+[arXiv:1506.04087 [hep-th]].
+[Hay10] A. Haydys, “Fukaya-Seidel category and gauge theory,” J. Sympl. Geom. 13, 151-
+207 (2015) doi:10.4310/JSG.2015.v13.n1.a5 [arXiv:1010.2353 [math.SG]].
+[Jin21] X. Jin, “Representing the Big tilting sheaves as holomorphic Morse Branes,” Ad-
+vances in Mathematics, Volume 345 (2019) 845-860, [arXiv:1602.07382 [math.SG]].
+[KKS14] M. Kapranov, M. Kontsevich and Y. Soibelman, “Algebra of the infrared and
+secondary polytopes,” Adv. Math. 300, 616-671 (2016) doi:10.1016/j.aim.2016.03.028
+[arXiv:1408.2673 [math.SG]].
+[Seid] P. Seidel “Fukaya categories and Picard-Lefschetz theory,” Zurich Lectures in Ad-
+vanced Mathematics. European Mathematical Society (EMS), Z¨urich
+[Seid04] P. Seidel “Exact Lagrangian submanifolds of T ∗Sn and the graded Kronecker
+quiver,” [arXiv:math/0401212 [math.SG]].
+12
+
+[SV18] J.
+P.
+Solomon
+and
+M.
+Verbitsky,
+“Locality
+in
+the
+Fukaya
+category
+of
+a
+hyperk¨ahler
+manifold,”
+Compos.
+Math.
+155,
+no.10,
+1924-1958
+(2019)
+doi:10.1112/S0010437X1900753X [arXiv:1805.00102 [math.SG]].
+[Wit82] E. Witten, “Supersymmetry and Morse theory,” J. Diff. Geom. 17, no.4, 661-692
+(1982)
+13
+

IdAyT4oBgHgl3EQfffja/content/tmp_files/load_file.txt

@@ -0,0 +1,234 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf,len=233
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='00343v1  [hep-th]  1 Jan 2023  On the A∞-Category of a Holomorphic Moment Map  Ahsan Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Khan∗  School of Natural Sciences  Institute for Advanced Study  Einstein Drive, Princeton NJ 08540  January 3, 2023  Abstract  Let M be a hyperK¨ahler manifold equipped with a U(1) hyperK¨ahler isometry, and  let I be a complex structure on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' In this note, we study the A∞-category of A-branes  for the Landau-Ginzburg model with target space (M, I), and superpotential being the  I-holomorphic moment map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' We show that if I is a generic complex structure, the  A∞-category is semi-simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' For exceptional complex structures, though typically not  semi-simple, the category still has no instanton corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  We illustrate the A∞-  category at both generic and exceptional loci when M is the cotangent bundle of the  projective line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  The Morse-Smale-Witten (MSW) complex is a cochain complex associated to a real func-  tion1 h on a Riemannian manifold (M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' From a physics perspective, the MSW complex  is the space of perturbative ground states of an N = 2 supersymmetric quantum mechan-  ics (SQM) system [Wit82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The differential on this complex is constructed from instanton  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  There is a special situation in which the MSW complex simplifies rather dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  This is when (M, g) admits a g-compatible complex structure I (so that (M, g, I) is a K¨ahler  manifold), and h is the real part of an I-holomorphic function W on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' If this is the case,  it is well-known that all critical points of h have the same Morse index, equal to the complex  dimension of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The MSW complex is thus simply a vector space concentrated in a single  degree, with one basis vector for each critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  This immediately implies that the  differential must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' In terms of physics, there is also something special that happens  ∗khan@ias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='edu  1Throughout this paper we assume that all critical points of h are isolated and non-degenerate  1  in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The N = 2 supersymmetry of the SQM with target space (M, g) and  superpotential h enhances to N = 4 precisely when M is K¨ahler and h = Re W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Systems  with twice the supersymmetry often have simpler properties for their supersymmetric ground  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Remark  Another case where the supersymmetry enhances to N = 4 is when h is the  moment map of a continuous K¨ahler isometry of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' We will use a complex version of this  statement later in the note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  We conclude that the MSW complex of Re W is not a very interesting invariant of the  pair  �  (M, g, I), W  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' There is, however, a richer invariant of the same pair which, in a sense  that can be made precise, categorifies the MSW complex of Re W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The categorification of  the MSW complex of Re W is known (to mathematicians) as the Fukaya-Seidel A∞-category  of (M, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' In terms of physics what’s responsible for the existence of this categorification  is the fact that there is an uplift of the N = 4 SQM to a two-dimensional N = (2, 2)  Landau-Ginzburg (LG) model such that the original SQM supercharge lifts to a topological  supercharge2 of the LG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The Fukaya-Seidel A∞-category is then simply the category  of boundary conditions of the Landau-Ginzburg model that preserve this topological super-  charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' These boundary conditions are known as A-branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The MSW complex of Re W  can be recovered upon taking the Hochschild homology of the Fukaya-Seidel category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  To be more precise, there are a few different versions of the A∞-category associated to  (M, W) (all conjectured to be A∞-equivalent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Usually the term “Fukaya-Seidel category”  refers to a formulation by Seidel based on vanishing cycles and their intersections [Seid].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  We will be working mostly with a formulation developed by Gaiotto, Moore and Witten,  which is based on particular kinds of solutions to the ζ-soliton and ζ-instanton equations3 4  [GMW15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  It is natural to wonder if there is a class of K¨ahler manifolds and superpotentials for  which the associated A∞-category simplifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' A possible criteria for such pairs could be that  the supersymmetry of the Landau-Ginzburg model of (M, W) enhances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' There is indeed a  special situation where this is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Suppose that the target space (M, g, I) admits an  I-holomorphic two-form Ω satisfying  Ωg−1Ω + g = 0,  ∇Ω = 0,  (1)  2A supercharge Q is called topological if all translations are in its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  3A similar (but not identical) formulation was also developed by Haydys [Hay10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  4See also [KKS14] for a reformulation of the formalism of [GMW15] that generalizes to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  2  so that the space (M, g, I, Ω) is a hyperK¨ahler manifold5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Moreover, suppose (M, g, I, Ω)  admits a hyperK¨ahler isometry generated by a vector field V , so that  LV g = LV I = LV Ω = 0,  (2)  and W is the corresponding holomorphic moment map  dW = ιV Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (3)  If this is the case, the Landau-Ginzburg model for the pair  �  (M, g, I), W  �  has an additional  symmetry6 coming from rotating the fermions of the theory by the endomorphism  J = g−1Re  �  Ω  �  (4)  of the tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The commutator of this symmetry transformation with the N =  (2, 2) supersymmetries generates four additional fermionic symmetries resulting in a total of  eight supercharges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The supersymmetry of the Landau-Ginzburg model of (M, W) therefore  enhances from N = (2, 2) to N = (4, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  The above example in fact gives us an infinite family of pairs of K¨ahler manifolds and  holomorphic functions for which the supersymmetry enhances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  In addition to I we can  define the integrable complex structures J = g−1Re(Ω) and K = g−1Im(Ω) that moreover  satisfy the quaternion relations  I2 = J2 = K2 = IJK = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (5)  This brings us to the well-known fact that a hyperK¨ahler manifold M has a two-sphere’s  worth of complex structures, since for any v = (v1, v2, v3) ∈ S2, the endomorphism  I(v) = v1I + v2J + v3K  (6)  is an integrable complex structure on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Moreover, given a hyperK¨ahler isometry generated  by a vector field V , there is a holomorphic moment map  W (v) : M → C  5Recall that one definition of a hyperK¨ahler is as a K¨ahler manifold with a compatible parallel holomor-  phic symplectic form Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  6The axial R-symmetry FA, the symmetry coming from rotating by J, and their commutator, generate  an so(3) R-symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Another way of seeing both the supersymmetry enhancement and this SO(3)  R-symmetry is to note that the two-dimensional theory comes from reducing a six dimensional hyperK¨ahler  sigma model, where the hyperK¨ahler isometry is gauged by a six-dimensional U(1) vector multiplet, to two  dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' When doing the reduction, we set the gauge fields in the four internal directions to a non-zero  vector in R4, while setting all other vector multiplet fields to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The choice of a non-zero vector breaks  the SO(4) R-symmetry coming from the internal directions to SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  3  in every complex structure I(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Explicitly, W (v) can be obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Let µI be the  moment map corresponding to the real symplectic form, ωI = gI and let  W = µJ + iµK,  Ω = ωJ + iωK,  (7)  so that the triple (µI, µJ, µK) satisfies  dµI = ιV ωI,  (8)  dµJ = ιV ωJ,  (9)  dµK = ιV ωK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (10)  We can obtain the complex structure I(v) from I by a hyperK¨ahler rotation: there is a  quaternion  q = q0 + q1i + q2j + q3k ∈ H  (11)  such that  ¯q i q = v1i + v2j + v3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (12)  The quaternion q is unique up to a redefinition by a phase q → eiθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Organize the hy-  perK¨ahler moment map in terms of the imaginary quaternions  ⃗µ = µIi + µJj + µKk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (13)  With respect to the complex structure I, the decomposition of the hyperK¨ahler moment  map into real and complex parts is given by  ⃗µ = h i + Wj,  (14)  so that h = µI is the real moment map and W = µJ +iµK is the complex moment map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The  real and holomorphic moment maps with respect to I(v) are then given by doing a rotation  of µ with respect to q:  q ⃗µ q = h(v)i + W (v)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (15)  Redefinition of q by a phase,  q → eiθq  (16)  leaves h(v) invariant whereas it transforms  W (v) → e2iθW (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (17)  Since redefinition of W (v) by a phase does not change the associated Landau-Ginzburg model,  we get an LG model ((M, g, I(v)), W (v)) for every point on the unit two-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Because the  supersymmetry enhances for all such models, it is natural to expect that the A∞-category  of each such pair  �  (M, g, I(v)), W (v)�  simplifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  4  The purpose of this note is to show that this is indeed the case7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  For the rest of the note we assume that the hyperK¨ahler moment map has isolated and  non-degenerate critical points, and v ∈ S2 is such that the critical values of W (v) are in  general position on the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  The first basic simplification in the A∞-category of a moment map is the lack of instan-  ton corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  This can be seen as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Recall that in the Gaiotto-Moore-Witten  formulation of the A∞-category of W, all instanton corrections come from solutions of the  ζ-instanton equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The ζ-instanton equation for a map φ : C → M is  ∂φi  ∂¯z = ζgi¯j ∂W  ∂ ¯φ¯j ,  (18)  where ζ = eiθ is a phase we are required to choose in order to define the category, (φi, φ  ¯i) are  local complex coordinates on M and z is the standard complex coordinate on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The par-  ticular solutions that contribute to the A∞-structure are solutions with “fan-like” boundary  conditions at infinity, that are rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Rigidity means that a solution has no moduli other  than overall translations of the Euclidean spacetime complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' That there cannot be  any such solution in the hyperK¨ahler case can be easily seen from the fact that the operator  obtained from linearizing the ζ-instanton equation has a zero-mode  δφi = V i  (19)  in addition to the translational zero-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' There are therefore no non-trivial rigid instan-  tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Next we prove that at a generic point v on the two-sphere of complex structures, a stronger  statement holds: the A∞-category of the pair  �  (M, g, I(v)), W (v)�  is semi-simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' What we  mean by this is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Recall that A∞-category of a superpotential W has a generating  set of “thimble” objects {Ti} that are in one-to-one correspondence with critical points of  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The A∞-category is said to be semi-simple if the morphism spaces between the thimble  objects satisfy  Hom(Ti, Tj) = δijC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (20)  This property follows from an elementary Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Lemma  Let (M, g, I, J, K) be a hyperK¨ahler manifold,  ⃗µ : M → R3  7For previous work that is related to the setting of the present note see [SV18] and [Jin21]  5  be a hyperK¨ahler moment map, and ⃗n = (n1, n2, n3) ∈ S2 be a point on the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Suppose φ : R → M is a (non-constant) solution to the gradient flow equation  dφA  dy = gAB ∂(⃗n · ⃗µ)  ∂φB  (21)  where  ⃗n · ⃗µ = n1µI + n2µJ + n3µK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (22)  Then the composition ⃗µ ◦ φ : R → R3 is an embedding with image a straight line in the  ⃗n-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' We show the claim for ⃗n = (0, 1, 0), so that we study the gradient flow equation for  µJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Since µJ is the real part of the I-holomorphic function  WI = µJ + iµK,  (23)  the Cauchy-Riemann equation  dµJ = ItdµK  (24)  implies that the the gradient vector field of µJ is equivalent to the Hamiltonian vector field  of µK with respect to the symplectic form ωI  g−1dµJ = ω−1  I dµK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (25)  Therefore µK is constant along a flow line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' But µJ is also the real part of the K-holomorphic  function  WK = µJ − iµI  and so the Cauchy-Riemann equation for WK implies that the gradient flow of µJ is also  equivalent to the Hamiltonian flow for −µI with respect to the symplectic form ωK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Therefore  µI is also constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Since µJ ◦ φ is monotonic, the image of the ⃗µ ◦ φ is a line parallel to the  J-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The case when ⃗n is arbitrary can be obtained by a hyperK¨ahler rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Consider now the Landau-Ginzburg model with target space (M, g, I(v)) and superpoten-  tial given by the I(v)-holomorphic function W (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' A fundamental role in Landau-Ginzburg  models is played by BPS solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Let φi and φj be critical points of W (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Recall that an  ij-BPS soliton is a solution of the gradient flow equation for Re(e−iθijW (v)) where  eiθij = W (v)(φi) − W (v)(φj)  |W (v)(φi) − W (v)(φj)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (26)  Let h(v) be the real moment map in the complex structure I(v), explicitly  h(v) = v1µI + v2µJ + v3µK = ⃗v · ⃗µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (27)  6  According to the Lemma, the real moment map h(v) is conserved along the gradient flow of  Re  �  eiθW (v)�  (as is Im  �  eiθW (v)�  ) for any value of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' An ij soliton can therefore only exist if  h(v)(φi) = h(v)(φj),  (28)  Letting ⃗µi ∈ R3 be the critical value of the hyperK¨ahler moment map in the ith vacuum,  this condition is simply saying that  ⃗v · (⃗µi − ⃗µj) = 0,  (29)  so that ⃗v is perpendicular to the vector  ⃗Zij = ⃗µi − ⃗µj  (30)  pointing from the jth critical value to the ith critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' If this does not happen for any  pair of distinct critical points, the soliton spectrum of the Landau-Ginzburg theory is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  In the notation of [GMW15], the soliton space Rij is trivial  Rij = {0},  (31)  for each pair of distinct critical points (φi, φj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Since, in the Gaiotto-Moore-Witten formal-  ism, morphism spaces between different thimble objects are given as direct sums of tensor  products of soliton spaces, we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Corollary  Suppose ⃗v ∈ S2 is such that ⃗v · (⃗µi − ⃗µj) ̸= 0 for any pair of critical points  (φi, φj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Then the A∞-category of  �  (M, g, I(v)), W (v)�  is semi-simple:  Hom(Ti, Ti)  =  C, for all i,  (32)  Hom(Ti, Tj)  =  0  for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (33)  One can also rephrase the discussion in terms Lefschetz thimbles and their intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  The Lefschetz thimble Li(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' v) for a critical point φi, and a given angle θ, is the space of  all gradient flow trajectories for the function Re(e−iθW (v)) that go to φi in the far past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' We  are simply saying that if v is away from the exceptional locus, then the Lefschetz thimbles  Li(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' v) and Lj(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' v) for a given angle θ, even when slightly rotated away from θ so that their  images in the W (v)-plane intersect, will have an empty intersection  Li(θ ± ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' s) ∩ Lj(θ ∓ ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' s) = ∅  (34)  in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  It is interesting to consider the exceptional loci, namely the points on the sphere of complex  structures where ⃗v · ⃗Zij = 0 for some pair (i, j) of vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The exceptional locus is typically a  set of great circles on the sphere, one such great circle Sij for each pair (i, j) of vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Along  a point on the great circle Sij there can be an ij-soliton, and so a morphism space between  distinct objects can be non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  7  To get a better feeling for what happens at these exceptional points, we consider an  example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Let M be the cotangent bundle of the projective line M = T ∗P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Consider the  complex structure induced from the complex structure on P1, and consider the holomorphic  coordinates (p, q) in one patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' We call this complex structure I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The real and holomorphic  symplectic forms in this complex structure are given by  ω  =  2i dq ∧ d¯q  (1 + |q|2)2 − 2i(1 + |q|2)2dp ∧ d¯p,  (35)  Ω  =  dp ∧ dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (36)  The real and complex moment maps corresponding to the hyperK¨ahler isometry  (p, q) → (e−iθp, eiθq),  (37)  are given by  hI  =  1 − |q|2  1 + |q|2 − (1 + |q|2)2|p|2  (38)  WI  =  ipq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (39)  There are two critical points, φ1 and φ2 given by (p, q) = (0, 0) and (p, q) = (0, ∞) respec-  tively, with corresponding critical values  (hI, WI)|(0,0)  =  (1, 0),  (40)  (hI, WI)|(0,∞)  =  (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (41)  In the complex structure I, the critical values of the real moment map are distinct, so that it  satisfies the criteria of the Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The Landau-Ginzburg superpotential in this complex  structure,  WI = ipq  (42)  has no flows between the distinct critical points, since the value of WI on both critical points  is the same, being equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Therefore the A∞-category of the pair  �  (T ∗P1, g, I), WI  �  is  indeed semi-simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' On the other hand, consider the point on the sphere corresponding to  the complex structure K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The holomorphic moment map in this complex structure is  WK = hI + iRe(WI),  (43)  and the real moment map is  hK = Im(WI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (44)  For the complex structure K we indeed find that the real moment map hK has the same  (vanishing) critical value for both critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The critical values of WK on the other  hand are  WK(φ1) = 1,  WK(φ2) = −1,  (45)  8  so the phase eiθ12 of a BPS soliton interpolating between the critical points φ1 and φ2 must  be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' We therefore study the gradient flow equation for Re(WK) = hI, or equivalently, the  Hamiltonian flow equation for Re(WI) with respect to ωK = Im(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' This is the equation  dq  dy  =  −i∂WI  ∂p ,  (46)  dp  dy  =  i∂WI  ∂q ,  (47)  which simply becomes  dq  dy = q,  dp  dy = −p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (48)  There is a family of solutions: for each q0 ∈ C\\{0} we have  q(y)  =  q0ey,  (49)  p(y)  =  0,  (50)  is a soliton interpolating between the critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Writing q0 = ey0+iα we find that y0  corresponds to an overall translation of the center, and α is the internal collective coordinate  corresponding to the U(1) isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Quantizing the latter collective coordinate, we conclude  that the soliton space R12 is isomorphic to the deRham cohomology of a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  To give a little more detail on the latter point, note that the space of one-particle BPS  states M12, namely the subspace of the Hilbert space H12 annihilated by the A-type su-  percharge (with ζ = 1) and its adjoint is four-dimensional, since there are four fermion  zero-modes: the superpartners b, b to the translational mode, and the superpartners c, ¯c to  the U(1) global symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' We can also see these as coming from the fact that the BPS  soliton equation is half-BPS in an eight-supercharge theory, giving us four broken super-  symmetries which we identify as the fermion zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' These fermion zero modes act on  the vacuum in H12 to generate an irreducible representation of a Clifford algebra with two  creation and two annihilation operators  {b, b} = {c, c} = 1,  (51)  thus giving us the four-dimensional vector space M12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' To obtain R12, as done in [GMW15],  we factor out the Clifford module corresponding to the translational mode b, so that  M12 =  �  C ⊕ C[1]�  ⊗ R12  (52)  leaving us with  R12 = C ⊕ C[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (53)  9  In a theory with two vacua 1 and 2, and ζ chosen so that T1 < T2, the morphism space  �R12 := Hom(T2, T1),  coincides with the soliton space R12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' We therefore conclude that at the exceptional locus,  the thimble objects T1, T2 satisfy  �R12 = C ⊕ C[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (54)  The A∞-structure on the category with objects T1 and T2 is the obvious one: the only  non-trivial compositions come from the units in Hom(T1, T1) and Hom(T2, T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  What happens at the complex structure K, and indeed for any I(0,a,b) = aJ + bK where  a2 + b2 = 1, is that we are considering an A-model for a real symplectic form  ω(0,a,b) = Im  �  (a + ib)Ω  �  ,  (55)  that is exact (since Ω is an exact form on T ∗P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' On the other hand, if we work with a complex  structure I(c,a,b) with c ̸= 0 (such as the complex structure I), then the real symplectic form  ω(c,a,b) is non-exact (since there’s a non-zero contribution from the non-exact symplectic form  ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The A∞-categories for exact and non-exact symplectic forms indeed behave differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  On the exceptional circle the symplectic manifold (M, ω(0,a,b)) is symplectomorphic to the  real cotangent bundle T ∗S2, in which the non-trivial morphism space Hom(T2, T1) was first  worked out in [Seid04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Another noteworthy feature of the exact locus is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Consider the zero-section  of T ∗P1, which we denote as C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Note that C is an I-holomorphic submanifold of T ∗P1 such  that the holomorphic symplectic form Ω vanishes when restricted to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' C is therefore (the  support of) what is known as a (B, A, A) brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' In particular it is Lagrangian with respect  to the symplectic structure ωK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' We therefore expect it to be an object in the A∞-category  of the pair  �  (T ∗P1, g, K), WK  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Recall that a general object or brane in the A∞-category  generated by thimbles is given by a twisted complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' A twisted complex B is specified by  a choice of graded vector space Vi for each thimble object Ti, along with a Maurer-Cartan  element (also known as a boundary amplitude): an element  γB ∈ ⊕i,jVi ⊗ �Rij ⊗ V ∨  j  (56)  of degree +1 that solves the Maurer-Cartan equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' We write such a brane as  B = [⊕iVi ⊗ Ti, γB].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (57)  We claim that the Lagrangian brane with support C is equivalent to the following twisted  complex:  BC = [T [1]  1 ⊕ T2, γC]  (58)  10  where T [1] denotes a choice of a multiplicity space of T being a one-dimensional vector space  in degree +1  T [1] = C[1] ⊗ T,  (59)  and γC is a Maurer-Cartan element  γC ∈ Hom(T [1]  1  ⊕ T2, T [1]  1 ⊕ T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (60)  The degree one part of this space is one-dimensional and we let γC be a spanning vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Since  m2(γC, γC) = 0,  (61)  it is a solution to the Maurer-Cartan equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' An elementary computation then shows that  the cohomology of the endomorphism space of this object is  H∗�  Hom(BC, BC)  �  = C ⊕ C[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  (62)  Moreover as an A∞-algebra this is nothing but the ring C[x]/x2 where x is an element  carrying homological degree +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Thus the cohomology of Hom(BC, BC) is the deRham  cohomology ring of C, as we expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' We see that at an exceptional point the A∞-category  admits a Lagrangian sphere as an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  In summary, for M = T ∗P1, the exceptional locus is a great circle on the twistor sphere  corresponding to when the real symplectic form is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' If we are away from this locus, the  A∞-category of the holomorphic moment map is semi-simple, whereas on the exceptional  locus there is a non-trivial morphism space isomorphic to the cohomology of a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' It is  also precisely at the exceptional locus that the A∞-category admits a Lagrangian sphere as  an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Going back to the general case, at the exceptional locus, we see that the A∞-category de-  pends on the spectrum of solitons for which in general there is no universal answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' However,  there is one feature which we comment on: A BPS soliton in the hyperK¨ahler moment map  setting is the critical point of a holomorphic functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Indeed, the gradient flow equation  for µI is equivalent to both the ωJ-flow for µK and the ωK-flow for −µJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' We can obtain the  latter as the critical locus of the following holomorphic functional on the space of maps from  R to M:  W[ϕ] =  � �  ϕ∗(Λc) + iµc dy  �  ,  (63)  where ϕ : R → M, Λc is the Liouville form for Ωc = ωJ + iωK and µc = µJ + iµK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' The  functional W is indeed holomorphic in the complex structure on Map(R, M) induced from I,  and a critical point of Re W obeys the ωJ-flow equation for µK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Thus we find that a soliton  is now the critical point of a holomorphic functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' This gives another argment for why  there are no instanton corrections even when there are non-trivial solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  11  To conclude, this note studies the A∞-category of a holomorphic moment map in some  complex structure of a hyperK¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Away from exceptional loci on the two-sphere  of complex structures, the A∞-category is semi-simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' At the exceptional loci, while the  category is not semi-simple in general, there are still no instanton corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' We therefore  find that the A∞-category of a holomorphic moment map is not very interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Just like the A∞-category of a holomorphic function categorifies the MSW complex of  its real part, it is natural to wonder if there is a categorification of the A∞-category of a  holomorphic moment map that is richer, and more intrinsic to hyperK¨ahler moment maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Such a proposal is supported by the fact that there is an uplift of the N = (4, 4) theory to  a three-dimensional theory with N = 4 supersymmetry, where the A-type supercharge lifts  to a topological supercharge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' This would suggest that the natural invariant associated to a  hyperK¨ahler moment map is a suitable version of a 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  The conjectural 2-category associated to (M, ⃗µ) is currently under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Acknowledgements  I thank Justin Hilburn, Mikhail Kapranov, Semon Rezchikov, Paul Seidel, and Edward  Witten for stimulating discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' This work is supported by the Institute for Advanced  Study and the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' PHY-2207584.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
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+page_content=' Kontsevich and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
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+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='aim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='028  [arXiv:1408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='2673 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='SG]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  [Seid] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Seidel “Fukaya categories and Picard-Lefschetz theory,” Zurich Lectures in Ad-  vanced Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' European Mathematical Society (EMS), Z¨urich  [Seid04] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
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+page_content='SG]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  12  [SV18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Solomon  and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Verbitsky,  “Locality  in  the  Fukaya  category  of  a  hyperk¨ahler  manifold,”  Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
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+page_content='10,  1924-1958  (2019)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='1112/S0010437X1900753X [arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='00102 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='SG]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content='  [Wit82] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
+page_content=' Witten, “Supersymmetry and Morse theory,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAyT4oBgHgl3EQfffja/content/2301.00343v1.pdf'}
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JNE2T4oBgHgl3EQfAAax/content/tmp_files/2301.03587v1.pdf.txt

@@ -0,0 +1,523 @@
+MACHINE LEARNING APPLIED TO PERUVIAN
+VEGETABLES IMPORTS
+Ticona-Salluca Hugo
+Faculty of Statistic and Computer Engineering,
+Universidad Nacional del Altiplano de Puno, P.O. Box 291
+Puno - Peru.
+Email: [email protected]
+Torres-Cruz Fred
+Faculty of Statistic and Computer Engineering,
+Universidad Nacional del Altiplano de Puno, P.O. Box 291
+Puno - Peru.
+Email: [email protected]
+Tumi-Figueroa Ernesto Nayer
+Faculty of Statistic and Computer Engineering,
+Universidad Nacional del Altiplano de Puno, P.O. Box 291
+Puno - Peru.
+Email: [email protected]
+Abstract—The current research work is being developed as a
+training and evaluation object. the performance of a predictive
+model to apply it to the imports of vegetable products into
+Peru using artificial intelligence algorithms, specifying for this
+study the Machine Learning models: LSTM and PROPHET.
+The forecast is made with data from the monthly record of
+imports of vegetable products(in kilograms) from Peru, collected
+from the years 2021 to 2022. As part of applying the training
+methodology for automatic learning algorithms, the exploration
+and construction of an appropriate dataset according to the
+parameters of a Time Series. Subsequently, the model with better
+performance will be selected, evaluating the precision of the
+predicted values so that they account for sufficient reliability to
+consider it a useful resource in the forecast of imports in Peru.
+Keywords—Machine Learning: forecasting; time series; im-
+ports;artificial intelligence;dataset
+I.
+INTRODUCTION
+The Machine Learning methodology is one of the most
+valuable technologies in the Industry 4.0, and advances in Ma-
+chine Learning have provided significant benefits in strategic
+decision-making in organizations[1]. In recent years, predictive
+models using Machine Learning algorithms have been imple-
+mented in real environments of different organizations, and
+as in this particular case that concerns us, it also operates in
+the agriculture industry and optimal results are expected from
+these technologies.
+Likewise, Machine Learning technology has a high poten-
+tial to optimize your processes and, consequently, make correct
+and anticipated decisions. For this reason, Learning Algorithms
+should also be considered as the next innovative direction to
+significantly improve the prediction of these use cases. [2].
+Therefore, this research will seek to demonstrate and
+encourage the use of Machine Learning applications in the
+field of agribusiness in Peru. Through the present work, the
+examination and modeling of real data will be developed in
+order to forecast future values with the least possible prediction
+error. In this sense, this applied research work covers the
+design, training, validation, and evaluation of 2 predictive
+models. The research models to be explored are LSTM and
+Prophet models, these being adequate and recommended to
+develop applied foresting, on the one hand, LSTM is rec-
+ognized for its hyperparameters which have the ability to
+capture non-linear patterns using the grid search method[3]
+and PROPHET for their modular regression with intuitive and
+adjustable interpretation parameters applied on Time Series[4].
+.
+II.
+DATASET
+According to the purpose of the study, the official records
+of the export volume (in kilograms) of vegetables in Peru were
+used, collected from 2021 to 2022 in the National Open Data
+Platform,[5], officially uploaded by the Ministry of Agrarian
+Development and Irrigation of Peru.
+Table 1
+Description of the variables in the Vegetable Import Dataset
+in Peru 2021-2022.
+Based on this dataset, it is proposed to make a reliable
+prediction, and according to the models to be developed, we
+will therefore need the continuous quantitative variable, PESO.
+Also according to the observed data, we find it necessary to
+adapt a proper timeline; therefore, we must join the variables
+A ˜NO and MES to foresee a more accurate seasonality, spec-
+ifying the variables to forecast. Subsequently, to get a more
+accurate overview of our data regarding products, we define
+IDs for each product.
+From the dataset, we highlight 3P: Soja, cake with 23.1%
+in 1st position, in second position 5P: Apple, fresh fruit
+with 6.3% and in third position 27P: Soja, grain with 4.2%.
+The data ranges from 2021-05-01 to 2022-06-01, with 34,364
+observations on imports of a total of 848 plant products.
+1 | P a g e
+arXiv:2301.03587v1  [cs.LG]  8 Jan 2023
+
+VAR
+SCALA
+DESCRIPTION
+1
+ANO
+CUANTITATIVA DISCRETA
+Anodelaimportacion
+2
+MES
+CUANTITATIVA DISCRETA
+Mes de la importacion
+3
+SEDE
+CUALITATIVA NOMINAL
+Sede donde se registra de la Importacion
+4
+PRODUCTO
+CUANTITATIVA DISCRETA
+Productoimportado
+7
+PESO
+CUANTITATIVA CONTINUA
+Pesoenkilogramosdelproductc
+8
+TIPO.ENVASE
+CUALITATIVA NOMINAL
+Envasecontenedordelaimportacion
+PAIS.ORIGEN
+CUALITATIVA NOMINAL
+Origendelaimportacion del productoFig. 1.
+Percentage description of each vegetable product imported in Peru
+III.
+METHODS
+In this section, we will describe the models used for the
+predictions.
+A. LSTM MODEL
+The LSTM name refers to the analogy that a standard RNN
+has both ’long-term memory’ and ’short-term memory’. The
+connection weights and biases in the network change once
+per training episode, analogous to how physiological changes
+in synaptic strengths store long-term memories; activation
+patterns in the network change once per time step, analogous
+to how moment-to-moment changes in electrical activation
+patterns in the general brain store short-term memories. The
+LSTM architecture aims to provide a short-term memory for
+RNNs that can last for thousands of time steps and continue
+to be reliable, hence ’short-term long-term memory’[6].
+LSTM is considered a special type of recurrent neural
+network (RNN), developed to solve the potential problem of
+descending gradient found in traditional RNN training, and is
+able to learn both short-term and long-term dependencies[7]
+and is constructed of four main components: an entry gate, an
+exit gate, memory cell and a forget gate.
+Input Gate: controls the sending addition of information to
+the cell state. In other words, the gateway will consider what
+information needs to be added to the cell state to ensure that
+important information is added and that there is no redundant
+information or noise.
+Memory Cell: controls the value that might be deleted or
+updated, and contains a value that might need to be kept as
+additional information for many other time steps.
+Output Gate: controls the selection of useful learning
+information from the current state of the cell as output.
+Forget Gate: controls the removal of information that is no
+longer needed for LTSM to learn things or, less importantly,
+the state of the cell. This helps to optimize the performance
+of the LSTM proposed network.
+Also, The LSTM model follows the sequence:
+1. Decide to discard the cell state information by a Sigmoid
+layer called ”forget gate”.
+2. Decide new information to store in the cell state. The
+Sigmoid layer called the ”gateway layer” decides which values
+will be updated. the tan coat creates a vector of new candidate
+values that could be added to the state.
+3. Update the old cell state to the new cell state.
+4. Decide the filtered output based on the state of the cell.
+Fig. 2.
+MODEL LSTM
+Considered for a normal LSTM model is the Tanh property,
+which is a non-linear activation function. It regulates the values
+that flow through the network, keeping the values between -
+1 and 1. To avoid information fading, a function is needed
+whose second derivative can survive longer. There might be a
+case where some values become huge, causing the values to
+be irrelevant.
+And of course an LSTM principal property: The Sigmoid
+that belongs to the family of nonlinear activation functions.
+Unlike Tanh, the Sigmoid holds values between 0 and 1. It
+helps the network update or forget data. If the multiplication
+results in 0, the information is considered forgotten. Similarly,
+the information is kept if the value is relevant[8].
+This will assist the network in determining what data can
+be lost and what data should be kept.
+B. PROPHET MODEL
+PROPHET is a procedure for forecasting time series data
+based on an additive model in which nonlinear trends are
+adjusted for annual, weekly, and daily seasonality. It works best
+with time series that have strong seasonal effects and multiple
+seasons of historical data.
+2 | P a g e
+
+3P
+23.1%
+6
+iF
+20
+6.3%
+5P
+145P
+43P
+10P
+239P
+56P
+4.2%
+86P
+11p
+106P
+31P
+3.5%
+104P
+1P
+0.8%
+42P
+27P
+3.2%
+29P
+64P
+2.8%
+59P
+2.4%
+77P
+61P
+23P
+8P
+91P
+17P
+84P
+6P
+18P
+34p
+57P
+12P
+44P
+89P
+22p
+9P
+28P
+41P
+70P
+69P
+26P
+16P
+50PForget gate
+ht
+Memory cell Ct
+Ct-1
+文
+Ct
+x
+tanh
+a
+tanh
+Hidden state ht
+ht-1 -
+X
+>ht
+Xt
+Input gate
+Output gateit=o(atU+ht-iW)
+x, input
+ft=o(atUf +ht-1Wf)
+f. forget gate
+i，
+inputgate
+Ot = o(atUo + ht-1Wo)
+cellupdate
+c, cell state
+Ct =tanh (ctU9+ht-iW9)
+o,outputgate
+Ct = o(ft* Ct-1 + it*Ct)
+h,output
+G sigmoidactivationfunction
+ht = tanh(Ct) * Ot
+tanh Tanh activationfunctionIn the specification of this Prophet model, there are several
+places where we can alter the model to apply your experience
+and external expert knowledge without needing to understand
+the underlying statistics.[4]
+For the Prophet model with general theory:
+We use a mathematical decomposable time series model[9]
+and this model has these components: trend, seasonality, and
+holidays. They are combined into the following equation:
+Series Model Formula.
+Here g(t) is the trend function that models non-periodic
+changes in the value of the time series, s(t) represents periodic
+changes (for example, weekly and yearly seasonality), and h(t)
+represents the effects of the seasons, The error term represents
+any idiosyncratic changes that do not fit the model.
+In PROPHET we incorporate trend changes into the growth
+model by explicitly determining the change points where the
+growth rate is allowed to change. Suppose there are exchange
+points at moments j,j= 1,..., S. We give N a vector of adjust-
+ments, where only the rate change occurs at moment j. The
+rate at any time is then the base rate k, plus all adjustments
+up to the point of This is best represented by the vectors as
+follows:
+Adjustments represented.
+When rate k is adjusted, the parameter set must also be
+adjusted to connect the endpoints of the segments. The correct
+fit at the shift point is easily calculated as:
+That would be the Adjust at shift point.
+The logistic piece of the growth model then looks as
+follows:
+The Fourier series is also applied in Prophet to provide
+a flexible model of periodic effects[10]. Let P be the regular
+period that we expect the time series to have (for example, P=
+365:25 for annual data or P= 7 for weekly data, when we scale
+our indices of time variables). We can approximate arbitrary
+uniform seasonal effects with this definition:
+Fourier Definition
+Sometimes we can’t just randomly split the data accord-
+ingly. PROPHET develops simulated historical forecasts by
+producing K forecasts at various cut-off points in history,
+chosen such that the horizons are within the historical record
+and the total error can be evaluated.
+This procedure is based on the classic ’rolling source’ fore-
+cast evaluation procedures[11], but uses only a small sequence
+of target dates instead of forecasting by historical date) is that
+it saves on computation and provides less correlated precision
+measurements.
+C. Work Sequence
+•
+We apply an EDA (Exploratory Data Analysis) it is
+necessary to clean the data and adapt it for the job.
+•
+Normalization of the data, we adapt the data to follow
+a supervised sequence model according to a time
+series.
+•
+Let’s divide our dataset into proof and validation tests.
+•
+Coding and implementation of the models according
+to the Prophet or LSTM case.
+•
+Adjustment of the parameters and extraction of pre-
+dictions with their respective evaluation metrics.
+IV.
+RESULTS
+The results using PROPHET and LSTM are shown in this
+section together with the general comparison of the mentioned
+models.
+We will also appreciate the comparisons and metrics de-
+veloped according to the models applied to the study variable:
+PESO.
+Result according to the validation of the LSTM model:
+Fig. 3.
+LSTM VALIDATION
+3 | P a g e
+
+PESO
+600000
+LSTM_Predictions
+500000
+400000
+300000
+20000
+100000
+0
+00:45:38
+00:45:40
+00:45:42
+00:45:44
+00:45:46
+00:45:48
+Fechay(t) = g(t) + s(t) + h(t) + Et1,
+if t≥sj,
+a;(t)
+0,
+otherwise.=(s-m-)(1
+1-)C(t)s(t)-(an cos()+bsin())Result according to the validation of the PROPHET model:
+Fig. 4.
+PROPHET VALIDATION
+Comparison to the validation of the PROPHET and LSTM
+model:
+Fig. 5.
+LSTM AND PROPHET COMPARISON
+MSE(mean squared error) and the RMSE(root mean
+squared error) of both models:
+We can appreciate each model according to its evaluation.
+Fig. 6.
+RMSE Y MSE
+V.
+DISCUSSION AND CONCLUSIONS
+In keeping with the theory, all machine learning algorithms
+are unique, which is the root cause of why the prediction
+results are different algorithms on the same data set differ.
+LSTM was proposed for being an improvement of the RNNs
+and Prophet for its versatility in data with less presence of
+seasonality.
+Given what was taken into consideration for LSTM, the
+applied model was with the minimum parameters and show
+results with much better conditioning than was expected,
+remembering that in different studies the superiority of LSTM
+is detailed over the basic algorithms that apply Recurrent
+Neural Networks. Furthermore, it is noted from the theory
+that the number of training times, known as the ’epoch’ in
+learning[12], did not take into account the expected effect on
+the performance of the trained forecast model and exhibited
+mostly random behavior. How intuitive The developed model
+based on LSTM incorporates additional ’gates’ in order to store
+longer sequences of input data.
+One of the main questions when developing and analyzing is
+whether the gates incorporated in the LSTM architecture would
+give a good prediction and if additional data training would
+be needed to further improve the prediction[13] and in this
+case, we can deduce that the quantity of data was acceptable
+but the predictions were affected by the not very well defined
+seasonality of the dataset. A more effective solution would be
+to add exact dates and continuous seasonalities.
+For the Prophet model, we should have some more
+intuitive results according to its theory, the application of
+the Fourier series could develop more precision. The model
+is expected to obtain a reasonable forecast on disordered
+data without too much manual effort, unlike LSTM, which
+has more hyperparameters. Prophet proved to be resistant
+to outliers, missing data, and drastic changes in its time
+series, the intention to fit the timeline is noted. Compared
+to other classical forecasting methods, Prophet should be
+fast and easy to apply to time series, which is what it
+was designed for in the first place; however, it could be
+considerably less accurate[14] and in this case, we confirm
+this appreciation by highlighting once again its intuitive factor.
+The Prophet procedure should include more parameters for
+users to modify and adjust the forecasts in a more effective
+way. Also, a hybrid model could improve significantly.
+According to the RMSE results of the import predictions,
+we can conclude that the LSTM model presents a significantly
+better performance and reliability with respect to the Prophet
+model, however, as we deduced previously, the seasonality
+of the dataset was an important key in the variation of the
+development of the models and their predictions. Therefore,
+increasing the size of the dataset and adapting an exact timeline
+for our dataset of vegetable imports from Peru should be a
+priority, in this way, we would undoubtedly obtain results
+with better relevance and reliability and, of course, the field of
+application with the use of machine learning techniques would
+be widely used and its results would be of a strongly necessary
+relevance.
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+5 | P a g e
+

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+page_content='unap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='pe  Torres-Cruz Fred  Faculty of Statistic and Computer Engineering,  Universidad Nacional del Altiplano de Puno, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Box 291  Puno - Peru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Email: ftorres@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='unap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='pe  Tumi-Figueroa Ernesto Nayer  Faculty of Statistic and Computer Engineering,  Universidad Nacional del Altiplano de Puno, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Box 291  Puno - Peru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Email: nayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='tumi@unap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='pe  Abstract—The current research work is being developed as a  training and evaluation object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' the performance of a predictive  model to apply it to the imports of vegetable products into  Peru using artificial intelligence algorithms, specifying for this  study the Machine Learning models: LSTM and PROPHET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  The forecast is made with data from the monthly record of  imports of vegetable products(in kilograms) from Peru, collected  from the years 2021 to 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' As part of applying the training  methodology for automatic learning algorithms, the exploration  and construction of an appropriate dataset according to the  parameters of a Time Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Subsequently, the model with better  performance will be selected, evaluating the precision of the  predicted values so that they account for sufficient reliability to  consider it a useful resource in the forecast of imports in Peru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Keywords—Machine Learning: forecasting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' time series;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' im-  ports;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='artificial intelligence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='dataset  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  INTRODUCTION  The Machine Learning methodology is one of the most  valuable technologies in the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='0, and advances in Ma-  chine Learning have provided significant benefits in strategic  decision-making in organizations[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' In recent years, predictive  models using Machine Learning algorithms have been imple-  mented in real environments of different organizations, and  as in this particular case that concerns us, it also operates in  the agriculture industry and optimal results are expected from  these technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Likewise, Machine Learning technology has a high poten-  tial to optimize your processes and, consequently, make correct  and anticipated decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' For this reason, Learning Algorithms  should also be considered as the next innovative direction to  significantly improve the prediction of these use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Therefore, this research will seek to demonstrate and  encourage the use of Machine Learning applications in the  field of agribusiness in Peru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Through the present work, the  examination and modeling of real data will be developed in  order to forecast future values with the least possible prediction  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' In this sense, this applied research work covers the  design, training, validation, and evaluation of 2 predictive  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' The research models to be explored are LSTM and  Prophet models, these being adequate and recommended to  develop applied foresting, on the one hand, LSTM is rec-  ognized for its hyperparameters which have the ability to  capture non-linear patterns using the grid search method[3]  and PROPHET for their modular regression with intuitive and  adjustable interpretation parameters applied on Time Series[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  DATASET  According to the purpose of the study, the official records  of the export volume (in kilograms) of vegetables in Peru were  used, collected from 2021 to 2022 in the National Open Data  Platform,[5], officially uploaded by the Ministry of Agrarian  Development and Irrigation of Peru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Table 1  Description of the variables in the Vegetable Import Dataset  in Peru 2021-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Based on this dataset, it is proposed to make a reliable  prediction, and according to the models to be developed, we  will therefore need the continuous quantitative variable, PESO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Also according to the observed data, we find it necessary to  adapt a proper timeline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' therefore, we must join the variables  A ˜NO and MES to foresee a more accurate seasonality, spec-  ifying the variables to forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Subsequently, to get a more  accurate overview of our data regarding products, we define  IDs for each product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  From the dataset, we highlight 3P: Soja, cake with 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='1%  in 1st position, in second position 5P: Apple, fresh fruit  with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='3% and in third position 27P: Soja, grain with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  The data ranges from 2021-05-01 to 2022-06-01, with 34,364  observations on imports of a total of 848 plant products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  1 | P a g e  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='03587v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='LG]  8 Jan 2023  VAR  SCALA  DESCRIPTION  1  ANO  CUANTITATIVA DISCRETA  Anodelaimportacion  2  MES  CUANTITATIVA DISCRETA  Mes de la importacion  3  SEDE  CUALITATIVA NOMINAL  Sede donde se registra de la Importacion  4  PRODUCTO  CUANTITATIVA DISCRETA  Productoimportado  7  PESO  CUANTITATIVA CONTINUA  Pesoenkilogramosdelproductc  8  TIPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='ENVASE  CUALITATIVA NOMINAL  Envasecontenedordelaimportacion  PAIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='ORIGEN  CUALITATIVA NOMINAL  Origendelaimportacion del productoFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Percentage description of each vegetable product imported in Peru  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  METHODS  In this section, we will describe the models used for the  predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' LSTM MODEL  The LSTM name refers to the analogy that a standard RNN  has both ’long-term memory’ and ’short-term memory’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' The  connection weights and biases in the network change once  per training episode, analogous to how physiological changes  in synaptic strengths store long-term memories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' activation  patterns in the network change once per time step, analogous  to how moment-to-moment changes in electrical activation  patterns in the general brain store short-term memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' The  LSTM architecture aims to provide a short-term memory for  RNNs that can last for thousands of time steps and continue  to be reliable, hence ’short-term long-term memory’[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  LSTM is considered a special type of recurrent neural  network (RNN), developed to solve the potential problem of  descending gradient found in traditional RNN training, and is  able to learn both short-term and long-term dependencies[7]  and is constructed of four main components: an entry gate, an  exit gate, memory cell and a forget gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Input Gate: controls the sending addition of information to  the cell state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' In other words, the gateway will consider what  information needs to be added to the cell state to ensure that  important information is added and that there is no redundant  information or noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Memory Cell: controls the value that might be deleted or  updated, and contains a value that might need to be kept as  additional information for many other time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Output Gate: controls the selection of useful learning  information from the current state of the cell as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Forget Gate: controls the removal of information that is no  longer needed for LTSM to learn things or, less importantly,  the state of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' This helps to optimize the performance  of the LSTM proposed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Also, The LSTM model follows the sequence:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Decide to discard the cell state information by a Sigmoid  layer called ”forget gate”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Decide new information to store in the cell state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' The  Sigmoid layer called the ”gateway layer” decides which values  will be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' the tan coat creates a vector of new candidate  values that could be added to the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Update the old cell state to the new cell state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Decide the filtered output based on the state of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  MODEL LSTM  Considered for a normal LSTM model is the Tanh property,  which is a non-linear activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' It regulates the values  that flow through the network, keeping the values between -  1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' To avoid information fading, a function is needed  whose second derivative can survive longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' There might be a  case where some values become huge, causing the values to  be irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  And of course an LSTM principal property: The Sigmoid  that belongs to the family of nonlinear activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Unlike Tanh, the Sigmoid holds values between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' It  helps the network update or forget data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' If the multiplication  results in 0, the information is considered forgotten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Similarly,  the information is kept if the value is relevant[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  This will assist the network in determining what data can  be lost and what data should be kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' PROPHET MODEL  PROPHET is a procedure for forecasting time series data  based on an additive model in which nonlinear trends are  adjusted for annual, weekly, and daily seasonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' It works best  with time series that have strong seasonal effects and multiple  seasons of historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  2 | P a g e  3P  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
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+page_content='2%  29P  64P  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='8%  59P  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='4%  77P  61P  23P  8P  91P  17P  84P  6P  18P  34p  57P  12P  44P  89P  22p  9P  28P  41P  70P  69P  26P  16P  50PForget gate  ht  Memory cell Ct  Ct-1  文  Ct  x  tanh  a  tanh  Hidden state ht  ht-1 -  X  >ht  Xt  Input gate  Output gateit=o(atU+ht-iW)  x, input  ft=o(atUf +ht-1Wf)  f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' forget gate  i，  inputgate  Ot = o(atUo + ht-1Wo)  cellupdate  c, cell state  Ct =tanh (ctU9+ht-iW9)  o,outputgate  Ct = o(ft* Ct-1 + it*Ct)  h,output  G sigmoidactivationfunction  ht = tanh(Ct) * Ot  tanh Tanh activationfunctionIn the specification of this Prophet model, there are several  places where we can alter the model to apply your experience  and external expert knowledge without needing to understand  the underlying statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' [4]  For the Prophet model with general theory:  We use a mathematical decomposable time series model[9]  and this model has these components: trend, seasonality, and  holidays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' They are combined into the following equation:  Series Model Formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Here g(t) is the trend function that models non-periodic  changes in the value of the time series, s(t) represents periodic  changes (for example, weekly and yearly seasonality), and h(t)  represents the effects of the seasons, The error term represents  any idiosyncratic changes that do not fit the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  In PROPHET we incorporate trend changes into the growth  model by explicitly determining the change points where the  growth rate is allowed to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Suppose there are exchange  points at moments j,j= 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' We give N a vector of adjust-  ments, where only the rate change occurs at moment j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' The  rate at any time is then the base rate k, plus all adjustments  up to the point of This is best represented by the vectors as  follows:  Adjustments represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  When rate k is adjusted, the parameter set must also be  adjusted to connect the endpoints of the segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' The correct  fit at the shift point is easily calculated as:  That would be the Adjust at shift point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  The logistic piece of the growth model then looks as  follows:  The Fourier series is also applied in Prophet to provide  a flexible model of periodic effects[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Let P be the regular  period that we expect the time series to have (for example, P=  365:25 for annual data or P= 7 for weekly data, when we scale  our indices of time variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' We can approximate arbitrary  uniform seasonal effects with this definition:  Fourier Definition  Sometimes we can’t just randomly split the data accord-  ingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' PROPHET develops simulated historical forecasts by  producing K forecasts at various cut-off points in history,  chosen such that the horizons are within the historical record  and the total error can be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  This procedure is based on the classic ’rolling source’ fore-  cast evaluation procedures[11], but uses only a small sequence  of target dates instead of forecasting by historical date) is that  it saves on computation and provides less correlated precision  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Work Sequence We apply an EDA (Exploratory Data Analysis) it is  necessary to clean the data and adapt it for the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Normalization of the data, we adapt the data to follow  a supervised sequence model according to a time  series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Let’s divide our dataset into proof and validation tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Coding and implementation of the models according  to the Prophet or LSTM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Adjustment of the parameters and extraction of pre-  dictions with their respective evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  RESULTS  The results using PROPHET and LSTM are shown in this  section together with the general comparison of the mentioned  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  We will also appreciate the comparisons and metrics de-  veloped according to the models applied to the study variable:  PESO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Result according to the validation of the LSTM model:  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  LSTM VALIDATION  3 | P a g e  PESO  600000  LSTM_Predictions  500000  400000  300000  20000  100000  0  00:45:38  00:45:40  00:45:42  00:45:44  00:45:46  00:45:48  Fechay(t) = g(t) + s(t) + h(t) + Et1,  if t≥sj,  a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='(t)  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='=(s-m-)(1  1-)C(t)s(t)-(an cos()+bsin())Result according to the validation of the PROPHET model:  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  PROPHET VALIDATION  Comparison to the validation of the PROPHET and LSTM  model:  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  LSTM AND PROPHET COMPARISON  MSE(mean squared error) and the RMSE(root mean  squared error) of both models:  We can appreciate each model according to its evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  RMSE Y MSE  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  DISCUSSION AND CONCLUSIONS  In keeping with the theory, all machine learning algorithms  are unique, which is the root cause of why the prediction  results are different algorithms on the same data set differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  LSTM was proposed for being an improvement of the RNNs  and Prophet for its versatility in data with less presence of  seasonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  Given what was taken into consideration for LSTM, the  applied model was with the minimum parameters and show  results with much better conditioning than was expected,  remembering that in different studies the superiority of LSTM  is detailed over the basic algorithms that apply Recurrent  Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Furthermore, it is noted from the theory  that the number of training times, known as the ’epoch’ in  learning[12], did not take into account the expected effect on  the performance of the trained forecast model and exhibited  mostly random behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' How intuitive The developed model  based on LSTM incorporates additional ’gates’ in order to store  longer sequences of input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  One of the main questions when developing and analyzing is  whether the gates incorporated in the LSTM architecture would  give a good prediction and if additional data training would  be needed to further improve the prediction[13] and in this  case, we can deduce that the quantity of data was acceptable  but the predictions were affected by the not very well defined  seasonality of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' A more effective solution would be  to add exact dates and continuous seasonalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  For the Prophet model, we should have some more  intuitive results according to its theory, the application of  the Fourier series could develop more precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' The model  is expected to obtain a reasonable forecast on disordered  data without too much manual effort, unlike LSTM, which  has more hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Prophet proved to be resistant  to outliers, missing data, and drastic changes in its time  series, the intention to fit the timeline is noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Compared  to other classical forecasting methods, Prophet should be  fast and easy to apply to time series, which is what it  was designed for in the first place;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' however, it could be  considerably less accurate[14] and in this case, we confirm  this appreciation by highlighting once again its intuitive factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  The Prophet procedure should include more parameters for  users to modify and adjust the forecasts in a more effective  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Also, a hybrid model could improve significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='  According to the RMSE results of the import predictions,  we can conclude that the LSTM model presents a significantly  better performance and reliability with respect to the Prophet  model, however, as we deduced previously, the seasonality  of the dataset was an important key in the variation of the  development of the models and their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content=' Therefore,  increasing the size of the dataset and adapting an exact timeline  for our dataset of vegetable imports from Peru should be a  priority, in this way, we would undoubtedly obtain results  with better relevance and reliability and, of course, the field of  application with the use of machine learning techniques would  be widely used and its results would be of a strongly necessary  relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
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+page_content='  4 | P a g e  1e6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='5  10  PESO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='5  00:45:40  00:45:45  Fecha1e6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'}
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+SCHOLAR RANKING 2023: RANKING OF COMPUTER SCIENCE
+DEPARTMENTS BASED ON FACULTY CITATIONS
+Sai Shi
+Department of Computer & Information Science
+Temple University
+Philadelphia, PA 19122
+[email protected]
+Aniruddha Maiti
+Department of Computer & Information Science
+Temple University
+Philadelphia, PA 19122
+[email protected]
+Ashis Kumar Chanda
+Department of Computer & Information Science
+Temple University
+Philadelphia, PA 19122
+[email protected]
+Slobodan Vucetic
+Department of Computer & Information Science
+Temple University
+Philadelphia, PA 19122
+[email protected]
+January 10, 2023
+ABSTRACT
+Scholar Ranking 2023 is the second edition of U.S. Computer Science (CS) departments ranking
+based on faculty citation measures. Using Google Scholar, we gathered data about publication
+citations for 5,574 tenure-track faculty from 185 U.S. universities. For each faculty, we extracted
+their t10 index, defined as the number of citations received by their 10th highest cited paper. For each
+department, we calculated four quality metrics: median t10 (m10), the geometric mean of t10 (g10),
+and the number of well-cited faculty with t10 above 40% (c40) and 60% (c60) of the national average.
+We fitted a linear regression model using those four measures to match the 2022 U.S. News ranking
+scores of CS doctoral programs. The resulting model provides Scholar Ranking 2023, which can be
+found at https://chi.temple.edu/csranking_scholar.
+1
+Introduction
+A previous version of the Scholar ranking [1] was published in the spring of 2017, based on citation data collected
+during the fall of 2016. This previous effort demonstrated that it is possible to learn a simple formula from citation
+measures that has a high correlation with peer assessment scores of CS doctoral programs published by the U.S. News
+(USN).
+A few years have passed since the last publication of the Scholar ranking, and a new U.S. News ranking came out
+in 20221. Given the fact that the data on which the last ranking was performed is a few years old, we felt that it would
+be helpful to conduct another round of data collection and validate our method with the recent U.S. News ranking. The
+first objective is to refine the data collection method and collect a new set of high-quality faculty citation data. The
+second objective is to use the 2022 U.S. News ranking to validate the method proposed in the first version of the scholar
+ranking and observe changes in the ranking. The third objective is to analyze the trends in aggregated metrics used to
+perform the ranking given the data sets, with the first collected during the fall of 2016 and the other during the fall of
+2021.
+1https://www.usnews.com/best-graduate-schools/top-science-schools/computer-science-rankings
+arXiv:2301.03140v1  [cs.DL]  9 Jan 2023
+
+A PREPRINT - JANUARY 10, 2023
+2
+Data collection
+In this section, we explain the data collection process which took place from September 2021 to December 2021.
+The data collection team consisted of two CS graduate students and a CS professor.
+2.1
+U.S. News (USN) Data
+USN is well-known for producing several rankings. We gathered the scores from the most recent ranking of CS
+doctoral programs, Best Computer Science Schools, which was published in 2022. USN collected the names of those to
+be surveyed for the science doctoral surveys in the summer of 2021. We retained the scores from the 2013 version of
+Best Computer Science Schools from previous data collection.
+USN ranks programs using scores generated from surveys sent to academic professionals2. Only survey responses
+from fall 2021 and early 2022 were used to compute the scores. The surveys asked respondents to rate each program
+from 1 to 5, with one being marginal and five being outstanding. Respondents could skip programs and select "don’t
+know" if they were unfamiliar with them. Each program’s score is the average of its survey ratings if it has at least ten
+ratings. Programs with less than ten ratings are not scored. Unlike the scores reported in USN 2017, where the program
+is ranked if it has a score of at least 2.0, USN 2022 published and ranked the scores of programs that are lower than
+2.0. USN does not provide raw survey data or information about potential sources of bias in responses. USN does not
+attempt to fill in missing values.
+2.2
+Computer Science Faculty List Data
+We collected the data on 5,574 tenure-track CS professors from 185 departments ranked by USN. We identified
+2,011 faculty on our 2022 list but not on our 2017 list, including 1,750 new professors and 261 professors who joined
+another department. In contrast, 4,728 professors were collected in 2017 from 173 departments. The number of CS
+professors included in our list increased by 17.89%.
+We consider a professor part of a department if the professor is listed on the website’s faculty list. We found each
+website by performing Google searches on the school’s name followed by "computer science" or "cs." In most cases,
+lists of faculty and their appointments were on pages labeled "Directory," "People," or "Faculty." Some pages did not
+specify appointments. In these cases, we found a professor’s appointment by performing a Google search on their name
+and exploring their website or profile page.
+We only consider tenure-track professors, which would have the rank of assistant, associate, or full professor.
+We excluded professors who have the titles "Clinical", "Courtesy", "Adjunct", "Research", "Teaching", "Emeritus",
+"Visiting", or other additional labels that indicate that the professor was not a tenure-track professor.
+For universities with CS departments, we added all professors because they were in a department for CS professors
+only. In some universities, the computer science faculty are part of joint departments called "Electrical Engineering
+and Computer Sciences" or "Computer Science and Engineering." Some universities have colleges or departments
+of computing or informatics, which contain faculty in CS, library science, information sciences, or management
+information systems. These departments made it harder to distinguish who was a CS professor. We determined that
+professors with research interests and publications in CS topics will be CS professors. We looked at the publications or
+research interests on their department profile page or website. CS topics include artificial intelligence, machine learning,
+data science, human-computer interaction, bioinformatics, cybersecurity, and others. Some cases we do not consider
+within CS are sensor networks, hardware, genomics, signals, and others.
+There were some unique cases in choosing the departments. New York University has the Department of
+Computer Science and Department of Computer Science and Engineering within two separate colleges. We only
+considered the department within the Courant Institute of Mathematical Sciences. Case Western University has the
+Department of Computer and Data Sciences and Department of Electrical, Computer and Systems Engineering. We
+only included professors within The Department of Computer and Data Sciences because the faculty of the other
+department generally had research interests that we excluded. Rochester Institute of Technology has a big college of
+Computing and Information Sciences, which consists of several departments, such as Computer Science, Computing
+and Information Sciences, Software Engineering, etc. We only included faculty from the department of Computer
+Science since they listed their faculty in separate departments.
+Another issue was affiliated professors and professors who have non-primary appointments. We did not include
+affiliated faculty if they were in a separate section from the main faculty or labeled as having a joint or secondary
+2https://www.usnews.com/education/best-graduate-schools/articles/science-schools-methodology
+2
+
+A PREPRINT - JANUARY 10, 2023
+appointment. In cases when affiliated, joint, or secondary appointed professors were mixed in with primary faculty, we
+added all professors who were on the list because the department listed them as its professors.
+Outside these departments, some professors have made significant contributions in CS venues but have primary
+appointments within engineering, biology, statistics, business, or other departments. We did not include non-CS
+professors because we needed a time-effective and unbiased method of finding these professors.
+In 2017, we found that 23.6% (1114/4728) of CS faculty are assistant professors, and that percentage has changed
+to 29.2% (1630/5574) in 2022. Since assistant professors are starting to establish their publications, we treat them
+differently from associate and full professors. We refer to associate and full professors as senior faculty, and our
+collection of Google Scholar data focused on senior faculty.
+2.3
+Google Scholar Data
+After determining what professors are in each department, we identified each professor’s Google Scholar page
+to collect their citation measures3. Despite some limitations of automated web crawling [2][3], the quality of Google
+Scholar data is comparable to the data coming from the subscription-based services for journal publications such as
+Web of Science [4]. A Google Scholar page lists the individual’s publications with each respective number of citations
+and overall aggregate citation measures. A profile’s aggregate citation measures include the h-index and i10-index
+(i10). The h-index [5] is defined as the highest number x for which the individual has x number of papers with at least x
+number of citations. The i10 is the total number of papers with above ten citations by an individual.
+We found each page by searching Google Scholar for the professor’s name and university. Some professors have
+common names, and multiple people appeared in the search result. We ensured that the professor’s rank, university, and
+research topics aligned with the page. Sometimes, the Google Scholar page would list a university where the professor
+was a previous student or faculty member. We confirmed that it was the correct page by looking at past affiliations
+through their websites or department website pages.
+We found about 89.8% (5,005/5,574) of professors’ Google Scholar pages. About 85.7% (3,379/3,944) of senior
+faculty have a page. We determined that using the data of professors who have Google Scholar pages is biased because
+they tend to have higher citation measures. To prevent bias from affecting the results, we decided to collect citation
+measures for professors who did not have a page. We introduced the t10 index, a citation measure that would be easy to
+collect for professors with or without a Google Scholar page. This measure is explained in the next section.
+2.4
+t10 Index
+The t10 index (t10) is defined as the number of citations of one’s 10th most cited paper. Identifying this index is
+more convenient and less prone to error than the h-index when performing a manual search. The t10 is obtained by
+identifying an individual’s ten most cited papers. In contrast, the h-index is obtained by identifying the top x number
+of papers that have at least x citations. Because assistant professors are starting to build their publication records, we
+decided to focus on collecting the t10 of the senior faculty.
+We obtained the t10 of the associate and full professors with a Google Scholar page using a web scraping program
+that takes the 10th highest paper on each page. This is a simple process because a Google Scholar page lists the
+individual’s highest-cited articles in descending order. We collected them manually for professors who do not have a
+Google Scholar page. To save time, we took the t10 that were gathered in 2017 and matched them to each professor
+who did not have a t10.
+To manually gather the t10, we searched Google Scholar for the professor’s name. The search engine typically
+retrieves publications with the name in the author list by descending the number of citations. We looked for the 10th
+highest cited paper from the search results with the professor’s name in the author list. For professors with common
+names, the search results would show publications from multiple people. We checked each publication to ensure the
+author was the correct professor.
+We identified the t10 of 5,553 of the 5,574 CS faculty (99.6% coverage) and for 3,932 of the 3,944 senior CS
+faculty (99.7% coverage) by manually searching Google Scholar. However, when a faculty has a common name or a
+faculty name listed on the people pages does not precisely match the name listed in their papers, obtaining t10-index
+can be too time-consuming. To save time, the curators were instructed to abort the extraction if it took more than 5
+minutes. As a result, we did not collect t10 for 21 of the 5,574 CS faculty (0.4%). Since a faculty’s name should not
+have an influence on their citation count, the resulting sample of faculty with known t10 can be treated as an unbiased
+sample of the senior CS faculty.
+3https://scholar.google.com/
+3
+
+A PREPRINT - JANUARY 10, 2023
+Furthermore, among 3,416 faculty in both the 2017 and 2022 data, we found that 2,520 (73.8% coverage) have an
+h-index and t10. 459 of them have added google scholar profiles since 2017, and none of them removed their google
+scholar profile. 926 of them were promoted during the past few years. Among 1,750 new faculty in our 2022 data,
+1,636 have a google scholar profile, where 205 are full professors, 196 are associate professors, and 1,235 are assistant
+professors. 1,741 of these new faculty have t10, where 252 are full professors, 219 are associate professors, and 1,270
+are assistant professors.
+3
+Methods
+3.1
+Program Strength Measures
+We propose two approaches for using individual citation measures to calculate the strength of a program, averaged
+and cumulative citation measures, the same as those used in 2017.
+3.2
+Averaged citation measures
+The first method we use to measure the strength of a program is by averaging citations of its faculty members [6].
+We use three different averaging schemes. The first averaging scheme is the median of t10 values of senior CS faculty,
+which we denote as m10. The second is the geometric mean of (1+t10) values of senior CS faculty, which we denote
+as g10. The third is the average percentile of the senior faculty’s t10, which we denote as p10. We exclude assistant
+professors from the averaged measures because their citations are typically smaller, and their inclusion would hurt
+departments with more assistant professors.
+3.3
+Cumulative citation measures
+The second method to measure the strength of a program is to count the number of highly cited faculty in a
+program. We define a t10 threshold to determine which professors are highly cited. We introduce cN, which denotes the
+number of faculty whose t10 is higher than N% of senior faculty, with 0 < N < 100. For example, c40 is the count of
+professors within a department with a t10 higher than 40% of senior faculty. We include all faculty to calculate cN.
+3.4
+Regression Models
+We use regression models that combine the averaged and cumulative citation measures into an aggregated score.
+The regression models that we consider are of the type in formula 1:
+si = β0 + β1ai + β2ci,
+(1)
+si is the predicted USN CS score, ai is an aggregated citation measure, and ci is a cumulative citation measure of
+the i-th program. The regression parameters are β0, β1, and β2. Instead of learning the intercept parameter β0, we set it
+to β0 = 1 by default. The primary justification is that a program with ai = 0 and si = 0 does not have active research
+faculty; based on the peer assessment instructions by USN, this program would have a score of 1 ("marginal"). The
+secondary justification is that the resulting regression models would be simpler because they only require fitting two
+regression parameters, β1, and β2. We train one regression model for each combination of the three averaged and the
+ranges of cumulative measures. We average the individual regression models to create an aggregate score.
+4
+Results
+4.1
+Department size
+Fig.1 compares the number of assistant, associate, and full professors between our newly collected data and that
+from our 2017 paper. The percentage of assistant professors increased during the past four years, which indicates that
+more young professors are joining CS academia. The reason might be that recently popular areas, such as machine
+learning or deep learning, are attracting more young professors to contribute to research.
+Fig.2 shows the distribution of department sizes, defined as the number of tenure-track faculty in each of the 185
+CS programs. The median faculty size is 23, the mode is 20, the minimum is 3, and the maximum is 170 (Carnegie
+Mellon University). We also show the scatter plot between the department size in 2017 and 2022 in Fig.3. The colors
+represent the USN CS scores, and it can be seen there is an overall increase in department sizes. The department
+4
+
+A PREPRINT - JANUARY 10, 2023
+Figure 1: Trend of faculty size
+Figure 2: The distribution of the U.S. CS department size
+size in 2022 has a similar distribution to that in 2017, since there is a nearly linear relationship between the two. The
+correlation coefficient of the department sizes and the USN CS scores of the 185 programs is 0.755, which is higher
+than our previous finding (0.676), indicating that larger departments are more likely to be ranked higher.
+4.2
+Distribution of the Citation Indices
+As shown in Fig.4, the median value of the h-index, i10-index, and t10 index all increased compared to our 2017
+result. Particularly, the t10 index has a larger rise compared to the h-index and i10-index, indicating that it is a more
+sensitive citation metric.
+In Fig.5, we show the histogram of the t10 for the 3,944 senior faculty. We observe a similar distribution pattern
+compared to the 2017 result. Since the t10 distribution resembles a lognormal distribution, the histogram of t10 is
+shown in a log scale. We observe a bump at low values, representing the 70 senior faculty with a t10 of 0, meaning they
+have less than ten cited papers listed in Google Scholar. The median of t10 is 114, and the percentiles of t10 are shown
+in Table 1. Overall, the t10 values increased compared to our results from 2017.
+5
+
+60
+old data
+new data
+49.6%
+50
+46.8%
+40
+Percentage
+29.2%
+30
+26.9%
+24.0%
+23.6%
+20
+10
+0
+Full professor
+Associate professor
+Assistant professor50
+45
+40
+35
+30
+uno
+25
+20
+15
+10
+5
+0
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Faculty SizeA PREPRINT - JANUARY 10, 2023
+Figure 3: Department size in 2016 and 2022
+Figure 4: Trend of citation measurements (median value)
+The correlation coefficient between logarithms of h-index and t10 for the 3,379 senior faculty with both indices is
+0.943, which is close to our 2017 result. The sufficiently high correlation concludes that the t10 is a good proxy for the
+h-index.
+Table 1: Percentiles of t10
+Percentile
+t10
+10%
+21
+20%
+40
+30%
+60
+40%
+83
+50%
+115
+60%
+154
+Percentile
+t10
+70%
+212
+80%
+301
+90%
+493
+95%
+751
+98%
+1247
+99%
+1843
+6
+
+Score.
+175
+150
+(2022)
+125
+100
+3
+size
+Faculty 
+75
+2
+50 
+25
+1
+F 0
+20
+-0
+0
+40
+60
+80
+100
+120
+140
+Faculty size (2016)80
+old data
+75
+new data
+70
+62
+60
+indices
+50
+48
+le
+44
+40
+30
+27
+25
+20
+10
+0
+h-index
+il0-index
+tl0-indexA PREPRINT - JANUARY 10, 2023
+Figure 5: Histogram of t10 of associate and full CS professors
+Figure 6: Number of tenured CS faculty with (blue) and without (orange) Google scholar profile as a function of the t10
+percentile
+4.3
+Scholar profile bias
+While the median of t10 for the 3,932 senior CS faculty is 114, it increases to 131 among the 3,379 professors
+who have a Google scholar profile and drops to 33 among the 553 without a profile. In Fig. 6 we show a stacked bar
+plot of the numbers of faculty with and without Google scholar profiles as a function of their t10 percentile. These
+results are consistent with the observations in our 2017 paper, which indicate that CS faculty who have Google scholar
+profiles are a biased sample of the entire CS faculty and validate our effort to gather the t10 values and use them in our
+study instead of the h-index.
+4.4
+Scholar model
+According to the above analysis, it can be observed our 2022 data shows a similar pattern and trend compared to
+our 2017 data. Hence, we decided to keep the same linear regression scholar model used in 2017 and directly applied it
+to our 2022 data, which is shown in formula 2:
+s = 1 + 0.058
+√
+m10 + 0.059
+�
+g10 + 0.121
+√
+c40 + 0.127
+√
+c60
+(2)
+7
+
+600
+500
+400
+Count
+300
+200
+100
+0
+0.5
+1.5
+2
+2.5
+3
+3.5
+4
+log10(t10)450
+400
+350
+300
+Count
+250
+200
+150
+100
+50
+0
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+-
+Percentile RangeA PREPRINT - JANUARY 10, 2023
+Figure 7: Comparison of scholar model scores and USN CS scores of CS graduate programs.
+In Fig 7, we show a scatter plot of the USN CS scores and scholar model scores for the 185 CS programs. A
+closer look at the scatter plot reveals that two groups of CS programs can be distinguished with respect to the correlation
+between the USN CS scores and scholar model scores. The first group contains 73 programs that were scored 2.8
+and higher by the USN. The correlation between the USN CS scores and joint model scores in this group is 0.914.
+The second group contains 112 programs with USN scores between 0 and 2.7. The correlation between the USN CS
+scores and scholar model scores in this low-scoring group is 0.673, which is much higher than what we observed in
+2017 (0.360) but still lower than those 73 programs. We hypothesize that the CS programs whose USN CS scores are
+between 0 and 2.8 are not sufficiently well-known among the peers to provide objective and reliable peer assessment at
+the national level.
+4.5
+Comparison Study
+We first study the newly added professor list to compare our new ranking result with the previous one conducted
+in 2017. Among those 1,750 new professors, it is observed that 1,518 (86.7%) of them have a t10-index lower than
+the average t10-index of all faculty from the department they join in. 223 of them are higher, and 9 of them have a
+missing t10-index. Furthermore, among those 261 professors who transferred to another department, 114 have joined
+a higher-ranking department than their previous department, and 147 have joined a lower-ranking department. We
+also compared their t10-index in 2017 with the previous department average before they moved. It is an interesting
+finding that only 66 have a t10-index higher than the department average, and 176 have a t10-index lower or equal to
+the department average. 19 of them have a missing t10-index. Based on these observations, it can be inferred that new
+professors are primarily young, with a lower t10-index. It is also true that young professors at the starting stage of their
+careers are more likely to transfer to another university, and most of them tend to join a higher-ranking university. The
+story might be they built a stronger profile during the past 5 years and then joined a higher-ranking department.
+4.5.1
+2017 ranking vs. 2022 ranking
+To better understand how the ranking has changed during the past five years, we calculate the ranking difference
+between the results we produced in 2017 and 2022 using the same scholar model created in 2017. Fig. 8 shows the
+box plot of the absolute difference between our old (2017) and new (2022) scholar ranking results. It can be seen that
+the placements of top universities above rank 90 are more stable, while lower-ranked universities tend to have more
+significant variations. However, we observe some outliers. For example, Northeastern University jumped from rank 40
+to 21 because this department may have recruited many new professors since 2017.
+Another observation is that larger departments are not necessarily ranked higher. For example, Stanford University
+and Princeton University both have relatively small departments. Still, they are within the top 10 departments in our
+ranking, indicating that other factors, such as faculty citations, significantly influence the ranking result.
+8
+
+5.5
+5
+00
+O
+000
+4.5
+8
+o
+4
+900
+o
+8
+o
+8
+o
+08
+Model
+08
+80
+3
+000
+.
+2.5
+08
+009
+00
+00
+8000
+08
+1.59
+8000
+8
+o
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+5.5
+U.S.News ScoreA PREPRINT - JANUARY 10, 2023
+Figure 8: Absolute value of ranking difference between our 2017 and 2022 scholar ranking results
+Figure 9: Absolute value of ranking difference between USN 2017 and USN 2022 results
+Fig. 9 shows the absolute difference between the USN 2017 and USN 2022 ranking. It can be observed that the
+ranking difference has a similar pattern compared to what has been shown in Fig. 8. This indicates that our method
+using the t-10 index of faculty to rank the program produces similar ranking results compared to USN.
+4.5.2
+Scholar ranking vs. USN
+The new ranking results and a comparison to the USN scores are shown in the Appendix. To justify that it
+is appropriate to apply the linear regression model obtained from the 2017 data to the 2022 data, we calculated the
+correlations between our scholar ranking result and the USN ranking in 2017 and 2022, respectively. The results are
+shown in Table 2.
+Table 2: Correlation between USN ranking and our scholar ranking using the 2017 regression model
+R2
+Pearson
+Spearman
+USN 2017 vs. scholar ranking 2017
+0.8731
+0.9357
+0.8978
+USN 2022 vs. scholar ranking 2022
+0.8734
+0.9390
+0.9126
+Despite the fact that we used the same regression formula for both 2017 and 2022 data, our scholar ranking results
+show a high correlation with both USN 2017 and USN 2022 rankings. This result confirms that our practice of using
+the 2017 ranking formula on our 2022 data is justifiable.
+To further investigate the relationship between our scholar ranking and the USN ranking, we calculated the
+difference between our new ranking and the USN 2022 ranking. Fig. 10 shows the boxplot of the absolute value of
+difference. We separate universities into six groups based on their new scholar rank. It can be observed that our ranking
+9
+
+50
+40
+Rank difference
+30
+20
+10
+0
+Rank 1-30
+Rank 31-60
+Rank 61-90
+Rank 91-120
+University ranking (2022)50
+40
+0
+Rank difference
+30
+0
+20
+10
+0
+Rank 1-30
+Rank 31-60
+Rank 61-90
+Rank91-120
+University ranking (2022)A PREPRINT - JANUARY 10, 2023
+Figure 10: Absolute value of ranking difference between our 2022 scholar ranking and USN 2022 ranking
+Figure 11: Histogram plot of ranking score increase between 2022 and 2017
+model is more likely to match with the USN ranking for higher-ranking departments. For instance, all departments that
+rank 1-30 show a rank difference of less than 10.
+We also calculate the ranking score increase in 2022 compared to 2017 for both our scholar model and USN and
+show the histogram plots in Fig. 11. It can be seen that most departments have a score increase in both the scholar
+model and USN, and all of them are within the range between -0.4 and 0.8. There are a few extreme cases. For example,
+the ranking score of Northeastern University increased by 0.8; where the reason may be the department recruited many
+new faculty during the past few years. UNC Chapel Hill has a score decrease of 0.3. Their department size remains the
+same (32), but the m10, g10, c40, and c60 values drop in 2022.
+4.5.3
+Scholar ranking vs. CSRankings
+Table 3: Correlation between CSRankings, USN 2022 and our scholar ranking
+R2
+Pearson
+Spearman
+USN vs. scholar ranking
+0.8734
+0.9390
+0.9126
+USN vs. CSRankings
+0.8391
+0.9160
+0.9157
+scholar ranking vs. CSRankings
+0.8375
+0.9151
+0.8965
+USN vs. average model
+0.8462
+0.9199
+0.9305
+To further investigate the relationship between our ranking and some widely-used CS ranking results, we compared
+the CSRankings result with our ranking. Unlike USN ranking, CSRankings4 relies on publications in top-tier computer
+science conferences, as reported by DBLP, a computer science bibliography website 5. To study the relationship between
+CSRankings and our scholar ranking, we collected the current CSRankings result and calculated its correlations with
+4http://csrankings.org
+5https://dblp.org/
+10
+
+50
+40
+Rank difference
+30
+20
+10
+0
+Rank1-30
+Rank 31-60
+Rank 61-90
+Rank 91-120
+Universityranking(2022)Histogram of ranking score increase in 2022 compared to 2017
+Scholarranking
+USN ranking
+35
+30
+25
+20 
+Count
+15
+10
+5
+0
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+Ranking score increase
+Ranking score increaseA PREPRINT - JANUARY 10, 2023
+Figure 12: Absolute value of ranking difference between CSRankings and USN 2022
+Figure 13: Absolute value of ranking difference between CSRankings and our scholar ranking
+USN and our scholar ranking. Since the CSRanking score is based on a different scale, we applied log transformation
+to its score before computing the correlations. The result is shown in Table 3.
+It can be seen from the Table that our scholar ranking result has a higher correlation with the USN ranking, which
+indicates that it is better aligned with the USN ranking compared to CSRankings. This indication is further justified
+in Fig. 10 and Fig. 12, where we show the ranking difference between 1) our scholar ranking results with USN and
+2) CSRankings with USN. It can be seen that CSRankings shows an overall more significant ranking difference with
+USN compared to our scholar model ranking, especially for the top 60 departments. An interesting finding is that
+CSRankings is better aligned with USN than our scholar ranking for those lower-ranking departments (>90). The
+ranking difference between CSRankings and our scholar model ranking is shown in Fig. 13. The result shows that the
+difference increases with a larger variance as the ranking of the department decreases. This indicates that the ranking
+of top departments is more stable despite which model is being used, whereas lower-ranking departments are more
+sensitive to the ranking method. We also build an average model by computing the average score using the scholar
+model score and CSRankings score (shown in the last row of Table), which yields a higher spearman correlation with
+the USN ranking.
+References
+[1] Slobodan Vucetic, Ashis Kumar Chanda, Shanshan Zhang, Tian Bai, and Aniruddha Maiti. Faculty citation measures
+are highly correlated with peer assessment of computer science doctoral programs. ArXiv, abs/1708.05435, 2017.
+[2] Emilio Delgado López-Cózar, Nicolás Robinson-García, and Daniel Torres-Salinas. The Google scholar experiment:
+How to index false papers and manipulate bibliometric indicators. Journal of the Association for Information
+Science & Technology, 65(3):446–454, March 2014.
+[3] Péter Jacsó. Deflated, inflated and phantom citation counts. Online Inf. Rev., 30:297–309, 2006.
+11
+
+50
+40
+30
+0
+0
+20
+8
+10
+0
+Rank 1-30
+Rank 31-60
+Rank 61-90
+Rank 91-120
+Universityranking(2022)50
+40
+0
+30
+20
+0
+Rank
+0
+10
+0
+Rank 1-30
+Rank 31-60
+Rank 61-90
+Rank91-120
+University ranking (2022)A PREPRINT - JANUARY 10, 2023
+[4] Judit Bar-Ilan. Which h-index?—a comparison of wos, scopus and google scholar. Scientometrics, 74:257–271, 02
+2008.
+[5] J. E. Hirsch. An index to quantify an individual’s scientific research output. Proceedings of the National Academy
+of Sciences, 102(46):16569–16572, 2005.
+[6] Themis Lazaridis. Ranking university departments using the mean h-index. Scientometrics, 82:211–216, 2009.
+12
+
+A PREPRINT - JANUARY 10, 2023
+Table 4: List of 185 U.S. CS graduate programs: Ranking by our scholar model (Rank), University name (University),
+Number of tenured faculty with t10 score (Size), median t10 score of all the faculty (M10), geometric mean of t10 score
+of all faculty (G10), number of highly cited faculty based on c40 (C40) and c60 (C60), U.S. News CS score (USN),
+Scholar score (Scholar)
+Rank
+University
+Size
+M10
+G10
+C40
+C60
+USN
+Scholar
+USN 2016
+Scholar 2016
+1
+Carnegie Mellon University
+170
+280
+262
+121
+84
+4.9
+5
+5
+5
+1
+Cornell University
+102
+327
+315
+75
+58
+4.6
+5
+4.5
+4.4
+1
+Massachusetts Institute of Technology
+110
+304
+302
+92
+74
+5
+5
+5
+5
+1
+Stanford University
+64
+707
+706
+60
+53
+4.9
+5
+5
+5
+1
+University of California Berkeley
+82
+421
+461
+68
+60
+4.9
+5
+5
+5
+6
+University of Washington
+73
+345
+295
+58
+47
+4.6
+4.9
+4.5
+4.3
+7
+University of California San Diego
+63
+341
+330
+49
+39
+4.3
+4.8
+4
+4.2
+8
+Georgia Institute of Technology
+105
+203
+198
+75
+53
+4.6
+4.6
+4.3
+4.3
+8
+Princeton University
+48
+360
+345
+36
+30
+4.5
+4.6
+4.4
+4.1
+8
+University of Illinois Urbana Champaign
+76
+265
+248
+56
+43
+4.7
+4.6
+4.6
+4.1
+11
+University of California Los Angeles
+44
+317
+316
+35
+31
+4.3
+4.5
+4.1
+4.2
+11
+University of Pennsylvania
+67
+256
+238
+49
+39
+4.1
+4.5
+3.8
+3.7
+13
+Columbia University
+52
+249
+288
+42
+34
+4.3
+4.4
+4
+4.1
+13
+New York University
+44
+288
+254
+38
+29
+3.5
+4.4
+3.4
+4
+13
+University of Michigan Ann Arbor
+67
+251
+254
+47
+35
+4.3
+4.4
+4.1
+4.1
+16
+Harvard University
+34
+277
+286
+28
+22
+4.2
+4.2
+3.9
+3.7
+17
+Duke University
+56
+216
+195
+37
+26
+3.9
+4.1
+3.6
+3.6
+17
+Johns Hopkins University
+32
+258
+309
+23
+17
+4
+4.1
+3.5
+4
+17
+University of Maryland College Park
+56
+219
+186
+37
+29
+4.1
+4.1
+4
+4
+17
+University of Wisconsin Madison
+45
+285
+225
+28
+20
+4.1
+4.1
+4.2
+3.9
+21
+Northeastern University
+77
+165
+180
+45
+30
+3.6
+4
+2.7
+3.2
+21
+University of Southern California
+40
+272
+238
+26
+19
+3.9
+4
+3.7
+3.9
+21
+University of Texas Austin
+53
+182
+184
+38
+25
+4.5
+4
+4.3
+3.7
+24
+Brown University
+31
+224
+233
+22
+17
+3.8
+3.9
+3.7
+3.5
+24
+University of Massachusetts Amherst
+59
+191
+187
+33
+22
+3.9
+3.9
+3.6
+3.7
+26
+University of Chicago
+48
+174
+208
+29
+17
+3.7
+3.8
+3.3
+3.5
+26
+Yale University
+26
+243
+231
+19
+14
+4
+3.8
+3.7
+4
+28
+California Institute of Technology
+21
+240
+240
+16
+12
+4.3
+3.7
+4.2
+3.7
+28
+Pennsylvania State University University Park
+44
+200
+176
+25
+17
+3.6
+3.7
+3.4
+3.4
+28
+Rice University
+28
+213
+204
+18
+13
+3.7
+3.7
+3.7
+3.3
+28
+University of California Santa Barbara
+37
+183
+194
+25
+15
+3.7
+3.7
+3.3
+3.6
+28
+University of Minnesota Twin Cities
+46
+145
+186
+30
+15
+3.6
+3.7
+3.4
+3.4
+33
+Purdue University West Lafayette
+72
+137
+133
+31
+17
+4
+3.6
+3.7
+3.3
+33
+Stony Brook University SUNY
+45
+152
+137
+27
+17
+3.4
+3.6
+3.1
+3.3
+33
+University of California Davis
+37
+178
+162
+23
+18
+3.4
+3.6
+3.3
+3.5
+33
+University of California Irvine
+50
+152
+132
+28
+19
+3.7
+3.6
+3.4
+3.4
+33
+University of Virginia
+40
+193
+179
+20
+14
+3.7
+3.6
+3.4
+3.1
+38
+University of California Santa Cruz
+39
+185
+153
+20
+14
+3.2
+3.5
+2.8
+3.5
+39
+Boston University
+33
+141
+167
+18
+11
+3.3
+3.4
+3
+3.2
+39
+Michigan State University
+39
+151
+159
+20
+12
+3
+3.4
+2.8
+3
+13
+
+A PREPRINT - JANUARY 10, 2023
+39
+Northwestern University
+44
+122
+123
+26
+12
+3.7
+3.4
+3.3
+3.1
+39
+Rutgers University
+39
+151
+152
+18
+10
+3.5
+3.4
+3.3
+3.3
+39
+University of Arizona
+22
+197
+168
+14
+9
+3.2
+3.4
+3.1
+3.2
+39
+University of California Riverside
+34
+153
+133
+22
+12
+3
+3.4
+2.8
+3.3
+45
+Arizona State University
+54
+92
+122
+21
+16
+3.2
+3.3
+3
+2.9
+45
+University of Colorado Boulder
+51
+127
+100
+24
+12
+3.5
+3.3
+3.1
+3
+45
+University of Rochester
+18
+186
+160
+10
+7
+3.1
+3.3
+2.9
+3
+45
+University of Utah
+54
+117
+121
+22
+12
+3.4
+3.3
+3.1
+3
+45
+Vanderbilt University
+26
+164
+155
+13
+10
+3.1
+3.3
+2.8
+2.9
+45
+Washington University in St Louis
+28
+135
+150
+17
+10
+3.4
+3.3
+3.1
+2.9
+51
+University of North Carolina Chapel Hill
+32
+139
+122
+16
+10
+3.8
+3.2
+3.6
+3.5
+51
+University of Notre Dame
+27
+126
+144
+17
+7
+3.2
+3.2
+2.7
+2.7
+53
+Colorado State University
+22
+115
+136
+14
+7
+2.5
+3.1
+2.4
+3
+53
+George Mason University
+47
+100
+99
+21
+10
+2.9
+3.1
+2.5
+2.9
+53
+Indiana University Bloomington
+36
+103
+99
+17
+9
+3.1
+3.1
+2.9
+3
+53
+Ohio State University
+44
+99
+84
+22
+11
+3.6
+3.1
+3.3
+3.1
+53
+Texas AM University College Station
+48
+94
+90
+22
+9
+3.5
+3.1
+3.1
+2.9
+53
+University of Central Florida
+37
+89
+100
+18
+10
+2.8
+3.1
+2.2
+2.6
+53
+University of Tennessee Knoxville
+29
+125
+114
+13
+9
+2.5
+3.1
+2.4
+3
+53
+University of Texas Dallas
+52
+88
+76
+24
+15
+2.9
+3.1
+2.4
+2.9
+53
+Virginia Tech
+56
+93
+99
+24
+9
+3.5
+3.1
+3.1
+3
+62
+College of William and Mary
+21
+136
+140
+9
+6
+2.8
+3
+2.4
+2.8
+62
+Lehigh University
+20
+148
+129
+9
+5
+2.5
+3
+2.1
+2.7
+62
+Temple University
+23
+130
+103
+12
+7
+2.4
+3
+2
+2.6
+62
+University at Buffalo SUNY
+39
+96
+90
+17
+9
+2.9
+3
+2.6
+3
+62
+University of Florida
+47
+82
+79
+18
+11
+3.4
+3
+3
+2.7
+67
+North Carolina State University
+43
+82
+83
+16
+9
+3.2
+2.9
+3
+2.9
+67
+Rensselaer Polytechnic Institute
+19
+89
+134
+9
+6
+3.1
+2.9
+2.9
+2.8
+67
+University of Maryland Baltimore County
+29
+94
+101
+12
+6
+2.7
+2.9
+2.4
+2.7
+67
+University of Pittsburgh
+23
+104
+104
+11
+6
+3.1
+2.9
+2.9
+2.8
+71
+CUNY Graduate School and University Center
+99
+37
+43
+25
+13
+2.1
+2.8
+2.3
+2.6
+71
+Dartmouth College
+20
+97
+110
+8
+3
+3.2
+2.8
+3.1
+2.7
+71
+Oregon State University
+49
+84
+69
+17
+4
+2.9
+2.8
+2.5
+2.3
+71
+University of Houston
+24
+95
+88
+13
+4
+2.2
+2.8
+2.1
+2.4
+71
+University of Illinois Chicago
+39
+85
+100
+13
+5
+3
+2.8
+2.7
+2.7
+71
+University of Texas Arlington
+37
+85
+69
+13
+6
+2.5
+2.8
+2.2
+2.7
+71
+Wayne State University
+22
+109
+89
+11
+3
+2.3
+2.8
+2
+2.4
+78
+New Jersey Institute of Technology
+30
+70
+70
+11
+6
+2.5
+2.7
+2.2
+2.4
+78
+Portland State University
+19
+94
+71
+7
+6
+2
+2.7
+0
+2.7
+78
+Tufts University
+21
+81
+95
+8
+3
+2.8
+2.7
+2.4
+2.4
+78
+University of Memphis
+18
+88
+99
+7
+5
+1.8
+2.7
+0
+2.4
+78
+University of New Mexico
+16
+93
+89
+7
+3
+2.4
+2.7
+2.3
+2.4
+78
+University of California Merced
+20
+91
+106
+7
+4
+2.2
+2.7
+84
+Binghamton University SUNY
+27
+76
+56
+11
+4
+2.4
+2.6
+2
+2.2
+14
+
+A PREPRINT - JANUARY 10, 2023
+84
+Drexel University
+15
+94
+80
+6
+2
+2.7
+2.6
+2.2
+2.4
+84
+Illinois Institute of Technology
+23
+74
+65
+9
+4
+2.4
+2.6
+2.1
+2.5
+84
+University of Connecticut
+28
+102
+78
+9
+1
+2.7
+2.6
+2.3
+2.3
+84
+University of Georgia
+24
+81
+65
+8
+4
+2.5
+2.6
+2.2
+2.2
+84
+University of Massachusetts Lowell
+26
+90
+47
+7
+5
+2.1
+2.6
+0
+2.1
+84
+University of Missouri
+29
+62
+65
+11
+4
+2.3
+2.6
+2.1
+2.4
+84
+University of Nebraska Lincoln
+31
+78
+68
+10
+3
+2.6
+2.6
+2.4
+2.6
+84
+University of Oregon
+20
+97
+93
+7
+2
+2.7
+2.6
+2.6
+2.2
+84
+University of South Florida
+26
+60
+77
+7
+6
+2.3
+2.6
+2.1
+2.5
+84
+Washington State University
+22
+75
+85
+8
+3
+2.7
+2.6
+2.4
+2
+84
+West Virginia University
+11
+117
+62
+6
+4
+2
+2.6
+2
+2.3
+84
+Worcester Polytechnic Institute
+32
+69
+76
+9
+4
+2.5
+2.6
+2.2
+2.4
+97
+Case Western Reserve University
+16
+91
+68
+7
+2
+2.9
+2.5
+2.4
+2.4
+97
+Florida Atlantic University
+27
+53
+55
+11
+4
+1.9
+2.5
+0
+2.1
+97
+Florida International University
+34
+60
+42
+12
+4
+2.1
+2.5
+0
+2.2
+97
+Georgia State University
+28
+59
+65
+9
+3
+2.1
+2.5
+2
+2.3
+97
+Iowa State University
+29
+70
+63
+8
+3
+2.9
+2.5
+2.6
+2.2
+97
+Syracuse University
+25
+86
+48
+9
+2
+2.8
+2.5
+2.5
+2.2
+97
+University of Delaware
+31
+73
+54
+6
+4
+2.6
+2.5
+2.4
+2.5
+97
+University of Iowa
+21
+79
+82
+6
+2
+2.8
+2.5
+2.6
+2.3
+97
+University of Massachusetts Boston
+17
+81
+82
+6
+1
+2.2
+2.5
+0
+2
+97
+University of South Carolina
+28
+65
+73
+10
+1
+2.3
+2.5
+2.1
+2.2
+97
+University of Vermont
+11
+113
+95
+2
+2
+1.9
+2.5
+108
+Brandeis University
+18
+67
+36
+7
+4
+2.3
+2.4
+2.3
+2.3
+108
+Brigham Young University
+36
+66
+32
+8
+3
+2.4
+2.4
+2.2
+2.3
+108
+Clemson University
+38
+62
+51
+8
+3
+2.6
+2.4
+2.3
+2.2
+108
+Florida State University
+24
+74
+62
+8
+1
+2.6
+2.4
+2.3
+2.1
+108
+Kansas State University
+16
+66
+61
+6
+2
+2.3
+2.4
+2.2
+1.9
+108
+University of Alabama Birmingham
+9
+94
+80
+3
+1
+2
+2.4
+0
+2
+108
+University of North Texas
+33
+61
+63
+8
+2
+1.9
+2.4
+0
+1.8
+115
+George Washington University
+12
+67
+59
+3
+2
+2.7
+2.3
+2.3
+2.1
+115
+Louisiana State University
+18
+54
+46
+5
+2
+2.1
+2.3
+2.1
+2.2
+115
+University of Nevada Reno
+18
+61
+64
+3
+1
+1.8
+2.3
+0
+1.9
+115
+University of North Carolina Charlotte
+19
+53
+55
+4
+2
+2.4
+2.3
+2.1
+1.9
+115
+University of Oklahoma
+24
+51
+54
+4
+2
+2.2
+2.3
+2
+1.9
+115
+Utah State University
+17
+65
+70
+4
+1
+2
+2.3
+0
+1.8
+115
+Virginia Commonwealth University
+23
+58
+50
+7
+1
+2.1
+2.3
+0
+2
+115
+Emory University
+14
+64
+45
+3
+3
+2.7
+2.3
+123
+Colorado School of Mines
+15
+70
+61
+4
+0
+2.6
+2.2
+2.1
+2
+123
+Missouri University of Science Technology
+14
+46
+57
+3
+1
+2.1
+2.2
+2
+2.2
+123
+University of Arkansas Fayetteville
+22
+59
+51
+4
+1
+1.9
+2.2
+0
+2
+123
+University of Hawaii Manoa
+19
+29
+39
+7
+3
+2
+2.2
+0
+2.2
+123
+University of Kansas
+19
+60
+56
+2
+1
+2.5
+2.2
+2.3
+2.3
+123
+University of Texas San Antonio
+20
+54
+52
+3
+1
+2.1
+2.2
+0
+2.3
+15
+
+A PREPRINT - JANUARY 10, 2023
+129
+Mississippi State University
+16
+48
+38
+2
+1
+1.9
+2.1
+0
+1.6
+129
+Naval Postgraduate School
+20
+45
+39
+2
+1
+0
+2.1
+2.4
+2
+129
+Old Dominion University
+21
+34
+46
+4
+1
+2
+2.1
+0
+2
+129
+Oregon Health and Science University
+7
+66
+76
+1
+0
+1.8
+2.1
+2.2
+1.8
+129
+Stevens Institute of Technology
+20
+54
+53
+3
+0
+2.6
+2.1
+2.1
+2.1
+129
+University of Missouri Kansas City
+16
+61
+52
+3
+0
+1.8
+2.1
+0
+1.6
+129
+University of Texas El Paso
+19
+34
+40
+4
+1
+1.6
+2.1
+0
+1.9
+129
+University of Tulsa
+15
+37
+53
+3
+1
+1.8
+2.1
+0
+2
+129
+Western Michigan University
+10
+51
+34
+2
+1
+1.5
+2.1
+0
+1.6
+138
+Auburn University
+22
+39
+29
+2
+1
+2.5
+2
+2.2
+1.7
+138
+Claremont Graduate University
+5
+66
+43
+2
+0
+1.6
+2
+0
+1.9
+138
+DePaul University
+51
+24
+22
+4
+2
+1.9
+2
+0
+2
+138
+Kent State University
+20
+39
+27
+3
+1
+1.9
+2
+0
+1.7
+138
+New Mexico State University
+14
+59
+50
+2
+0
+2
+2
+0
+1.9
+138
+Texas Tech University
+17
+41
+34
+1
+1
+2.1
+2
+0
+1.7
+138
+University at Albany SUNY
+14
+62
+47
+1
+0
+2.2
+2
+2.1
+2.2
+138
+University of Alabama
+16
+24
+29
+4
+2
+2.3
+2
+0
+2
+138
+University of Colorado Colorado Springs
+15
+32
+39
+2
+2
+2
+2
+0
+1.9
+138
+University of Denver
+9
+56
+51
+1
+0
+1.9
+2
+0
+1.8
+138
+University of Kentucky
+19
+59
+32
+3
+0
+2.3
+2
+2.2
+2.2
+138
+University of Louisville
+18
+32
+25
+4
+2
+1.8
+2
+0
+1.8
+138
+Ohio University
+16
+49
+33
+3
+0
+1.9
+2
+151
+University of Louisiana Lafayette
+22
+30
+24
+2
+1
+1.9
+1.9
+0
+1.8
+151
+University of Wisconsin Milwaukee
+12
+57
+44
+1
+0
+2
+1.9
+0
+1.8
+153
+Florida Institute of Technology
+27
+19
+18
+1
+1
+1.8
+1.8
+0
+1.7
+153
+Oakland University
+23
+16
+25
+2
+1
+1.5
+1.8
+0
+1.9
+153
+Oklahoma State University
+10
+18
+24
+2
+1
+2
+1.8
+0
+1.4
+153
+Southern Methodist University
+8
+49
+20
+2
+0
+2.1
+1.8
+2
+1.6
+153
+University of Cincinnati
+20
+33
+29
+2
+0
+2.2
+1.8
+2
+1.8
+153
+University of Maine
+9
+33
+31
+2
+0
+1.8
+1.8
+0
+2
+153
+University of Mississippi
+7
+33
+32
+2
+0
+1.9
+1.8
+0
+1.6
+153
+Clarkson University
+10
+37
+31
+2
+0
+1.7
+1.8
+161
+Montana State University
+10
+30
+34
+0
+0
+1.7
+1.7
+0
+1.6
+161
+North Dakota State University
+12
+38
+32
+0
+0
+1.5
+1.7
+0
+1.5
+161
+University of Idaho
+14
+33
+31
+0
+0
+1.8
+1.7
+0
+1.5
+161
+University of New Orleans
+13
+30
+22
+1
+0
+1.5
+1.7
+0
+1.7
+161
+Rochester Institute of Technology
+24
+26
+20
+2
+0
+2.7
+1.7
+161
+San Diego State University
+13
+41
+15
+1
+0
+1.7
+1.7
+167
+Michigan Technological University
+22
+22
+28
+0
+0
+2.1
+1.6
+0
+1.5
+167
+New Mexico Institute of Mining and Technology
+7
+24
+25
+0
+0
+0
+1.6
+0
+1.4
+167
+Nova Southeastern University
+17
+14
+12
+3
+0
+1.3
+1.6
+0
+1.4
+167
+Towson University
+31
+15
+13
+1
+0
+1.6
+1.6
+0
+1.5
+167
+University of Southern Mississippi
+12
+10
+13
+1
+1
+1.4
+1.6
+0
+1.5
+167
+University of Wyoming
+7
+30
+25
+0
+0
+1.6
+1.6
+0
+1.6
+16
+
+A PREPRINT - JANUARY 10, 2023
+167
+Southern Illinois University
+12
+27
+28
+0
+0
+1.6
+1.6
+167
+University of New Hampshire
+9
+24
+23
+0
+0
+2
+1.6
+167
+University of Puerto Rico Mayaguez
+9
+12
+14
+2
+0
+1.4
+1.6
+167
+University of Rhode Island
+10
+17
+14
+1
+0
+2
+1.6
+177
+Air Force Institute of Technology
+6
+17
+18
+0
+0
+1.8
+1.5
+0
+1.5
+177
+Louisiana Tech University
+6
+25
+15
+0
+0
+1.6
+1.5
+0
+1.3
+177
+University of Alabama Huntsville
+13
+21
+23
+0
+0
+1.8
+1.5
+0
+1.7
+177
+University of Colorado Denver
+14
+13
+11
+1
+0
+1.9
+1.5
+0
+1.4
+181
+University of Nebraska Omaha
+16
+15
+11
+0
+0
+1.7
+1.4
+0
+1.4
+182
+University of Arkansas Little Rock
+6
+10
+7
+0
+0
+1.7
+1.3
+0
+1.7
+183
+Bowie State University
+12
+2
+5
+0
+0
+1.4
+1.2
+184
+Indiana State University
+6
+0
+2
+0
+0
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+Efficient cooling of high-angular-momentum systems
+Mark G. Raizen and Logan E. Hillberry
+Department of Physics, The University of Texas at Austin, Austin, Texas, 78712, USA
+Dmitry Budker
+Johannes Gutenberg-Universit¨at Mainz, Helmholtz-Institut Mainz,
+GSI Helmholtzzentrum f¨ur Schwerionenforschung, 55128 Mainz, Germany and
+Department of Physics, University of California, Berkeley, California 94720, USA
+Simon M. Rochester
+Rochester Scientific, LLC, El Cerrito, California 94530, USA
+(Dated: January 31, 2023)
+We propose a highly efficient and fast method of translational cooling for high-angular-momentum
+atoms or molecules. Internal-state optical pumping and stimulated optical transitions, combined
+with magnetic forces, can be used to compress phase-space density, and the efficiency of each com-
+pression step increases with the angular momentum. Entropy is removed by spontaneously emitted
+photons, and particle number is conserved. This method may be an attractive alternative to evap-
+orative cooling of atoms and molecules in order to produce quantum degenerate gases.
+INTRODUCTION
+Laser cooling, first proposed almost half a century
+ago, remains the standard approach for producing ultra-
+cold atoms. This method relies on momentum transfer
+from light to atoms as photons are repeatedly scattered,
+enabling the production and study of ultracold atomic
+gases. Many improvements on basic laser cooling have
+advanced the state of the art, including Sisyphus cooling
+[1–4], and subrecoil cooling [5, 6].
+While laser cooling works extremely well, the require-
+ment of a closed, two-level transition has limited the ap-
+plicability of the method to a subset of elements in the
+periodic table. For those atoms, after many years of re-
+finement, laser cooling has reached saturation in its per-
+formance due to multiple scattering of resonant photons
+which create an effective repulsive interaction between
+the atoms, pushing them apart.
+An important figure-
+of-merit is the phase-space density, a dimensionless pa-
+rameter which is the product of number density and the
+third power of the average de Broglie wavelength. Laser
+cooling typically produces a phase-space density of 10−6.
+This is also the starting point for the formation of Bose-
+Einstein condensates through evaporative cooling in a
+trap, and the creation of the so-called atom laser [7–9].
+Evaporative cooling is even more restrictive than laser
+cooling, as it relies on elastic collisions between atoms to
+maintain thermal equilibrium as the hottest atoms are
+ejected.
+Inelastic channels create unwanted losses and
+often make evaporative cooling impossible. Even when
+working optimally, evaporative cooling is a slow process
+and results in a significant loss of atom number. Cooling
+of molecules has become the focus of much effort in re-
+cent years, but is severely hampered by the difficulty of
+implementing the above cooling methods.
+In recent years, alternative approaches to producing
+Optical
+pumping
+Stimulated
+transitions
+Magnetic
+forces
+(a)
+(b)
+FIG. 1. A schematic depiction of the MOP-cooling sequence
+for ensembles with (a) angular momentum J = 1 and (b)
+J = 2. The boxes represent atoms, initially trapped in a flat,
+hard-wall potential.
+The colors represent the atom’s mag-
+netic state (orange: mJ = −J to purple: mJ = 0 to teal:
+mJ = +J). A cycle begins with optically pumping all atoms
+to the same state. Then, stimulated transitions correlate the
+magnetic states with position along the direction of compres-
+sion. A sequence of one-dimensional magnetic kicks pushes
+atoms of oppositely-signed magnetic states together. The cy-
+cle is closed by optically pumping the compressed atoms back
+into the same magnetic state. A cycle’s compression factor is
+limited by the number of available magnetic states.
+cold atoms and molecules were developed (see [10] and
+references therein). The starting point for much of this
+work is a supersonic molecular beam where desired atoms
+or molecules can be entrained in the flow and stopped in
+a series of pulsed magnetic or electric fields.
+Alterna-
+tively, cold atoms and molecules are produced by buffer-
+arXiv:2301.13121v1  [physics.atom-ph]  30 Jan 2023
+
+2
+gas cooling. After stopping, these atoms and molecules
+can be trapped, typically in a magnetic field configura-
+tion that confines the low-field seekers to the center of
+the trap. In parallel, cooling of atoms with a one-way
+wall was proposed and demonstrated [11, 12], and re-
+lies on photon entropy, not momentum as in laser cool-
+ing. The one-way wall is the first practical realization
+of Maxwell’s demon for an ensemble of atoms.
+While
+one-way-wall cooling demonstrated a large increase in
+phase-space density, it did not conserve atom number.
+To address this limitation, we proposed [13] a variation
+which we called magneto-optical (MOP) cooling, relying
+on cycles of optical pumping and magnetic kicks.
+In this paper, we present a new and highly efficient
+version of MOP cooling that can work for high-angular-
+momentum systems of atoms and molecules and offers
+an attractive alternative to laser cooling and evaporative
+cooling. Our method is depicted schematically in Fig. 1
+and is described in detail in the following section. We
+conclude the paper with a discussion of possible limita-
+tions to our new method, how those limitations may be
+overcome, and the significance of MOP cooling in the
+atomic physics toolbox.
+MOP COOLING
+MOP cooling is a conceptually new method that does
+not rely on the momentum of the photon, making it
+completely different from laser cooling.
+The key ben-
+efit of this approach is its universality and simplicity,
+since it relies only on optical pumping of an atomic or
+molecular internal magnetic state, combined with mag-
+netic forces from pulsed coils. We evaluate the efficacy of
+MOP cooling through numeric simulations. In this sec-
+tion, our simulation methodology is described, followed
+by a brief review of MOP cooling for a spin-1/2 system,
+as was originally proposed [13]. Finally, a new and much
+more efficient version of MOP cooling for high-angular-
+momentum systems is considered.
+Our
+MOP-cooling
+simulations
+track
+the
+three-
+dimensional positions x, velocities v, and magnetic states
+mJ ∈ {−2J, −2J + 1, . . . , 2J} of a sample of N = 105
+atoms. Positions are initialized from a flat distribution
+of a 0.5 cm width.
+Velocities are initialized from the
+Maxwell-Boltzmann distribution corresponding to a tem-
+perature of 25×Trec where Trec = h2/2mkBλ2 is the recoil
+temperature imposed by, e.g., a magneto-optical trap op-
+erating at wavelength λ to trap a species of mass m. Here
+h is Planck’s constant and kB is Boltzmann’s constant.
+A fourth-order Runge-Kutta algorithm updates the po-
+sition and velocity of each atom subject to the force
+F(t) = −mJgJµB∇ |B(x, t)| where gJ is the Land´e g-
+factor of the atom, µB is the Bohr magneton, and B is
+the pulsed magnetic field arranged to provide the one-
+dimensional kick. The internal state of each atom is set
+in accordance with the MOP-cooling cycle. The magnetic
+field is produced with two sets of coaxial coil pairs, one
+in the Maxwell configuration to provide a strong gradient
+[14] and the other in the Helmholtz configuration to shift
+the zero-crossing of the field away from the trap center,
+thereby providing a nearly-one-dimensional kick. We use
+superposition of the exact solution for a current-carrying
+loop to model the full coil geometry. B is evaluated on
+a dense grid for unit current. Vector interpolation allows
+us to evaluate ∇ |B(x)| at arbitrary positions.
+Time-
+dependent current pulses are modeled by scaling the field
+gradient interpolation result by I(t) = I0 sin[(t−t0)/2πτ]
+for t ∈ [t0, t0 + τ] and I(t) = 0 otherwise, where I0 is the
+peak current, t0 is the pulse delay, and τ is the pulse
+width.
+Table I reports the atomic properties used in this
+study. Additionally, the simulated coil parameters are
+as follows.
+There are 7 turns × 2 layers, or 14 loops
+per Helmholtz coil, each with a nominal radius of
+RHH = 3.5 cm, axially-separated by RHH, and carry-
+ing identically-oriented currents . There are 5 turns ×
+2 layers, or 10 loops per Maxwell coil, each with a nom-
+inal radius of RHH/
+√
+3, separated by RHH, and carry-
+ing oppositely-oriented currents. The peak currents are
+I0,HH = 1000 A for the Helmholtz coils and I0,M = 500 A
+for the Maxwell coils.
+The shared midpoint between
+the coil pairs is displaced by 0.2 cm in the positive z-
+direction from the initial center of mass of the atomic
+sample (taken as the origin of the coordinate system).
+We find that this region provides a more uniform and
+one-dimensional kick.
+TABLE I. Atomic properties used for simulation purposes.
+The Ref. column provides an example of magneto-optical trap
+operation for each species. Values for gJ are provided in [15],
+except for Li for which we adopt the electron’s value.
+Atom
+m
+J
+gJ
+λ
+Trec Ref.
+(10−26 kg)
+(nm)
+(µK)
+Li
+1.15
+1/2
+2.002 32
+671
+76.2
+[16]
+Cr
+8.63
+3
+2.001 83
+425
+25.3
+[17]
+Er
+27.8
+6
+1.163 81
+583
+4.2
+[18]
+Dy
+27.0
+8
+1.241 59
+626
+3.7
+[19]
+For a specific example, consider atomic lithium (Li)
+trapped using standard techniques [16].
+At sufficient
+magnetic fields, the electronic spin (J = 1/2) decou-
+ples from the nuclear spin; the two electronic mJ states
+are denoted |1/2⟩ and |−1/2⟩ and it is in this high-field
+regime that we propose MOP cooling. The cooling se-
+quence starts with suddenly turning off the trap so that
+the atoms are free. In step (1) of MOP cooling, all of the
+atoms are optically pumped to the |1/2⟩ state.
+Then,
+half of the cloud is transferred to the |−1/2⟩ state by
+stimulated transitions , i.e., stimulated Raman adiabatic
+passage (STIRAP) sequences [20, 21], thereby creating
+
+3
+−2
+0
+2
+vz (cm/s)
+(a)
+−50
+0
+50
+(b)
+−50
+0
+50
+(c)
+−2
+0
+2
+(d)
+−0.25
+0.00
+0.25
+z (cm)
+−0.1
+0.0
+0.1
+0.2
+vz (cm/s)
+(e)
+−0.25
+0.00
+0.25
+z (cm)
+−20
+0
+20
+(f)
+−0.25
+0.00
+0.25
+z (cm)
+−20
+0
+20
+(g)
+−0.25
+0.00
+0.25
+z (cm)
+−0.1
+0.0
+0.1
+0.2
+(h)
+−J
+0
+J
+mJ
+FIG. 2.
+MOP cooling simulations for Li (a-d) and Dy (e-h) visualized in phase space. Each column of plots represents a
+snapshot in the cycle. First, the magnetic states are correlated with position along the z-axis through optical pumping followed
+by spatially-resolved coherent population transfer via stimulated transitions. The vertical dashed black lines in panels (a) and
+(e) mark the ideal boundary between different magnetic states. Second, a one-dimensional magnetic kick accelerates atoms to a
+velocity that is proportional to their magnetic state. Third, after waiting an optimized delay time the phase space distribution
+has been compressed in real space but remains extended in velocity space. Finally, a reverse kick returns the atom’s velocity
+distribution to near its original extent. More precisely, we find the standard deviation of the ensemble’s velocity is, at most,
+about 11% of its initial value for all four species simulated.
+two spatially-distinct populations [see Fig. 2 (a)]. In step
+(2), a magnetic-field-gradient kick is applied to the cloud,
+thereby causing the two halves to merge [Fig. 2 (b-c)],
+and then a reverse kick returns the atoms to their origi-
+nal velocity distribution [Fig. 2 (d)]. As proposed in [13]
+and demonstrated experimentally in [22], such magnetic
+kicks can be applied along a single axis while minimally
+affecting the other two dimensions.
+Two-dimensional
+slices (Fig. 3) of the magnetic field gradient used in our
+simulations clearly show the one-dimensional-nature of
+the magnetic kick. In step (3), all atoms are optically-
+pumped back to the |1/2⟩ state, thereby completing the
+cooling cycle. In principle, a factor of 2× in phase-space
+compression is possible.
+We now generalize the method to a system with an
+arbitrary total angular momentum J. There are 2J + 1
+states in this case, and we assume that the atoms are
+trapped in a hard-walled flat potential. Just as in the
+case of Li above, we turn off the trap to start the cooling
+sequence. Using optical pumping and stimulated transi-
+tions in step (1), the cloud is divided into 2J + 1 com-
+ponents, where the leftmost section is prepared in state
+|J, −J⟩, then |J, –J + 1⟩, and so on, to the rightmost sec-
+tion in state |J, J⟩. In step (2), a magnetic field gradi-
+ent kick is applied to the cloud, causing each sub-section
+to move at a velocity that is proportional to the mag-
+netic quantum number.
+Thus, the leftmost section in
+state |J, J⟩ will move the fastest to the right, and lower
+magnetic-quantum-numbered sections will move slower.
+The cloud will collapse to a single section after an opti-
+mal delay time, and a reverse magnetic kick will restore
+the original velocity distributions. Finally, in step (3),
+all atoms would be optically pumped to the |J, J⟩ state,
+and the cloud can be re-trapped.
+Figures 2 (e - h) show snapshots of the simulated phase
+space for MOP cooling of Dy atoms (J = 8). Compar-
+ing the final spatial distribution of Li [Fig. 2 (d)] to that
+of Dy [Fig. 2 (h)] clearly demonstrates how MOP cool-
+ing may leverage high-angular momentum systems for
+efficient phase-space compression. The delay time is op-
+timized in our numeric simulations and the results are
+shown for a variety of atoms in Fig. 4.
+As a figure of
+merit, we compute the compression factor as the ratio
+of standard deviations between the initial and final z-
+coordinate distributions in the cloud. The inset of Fig. 4
+shows the peak compression factor for each species is
+bound by the geometric limit 2J + 1. In the following
+section we consider limitations of MOP cooling and how
+they may be overcome in an experiment.
+DISCUSSION
+In MOP cooling, a maximum compression factor of
+2J + 1 per cycle may be approached. However, in prac-
+tice, the efficiency will be lower due to deviations from
+a flat density distribution, imperfect kicking fields, and
+photon-recoil heating during the optical pumping (OP)
+stage.
+The flat-density initial condition is quite different from
+
+4
+−0.5
+0.0
+0.5
+y (cm)
+−0.5
+0.0
+0.5
+z (cm)
+(a) x = −0.25 cm
+944
+952
+952
+952
+960
+960
+−0.5
+0.0
+0.5
+y (cm)
+(b)
+x = 0.00 cm
+936
+944
+952
+952
+952
+952
+960
+960
+−0.5
+0.0
+0.5
+y (cm)
+(c)
+x = 0.25 cm
+944
+952
+952
+952
+960
+960
+−0.5
+0.0
+0.5
+x (cm)
+−0.5
+0.0
+0.5
+y (cm)
+(d) z = −0.25 cm
+15
+30
+45
+45
+45
+45
+−0.5
+0.0
+0.5
+x (cm)
+(e)
+z = 0.00 cm
+10
+20
+30
+40
+40
+40
+40
+−0.5
+0.0
+0.5
+x (cm)
+(f)
+z = 0.25 cm
+8
+16
+24
+32
+40
+40
+40
+40
+900
+920
+940
+960
+980
+∇|B| (G/cm)
+0
+20
+40
+60
+80
+∇|B| (G/cm)
+FIG. 3. Two-dimensional slices of the magnetic field gradient used for MOP cooling simulations, evaluated at peak current.
+The red dashed line marks the initial extent of the atomic cloud. The quadrupole field provided by Maxwell coils is symmetric
+under a sign change of any coordinate axis. However, the bias field provided by the Helmholtz coils breaks the symmetry along
+the z-axis by shifting the center of the quadrupole off of the coordinate origin. The magnitude of the total field |B| varies
+primarily along z in a nearly-linear fashion.
+0
+10
+20
+30
+40
+Unkick delay (ms)
+0
+5
+10
+15
+Compression factor
+0
+4
+8
+J
+0
+5
+10
+15
+Li
+Cr
+Er
+Dy
+FIG. 4. Optimizing the wait time between the MOP cool-
+ing kick and unkick according to the compression factor. The
+open circle marks the optimal delay time for each species sub-
+ject to the initial conditions and magnetic forces described in
+the text. The inset shows the peak compression factor vs the
+species’ angular momentum J, compared to the geometric
+limit 2J + 1 (black line).
+that usually encountered with laser cooled atoms in a
+magneto-optical trap, but turns out to be important for
+the gains in efficiency that are predicted. For example,
+our simulations predict a compression factor of 1.9 for
+Li initialized with with a flat density or 1.66 for a Gaus-
+sian distribution. For Dy initialized with a flat density,
+a peak compression factor of 13.8 is observed, while for
+an initially-Gaussian distribution the factor reduces to
+only 4.3. To obtain a flat “boxlike” distribution in an ex-
+periment, the atoms can first be confined in a magnetic
+quadrupole trap. An optical box can be created around
+the atoms using a time-averaged optical dipole potential
+from beams that are moving rapidly in two dimensions.
+Such potentials were created in the past to study opti-
+cal billiards [23, 24] and BECs in painted potentials [25].
+After trapping, the box can be adiabatically expanded in
+three dimensions to a desired size, which will result in a
+nearly flat density profile of atoms. The optical setup for
+state preparation of each segment would enable multiple
+cycles of cooling in each dimension. Adiabatic expansion
+would lower the kinetic energy and MOP cooling would
+compress the cloud spatially. The process can be dynam-
+ically controlled with motorized zoom lenses [26].
+Ideally, a complete cycle of MOP cooling leaves the
+velocity distribution of the atomic sample unchanged.
+This means that the momentum imparted on the atoms
+in the kick phase must be nulled in the reverse-kick
+
+5
+phase.
+Inhomogeneities in the kicking field will result
+in a nonzero mean velocity for the cloud.
+Such inho-
+mogeneities are noticeable in Fig. 3 (b) that shows the
+peak magnetic field gradient in the x = 0 plane.
+For
+instance, atoms near z = 0.25 will be kicked downward
+with less force than their upward motion-arresting kick
+that is applied once they are near z = 0. This is why
+Fig. 2 (h) shows a net positive velocity, which is partic-
+ularly evident for mJ = J (teal). In our simulation, the
+final mean velocity of the Dy atoms is ∼ 0.03 cm/s. In
+practice, this is not a significant limitation because the
+resulting net velocity is still well within the capture range
+of any reasonable trap as it is smaller than the thermal
+velocity. Over the same time scale t, the relative ther-
+mal expansion of the cloud σ(t)/σ0 ≈
+�
+1 + t2kBT/mσ2
+0
+is insignificant compared to the MOP cooling compres-
+sion factor σ0/σ(t) ≈ 2J + 1. Moreover, there exist ad-
+vanced wiring-design-optimization techniques to gener-
+ate uniform bias or gradient fields with minimal induc-
+tance [14, 27]. Though originally developed for magnetic
+resonance imaging, MOP cooling could benefit from such
+analyses to improve switching times of the pulsed mag-
+netic fields and increase the uniformity of the required
+biased gradients.
+A more significant mean velocity is incurred due to
+free fall in the Earth gravitational field. For example,
+in the ∼ 19 ms of delay time required for MOP cooling
+of Cr, the cloud accelerates to nearly 19 cm/s and dis-
+places 0.18 cm.
+Depending on the details of the trap,
+such velocity or displacement may result in significant
+atom loss. To mitigate the free fall effects, one could use
+the MOP cooling setup to apply an additional uniform
+kicks against gravity.
+In general, optical pumping the cloud to the same mag-
+netic state is a lossy step due to, for instance, atoms de-
+caying to unobserved trap states. Fortunately, there ex-
+ist efficient techniques for optical pumping as discussed
+in [28], where it is shown that it is possible to perform
+optical pumping with only one spontaneous photon emit-
+ted per atom.
+The same physics lets us understand
+MOP cooling’s high phase-space compression efficiency
+in terms of the photon entropy carried away from the
+ensemble by spontaneous emission [29].
+The entropy associated with the motional degrees of
+freedom is given by Smo. = kB ln V , where V is the phase-
+space volume.
+This volume is reduced by a factor of
+(at most) 2J + 1 at each cooling step, so the maximum
+entropy reduction per step is ∆Smo. = kB ln (2J + 1).
+The magnetic kicks are reversible evolution, so they do
+not produce a net change in entropy. Therefore the en-
+tropy change must occur during the optical pumping
+step.
+Optical pumping of an unpolarized ensemble to
+produce a pure state reduces the polarization entropy
+by ∆Spol. = kB ln (2J + 1). Thus for each step, (1) OP
+takes unpolarized state to pure state, reducing polariza-
+tion entropy by kB ln (2J + 1); (2) kicks take pure state
+to unpolarized state by overlapping ensembles, increas-
+ing polarization entropy by kB ln (2J + 1) and reducing
+motional entropy by the same amount. The cooling will
+be limited by recoil heating.
+From Ref. [28], we know
+that OP can theoretically be done with only one sponta-
+neously emitted photon per atom. If this is achieved, the
+temperature limit will correspond to the recoil energy,
+i.e., the standard recoil limit Trec. For less efficient OP
+the temperature limit will be higher.
+CONCLUSION
+In this paper, we proposed a highly efficient method
+for phase-space compression of high-angular momentum
+atomic and molecular samples.
+This work extends an
+earlier MOP-cooling proposal from Li to general high-
+angular-momentum systems. We numerically tested our
+new MOP-cooling protocol on four atomic species of in-
+creasing angular momenta that have each already been
+cooled using traditional techniques. We find, for exam-
+ple, an impressive compression factor of 13.5 is conceiv-
+ably attainable in less than 13 ms for the case of Dy.
+ACKNOWLEDGEMENTS
+The
+work
+of
+MGR
+was
+supported
+by
+the
+Sid
+W. Richardson Foundation. The work of DB was sup-
+ported by the Deutsche Forschungsgemeinschaft (DFG)
+- Project ID 423116110 and by the Cluster of Excellence
+Precision Physics, Fundamental Interactions, and Struc-
+ture of Matter (PRISMA+ EXC 2118/1) funded by the
+DFG within the German Excellence Strategy (Project ID
+39083149).
+
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+

M9FPT4oBgHgl3EQflTVS/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf,len=534
+page_content='Efficient cooling of high-angular-momentum systems  Mark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Raizen and Logan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Hillberry  Department of Physics, The University of Texas at Austin, Austin, Texas, 78712, USA  Dmitry Budker  Johannes Gutenberg-Universit¨at Mainz, Helmholtz-Institut Mainz,  GSI Helmholtzzentrum f¨ur Schwerionenforschung, 55128 Mainz, Germany and  Department of Physics, University of California, Berkeley, California 94720, USA  Simon M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Rochester  Rochester Scientific, LLC, El Cerrito, California 94530, USA  (Dated: January 31, 2023)  We propose a highly efficient and fast method of translational cooling for high-angular-momentum  atoms or molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Internal-state optical pumping and stimulated optical transitions, combined  with magnetic forces, can be used to compress phase-space density, and the efficiency of each com-  pression step increases with the angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Entropy is removed by spontaneously emitted  photons, and particle number is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' This method may be an attractive alternative to evap-  orative cooling of atoms and molecules in order to produce quantum degenerate gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  INTRODUCTION  Laser cooling, first proposed almost half a century  ago, remains the standard approach for producing ultra-  cold atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' This method relies on momentum transfer  from light to atoms as photons are repeatedly scattered,  enabling the production and study of ultracold atomic  gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Many improvements on basic laser cooling have  advanced the state of the art, including Sisyphus cooling  [1–4], and subrecoil cooling [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  While laser cooling works extremely well, the require-  ment of a closed, two-level transition has limited the ap-  plicability of the method to a subset of elements in the  periodic table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' For those atoms, after many years of re-  finement, laser cooling has reached saturation in its per-  formance due to multiple scattering of resonant photons  which create an effective repulsive interaction between  the atoms, pushing them apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  An important figure-  of-merit is the phase-space density, a dimensionless pa-  rameter which is the product of number density and the  third power of the average de Broglie wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Laser  cooling typically produces a phase-space density of 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  This is also the starting point for the formation of Bose-  Einstein condensates through evaporative cooling in a  trap, and the creation of the so-called atom laser [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Evaporative cooling is even more restrictive than laser  cooling, as it relies on elastic collisions between atoms to  maintain thermal equilibrium as the hottest atoms are  ejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Inelastic channels create unwanted losses and  often make evaporative cooling impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Even when  working optimally, evaporative cooling is a slow process  and results in a significant loss of atom number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Cooling  of molecules has become the focus of much effort in re-  cent years, but is severely hampered by the difficulty of  implementing the above cooling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  In recent years, alternative approaches to producing  Optical  pumping  Stimulated  transitions  Magnetic  forces  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' A schematic depiction of the MOP-cooling sequence  for ensembles with (a) angular momentum J = 1 and (b)  J = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The boxes represent atoms, initially trapped in a flat,  hard-wall potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  The colors represent the atom’s mag-  netic state (orange: mJ = −J to purple: mJ = 0 to teal:  mJ = +J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' A cycle begins with optically pumping all atoms  to the same state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Then, stimulated transitions correlate the  magnetic states with position along the direction of compres-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' A sequence of one-dimensional magnetic kicks pushes  atoms of oppositely-signed magnetic states together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The cy-  cle is closed by optically pumping the compressed atoms back  into the same magnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' A cycle’s compression factor is  limited by the number of available magnetic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  cold atoms and molecules were developed (see [10] and  references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The starting point for much of this  work is a supersonic molecular beam where desired atoms  or molecules can be entrained in the flow and stopped in  a series of pulsed magnetic or electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Alterna-  tively, cold atoms and molecules are produced by buffer-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='13121v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='atom-ph]  30 Jan 2023  2  gas cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' After stopping, these atoms and molecules  can be trapped, typically in a magnetic field configura-  tion that confines the low-field seekers to the center of  the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' In parallel, cooling of atoms with a one-way  wall was proposed and demonstrated [11, 12], and re-  lies on photon entropy, not momentum as in laser cool-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The one-way wall is the first practical realization  of Maxwell’s demon for an ensemble of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  While  one-way-wall cooling demonstrated a large increase in  phase-space density, it did not conserve atom number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  To address this limitation, we proposed [13] a variation  which we called magneto-optical (MOP) cooling, relying  on cycles of optical pumping and magnetic kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  In this paper, we present a new and highly efficient  version of MOP cooling that can work for high-angular-  momentum systems of atoms and molecules and offers  an attractive alternative to laser cooling and evaporative  cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Our method is depicted schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 1  and is described in detail in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' We  conclude the paper with a discussion of possible limita-  tions to our new method, how those limitations may be  overcome, and the significance of MOP cooling in the  atomic physics toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  MOP COOLING  MOP cooling is a conceptually new method that does  not rely on the momentum of the photon, making it  completely different from laser cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  The key ben-  efit of this approach is its universality and simplicity,  since it relies only on optical pumping of an atomic or  molecular internal magnetic state, combined with mag-  netic forces from pulsed coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' We evaluate the efficacy of  MOP cooling through numeric simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' In this sec-  tion, our simulation methodology is described, followed  by a brief review of MOP cooling for a spin-1/2 system,  as was originally proposed [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Finally, a new and much  more efficient version of MOP cooling for high-angular-  momentum systems is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Our  MOP-cooling  simulations  track  the  three-  dimensional positions x, velocities v, and magnetic states  mJ ∈ {−2J, −2J + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' , 2J} of a sample of N = 105  atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Positions are initialized from a flat distribution  of a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='5 cm width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Velocities are initialized from the  Maxwell-Boltzmann distribution corresponding to a tem-  perature of 25×Trec where Trec = h2/2mkBλ2 is the recoil  temperature imposed by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=', a magneto-optical trap op-  erating at wavelength λ to trap a species of mass m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Here  h is Planck’s constant and kB is Boltzmann’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  A fourth-order Runge-Kutta algorithm updates the po-  sition and velocity of each atom subject to the force  F(t) = −mJgJµB∇ |B(x, t)| where gJ is the Land´e g-  factor of the atom, µB is the Bohr magneton, and B is  the pulsed magnetic field arranged to provide the one-  dimensional kick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The internal state of each atom is set  in accordance with the MOP-cooling cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The magnetic  field is produced with two sets of coaxial coil pairs, one  in the Maxwell configuration to provide a strong gradient  [14] and the other in the Helmholtz configuration to shift  the zero-crossing of the field away from the trap center,  thereby providing a nearly-one-dimensional kick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' We use  superposition of the exact solution for a current-carrying  loop to model the full coil geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' B is evaluated on  a dense grid for unit current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Vector interpolation allows  us to evaluate ∇ |B(x)| at arbitrary positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Time-  dependent current pulses are modeled by scaling the field  gradient interpolation result by I(t) = I0 sin[(t−t0)/2πτ]  for t ∈ [t0, t0 + τ] and I(t) = 0 otherwise, where I0 is the  peak current, t0 is the pulse delay, and τ is the pulse  width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Table I reports the atomic properties used in this  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Additionally, the simulated coil parameters are  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  There are 7 turns × 2 layers, or 14 loops  per Helmholtz coil, each with a nominal radius of  RHH = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='5 cm, axially-separated by RHH, and carry-  ing identically-oriented currents .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' There are 5 turns ×  2 layers, or 10 loops per Maxwell coil, each with a nom-  inal radius of RHH/  √  3, separated by RHH, and carry-  ing oppositely-oriented currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The peak currents are  I0,HH = 1000 A for the Helmholtz coils and I0,M = 500 A  for the Maxwell coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  The shared midpoint between  the coil pairs is displaced by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='2 cm in the positive z-  direction from the initial center of mass of the atomic  sample (taken as the origin of the coordinate system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  We find that this region provides a more uniform and  one-dimensional kick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Atomic properties used for simulation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  The Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' column provides an example of magneto-optical trap  operation for each species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Values for gJ are provided in [15],  except for Li for which we adopt the electron’s value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Atom  m  J  gJ  λ  Trec Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  (10−26 kg)  (nm)  (µK)  Li  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='15  1/2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='002 32  671  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='2  [16]  Cr  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='63  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='001 83  425  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='3  [17]  Er  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='8  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='163 81  583  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='2  [18]  Dy  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='0  8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='241 59  626  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='7  [19]  For a specific example, consider atomic lithium (Li)  trapped using standard techniques [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  At sufficient  magnetic fields, the electronic spin (J = 1/2) decou-  ples from the nuclear spin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' the two electronic mJ states  are denoted |1/2⟩ and |−1/2⟩ and it is in this high-field  regime that we propose MOP cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The cooling se-  quence starts with suddenly turning off the trap so that  the atoms are free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' In step (1) of MOP cooling, all of the  atoms are optically pumped to the |1/2⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Then,  half of the cloud is transferred to the |−1/2⟩ state by  stimulated transitions , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=', stimulated Raman adiabatic  passage (STIRAP) sequences [20, 21], thereby creating  3  −2  0  2  vz (cm/s)  (a)  −50  0  50  (b)  −50  0  50  (c)  −2  0  2  (d)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='25  z (cm)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='2  vz (cm/s)  (e)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='25  z (cm)  −20  0  20  (f)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='25  z (cm)  −20  0  20  (g)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='25  z (cm)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='2  (h)  −J  0  J  mJ  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  MOP cooling simulations for Li (a-d) and Dy (e-h) visualized in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Each column of plots represents a  snapshot in the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' First, the magnetic states are correlated with position along the z-axis through optical pumping followed  by spatially-resolved coherent population transfer via stimulated transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The vertical dashed black lines in panels (a) and  (e) mark the ideal boundary between different magnetic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Second, a one-dimensional magnetic kick accelerates atoms to a  velocity that is proportional to their magnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Third, after waiting an optimized delay time the phase space distribution  has been compressed in real space but remains extended in velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Finally, a reverse kick returns the atom’s velocity  distribution to near its original extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' More precisely, we find the standard deviation of the ensemble’s velocity is, at most,  about 11% of its initial value for all four species simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  two spatially-distinct populations [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 2 (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' In step  (2), a magnetic-field-gradient kick is applied to the cloud,  thereby causing the two halves to merge [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 2 (b-c)],  and then a reverse kick returns the atoms to their origi-  nal velocity distribution [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 2 (d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' As proposed in [13]  and demonstrated experimentally in [22], such magnetic  kicks can be applied along a single axis while minimally  affecting the other two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Two-dimensional  slices (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 3) of the magnetic field gradient used in our  simulations clearly show the one-dimensional-nature of  the magnetic kick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' In step (3), all atoms are optically-  pumped back to the |1/2⟩ state, thereby completing the  cooling cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' In principle, a factor of 2× in phase-space  compression is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  We now generalize the method to a system with an  arbitrary total angular momentum J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' There are 2J + 1  states in this case, and we assume that the atoms are  trapped in a hard-walled flat potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Just as in the  case of Li above, we turn off the trap to start the cooling  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Using optical pumping and stimulated transi-  tions in step (1), the cloud is divided into 2J + 1 com-  ponents, where the leftmost section is prepared in state  |J, −J⟩, then |J, –J + 1⟩, and so on, to the rightmost sec-  tion in state |J, J⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' In step (2), a magnetic field gradi-  ent kick is applied to the cloud, causing each sub-section  to move at a velocity that is proportional to the mag-  netic quantum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Thus, the leftmost section in  state |J, J⟩ will move the fastest to the right, and lower  magnetic-quantum-numbered sections will move slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  The cloud will collapse to a single section after an opti-  mal delay time, and a reverse magnetic kick will restore  the original velocity distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Finally, in step (3),  all atoms would be optically pumped to the |J, J⟩ state,  and the cloud can be re-trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Figures 2 (e - h) show snapshots of the simulated phase  space for MOP cooling of Dy atoms (J = 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Compar-  ing the final spatial distribution of Li [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 2 (d)] to that  of Dy [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 2 (h)] clearly demonstrates how MOP cool-  ing may leverage high-angular momentum systems for  efficient phase-space compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The delay time is op-  timized in our numeric simulations and the results are  shown for a variety of atoms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  As a figure of  merit, we compute the compression factor as the ratio  of standard deviations between the initial and final z-  coordinate distributions in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 4  shows the peak compression factor for each species is  bound by the geometric limit 2J + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' In the following  section we consider limitations of MOP cooling and how  they may be overcome in an experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  DISCUSSION  In MOP cooling, a maximum compression factor of  2J + 1 per cycle may be approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' However, in prac-  tice, the efficiency will be lower due to deviations from  a flat density distribution, imperfect kicking fields, and  photon-recoil heating during the optical pumping (OP)  stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  The flat-density initial condition is quite different from  4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content='5  y (cm)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content='5  z (cm)  (a) x = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='25 cm  944  952  952  952  960  960  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content='5  y (cm)  (b)  x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='00 cm  936  944  952  952  952  952  960  960  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content='5  y (cm)  (c)  x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='25 cm  944  952  952  952  960  960  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='5  x (cm)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content='5  y (cm)  (d) z = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='25 cm  15  30  45  45  45  45  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content='5  x (cm)  (e)  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='00 cm  10  20  30  40  40  40  40  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content='5  x (cm)  (f)  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='25 cm  8  16  24  32  40  40  40  40  900  920  940  960  980  ∇|B| (G/cm)  0  20  40  60  80  ∇|B| (G/cm)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Two-dimensional slices of the magnetic field gradient used for MOP cooling simulations, evaluated at peak current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  The red dashed line marks the initial extent of the atomic cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The quadrupole field provided by Maxwell coils is symmetric  under a sign change of any coordinate axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' However, the bias field provided by the Helmholtz coils breaks the symmetry along  the z-axis by shifting the center of the quadrupole off of the coordinate origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The magnitude of the total field |B| varies  primarily along z in a nearly-linear fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  0  10  20  30  40  Unkick delay (ms)  0  5  10  15  Compression factor  0  4  8  J  0  5  10  15  Li  Cr  Er  Dy  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Optimizing the wait time between the MOP cool-  ing kick and unkick according to the compression factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The  open circle marks the optimal delay time for each species sub-  ject to the initial conditions and magnetic forces described in  the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The inset shows the peak compression factor vs the  species’ angular momentum J, compared to the geometric  limit 2J + 1 (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  that usually encountered with laser cooled atoms in a  magneto-optical trap, but turns out to be important for  the gains in efficiency that are predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' For example,  our simulations predict a compression factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='9 for  Li initialized with with a flat density or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='66 for a Gaus-  sian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' For Dy initialized with a flat density,  a peak compression factor of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='8 is observed, while for  an initially-Gaussian distribution the factor reduces to  only 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' To obtain a flat “boxlike” distribution in an ex-  periment, the atoms can first be confined in a magnetic  quadrupole trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' An optical box can be created around  the atoms using a time-averaged optical dipole potential  from beams that are moving rapidly in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Such potentials were created in the past to study opti-  cal billiards [23, 24] and BECs in painted potentials [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  After trapping, the box can be adiabatically expanded in  three dimensions to a desired size, which will result in a  nearly flat density profile of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The optical setup for  state preparation of each segment would enable multiple  cycles of cooling in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Adiabatic expansion  would lower the kinetic energy and MOP cooling would  compress the cloud spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The process can be dynam-  ically controlled with motorized zoom lenses [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Ideally, a complete cycle of MOP cooling leaves the  velocity distribution of the atomic sample unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  This means that the momentum imparted on the atoms  in the kick phase must be nulled in the reverse-kick  5  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Inhomogeneities in the kicking field will result  in a nonzero mean velocity for the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Such inho-  mogeneities are noticeable in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 3 (b) that shows the  peak magnetic field gradient in the x = 0 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  For  instance, atoms near z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='25 will be kicked downward  with less force than their upward motion-arresting kick  that is applied once they are near z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' This is why  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' 2 (h) shows a net positive velocity, which is partic-  ularly evident for mJ = J (teal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' In our simulation, the  final mean velocity of the Dy atoms is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='03 cm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' In  practice, this is not a significant limitation because the  resulting net velocity is still well within the capture range  of any reasonable trap as it is smaller than the thermal  velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Over the same time scale t, the relative ther-  mal expansion of the cloud σ(t)/σ0 ≈  �  1 + t2kBT/mσ2  0  is insignificant compared to the MOP cooling compres-  sion factor σ0/σ(t) ≈ 2J + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Moreover, there exist ad-  vanced wiring-design-optimization techniques to gener-  ate uniform bias or gradient fields with minimal induc-  tance [14, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Though originally developed for magnetic  resonance imaging, MOP cooling could benefit from such  analyses to improve switching times of the pulsed mag-  netic fields and increase the uniformity of the required  biased gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  A more significant mean velocity is incurred due to  free fall in the Earth gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' For example,  in the ∼ 19 ms of delay time required for MOP cooling  of Cr, the cloud accelerates to nearly 19 cm/s and dis-  places 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='18 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Depending on the details of the trap,  such velocity or displacement may result in significant  atom loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' To mitigate the free fall effects, one could use  the MOP cooling setup to apply an additional uniform  kicks against gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  In general, optical pumping the cloud to the same mag-  netic state is a lossy step due to, for instance, atoms de-  caying to unobserved trap states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Fortunately, there ex-  ist efficient techniques for optical pumping as discussed  in [28], where it is shown that it is possible to perform  optical pumping with only one spontaneous photon emit-  ted per atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  The same physics lets us understand  MOP cooling’s high phase-space compression efficiency  in terms of the photon entropy carried away from the  ensemble by spontaneous emission [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  The entropy associated with the motional degrees of  freedom is given by Smo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' = kB ln V , where V is the phase-  space volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  This volume is reduced by a factor of  (at most) 2J + 1 at each cooling step, so the maximum  entropy reduction per step is ∆Smo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' = kB ln (2J + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  The magnetic kicks are reversible evolution, so they do  not produce a net change in entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Therefore the en-  tropy change must occur during the optical pumping  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  Optical pumping of an unpolarized ensemble to  produce a pure state reduces the polarization entropy  by ∆Spol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' = kB ln (2J + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Thus for each step, (1) OP  takes unpolarized state to pure state, reducing polariza-  tion entropy by kB ln (2J + 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' (2) kicks take pure state  to unpolarized state by overlapping ensembles, increas-  ing polarization entropy by kB ln (2J + 1) and reducing  motional entropy by the same amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The cooling will  be limited by recoil heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  From Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' [28], we know  that OP can theoretically be done with only one sponta-  neously emitted photon per atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' If this is achieved, the  temperature limit will correspond to the recoil energy,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=', the standard recoil limit Trec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' For less efficient OP  the temperature limit will be higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  CONCLUSION  In this paper, we proposed a highly efficient method  for phase-space compression of high-angular momentum  atomic and molecular samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  This work extends an  earlier MOP-cooling proposal from Li to general high-  angular-momentum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' We numerically tested our  new MOP-cooling protocol on four atomic species of in-  creasing angular momenta that have each already been  cooled using traditional techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' We find, for exam-  ple, an impressive compression factor of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='5 is conceiv-  ably attainable in less than 13 ms for the case of Dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  The  work  of  MGR  was  supported  by  the  Sid  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Richardson Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' The work of DB was sup-  ported by the Deutsche Forschungsgemeinschaft (DFG)  Project ID 423116110 and by the Cluster of Excellence  Precision Physics, Fundamental Interactions, and Struc-  ture of Matter (PRISMA+ EXC 2118/1) funded by the  DFG within the German Excellence Strategy (Project ID  39083149).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  6  [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' Dalibard and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Cohen-Tannoudji, JOSA B 6, 2023  (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Ungar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Weiss, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Riis, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Chu, JOSA B  6, 2058 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content='  [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Weiss, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Riis, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Shevy, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' Chu,  JOSA B 6, 2072 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' Quack, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Grimes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' Narevicius, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Auzinsh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Budker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' P´alffy,  and  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Keitel, Journal of Physics B: Atomic, Molecular  and Optical Physics 52, 202001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Nagornykh, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Lu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' Raizen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' Budker, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' Bartolotta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'}
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