diff --git "a/BtE0T4oBgHgl3EQfQAAH/content/tmp_files/load_file.txt" "b/BtE0T4oBgHgl3EQfQAAH/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/BtE0T4oBgHgl3EQfQAAH/content/tmp_files/load_file.txt" @@ -0,0 +1,746 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf,len=745 +page_content='Discovering Sound Free-choice Workflow Nets With Non-block Structures Tsung-Hao Huang[0000−0002−3011−9999] and Wil M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' van der Aalst[0000−0002−0955−6940] Process and Data Science (PADS), RWTH Aachen University, Aachen, Germany {tsunghao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='huang, wvdaalst}@pads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='rwth-aachen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='de Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Process discovery aims to discover models that can explain the behaviors of event logs extracted from information systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' While various approaches have been proposed, only a few guarantee desirable properties such as soundness and free-choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' State-of-the-art approaches that exploit the representational bias of process trees to provide the guar- antees are constrained to be block-structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Such constructs limit the expressive power of the discovered models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=', only a subset of sound free-choice workflow nets can be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' To support a more flexible structural representation, we aim to discover process models that provide the same guarantees but also allow for non-block structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Inspired by existing works that utilize synthesis rules from the free-choice nets the- ory, we propose an automatic approach that incrementally adds activities to an existing process model with predefined patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Playing by the rules ensures that the resulting models are always sound and free-choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Furthermore, the discovered models are not restricted to block struc- tures and are thus more flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The approach has been implemented in Python and tested using various real-life event logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The experiments show that our approach can indeed discover models with competitive quality and more flexible structures compared to the existing approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Keywords: Process Discovery · Free-choice Net · Synthesis Rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 1 Introduction Process discovery aims to construct process models that reflect the behaviors of a given event log extracted from information systems [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' As it is a non-trivial problem, many challenges remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' In most cases, the one and only "best model" does not exist as there are trade-offs among the four model quality metrics, namely fitness, precision, generalization, and simplicity [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' In addition to the quality metrics, there exist properties that one would like to have for the discov- ered models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' One of the important properties is being a sound workflow net as soundness ensures the absence of deadlocks, proper completion, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' [1] and it is a prerequisite for many crucial automated analyses such as conformance check- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The other desirable structural property is being free-choice [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' In free-choice nets, choices and synchronizations are separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' This provides an easy conver- sion between the discovered models and many process modeling languages such arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='02185v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='DB] 3 Jan 2023 2 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Huang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' van der Aalst as Business Process Modeling Notation (BPMN) since the equivalent constructs (dedicated split and join connectors) are naturally embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Furthermore, free- choice nets have been studied extensively and thus supported by an abundance of theories [10], which provide efficient analysis techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' While various discovery algorithms have been proposed, only a handful of them provides such guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' State-of-the-art discovery algorithms like the In- ductive Miner (IM) [15] are able to discover sound free-choice workflow nets by exploiting its representational bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' However, due to the same reason, the discov- ered models are constrained to be block-structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' This limits the expressive power of such models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=', only a subset of the sound free-choice workflow nets can be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' As an example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 1a shows a sound free-choice workflow net (with non-block structures) discovered by our approach1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The same language can never be expressed by the model discovered by IM, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' b c d f g a e h (a) A model discovered by our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The same language cannot be expressed by the models discovered using the Inductive Miner, which uses process trees internally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' b c d f g a e h (b) A model discovered by the IM using the log generated by the model in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The two branches before c need to be synchronized first before d can be executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 1: Examples showing the need for the non-block process models discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Note that the trace ⟨a, b, c, d, e, f, g, h⟩ that is possible in (a) cannot be replayed by (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' In this paper, we aim to discover sound free-choice workflow nets with non- block structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Inspired by the interactive process discovery approach in [11,12], we develop an automatic process discovery algorithm that incrementally adds activities to an existing net using synthesis rules [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Since checking the feasibil- ity for the application of the synthesis rules is computationally expensive, we use log heuristics to locate the most possible position for the to-be-added activity on the existing process model instead of evaluating all possible applications of synthesis rules as in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Additionally, we identify the need for an additional rule and extend the set of patterns introduced in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Playing by the rules ensures that the discovered process models by our ap- proach are guaranteed to be sound free-choice workflow nets [10,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Moreover, the discovered models are not constrained to block structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Last but not least, the level of replay fitness is guaranteed via a threshold set by the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The approach has been implemented in Python and evaluated using various public- available real-life event logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The evaluation shows that our approach is able to discover non-block structured models with competitive qualities compared to the state-of-the-art discovery algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 1 The proposed approach has dedicated silent transitions for start and end as defined later in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' We dropped them here for ease of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Discovering Sound Free-choice Workflow Nets With Non-block Structures 3 The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' We review the related work in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 2 and introduce necessary concepts in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 4 introduces the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 5 presents the experiment and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 6 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 2 Related Work An overview of process discovery is out of the scope of this paper, we refer to [7,14] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' In this section, we focus on process discovery techniques that guarantee soundness (and free-choice) properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Approaches like [6,8] can discover non-block structured models but cannot guarantee both properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' While Split Miner discovers models that are deadlock-free, they are not nec- essarily sound [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The family of Inductive Miner (IM) algorithms [15] guarantees sound and free-choice of the discovered models by exploiting the representational bias of the process tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' By design, a process tree represents a sound workflow net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' It is a rooted tree where the leaf nodes are activities and the non-leaf nodes are the operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The hierarchical representation has a straightforward translation to Petri net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' However, the resulting models are limited to being block-structured as a process tree can only represent process models that can be separated into parts that have a single entry and exit [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Consequently, process trees can only rep- resent a subset of sound workflow nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The same arguments hold for approaches that are based on process trees such as the Evolutionary Tree Miner (ETM) [9] and the recently developed incremental process discovery approach [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Applying the synthesis rules [10], the interactive process discovery approaches developed in [12,13,11] ensure soundness and free-choice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' A semi- automatic interactive tool, ProDiGy, is proposed in [12] to recommend the best possible ways to add an activity to an existing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Our approach differs from [12,13,11] in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' First, we adopt an automatic setting as the order of adding activities is predetermined and the best modification to the existing net is selected based on the model quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Second, we use log heuristics to locate the most suitable position for adding the new activity instead of evaluating all the possibilities of synthesis rules applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Moreover, we identify the need for a new rule as the desired models often cannot be discovered without going back and forth by a combination of reduction and synthesis rules [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Lastly, the set of patterns is extended and formally defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 3 Preliminaries We denote the set of all sequences over some set A as A∗, the power set of A as P(A), and the set of all multisets over A as B(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' For some multiset b ∈ B(A), b(a) denotes the number of times a ∈ A appears in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' For a given sequence σ = ⟨a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=', an⟩ ∈ A∗, |σ| = n is the length of σ and dom(σ) = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=', |σ|} is the domain of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' ⟨⟩ is the empty sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' σ(i) = ai denotes the i-th element of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Given sequences σ1 and σ2, σ1 · σ2 denotes the concatenation of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Let A be a set and X ⊆ A be a subset of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' For σ ∈ A∗ and a ∈ A, 4 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Huang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' van der Aalst we define ↾X∈ A∗→X∗ as a projection function recursively with ⟨⟩↾X = ⟨⟩, (⟨a⟩ · σ)↾X = ⟨a⟩ · σ↾X if a ∈ X and (⟨a⟩ · σ)↾X = σ↾X if a /∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' For example, ⟨x, y, x⟩↾{x,z} = ⟨x, x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Projection can also be applied to multisets of sequences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=', [⟨a, b, a⟩6, ⟨a, b, c⟩6, ⟨b, a, c⟩2]↾{b,c} = [⟨b⟩6, ⟨b, c⟩8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Definition 1 (Trace, Log).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' A trace σ ∈ U∗ A is a sequence of activities, where UA is the universe of activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' A log L ∈ B(U∗ A) is a multiset of traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Definition 2 (Log Properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Let L ∈ B(U∗ A) and a, b ∈ UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' – #(a, L) = Σσ∈L|{i ∈ dom(σ)|σ(i) = a}| is the times a occurred in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' – #(a, b, L) = Σσ∈L|{i ∈ dom(σ)\\{|σ|}|σ(i) = a∧σ(i+1) = b}| is the number of direct successions from a to b in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' – caus(a, b, L) = � #(a,b,L)−#(b,a,L) #(a,b,L)+#(b,a,L)+1 if a ̸= b #(a,b,L) #(a,b,L)+1 if a = b is the strength of causal rela- tion (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' – Apre c (a, L) = {apre ∈ UA|caus(apre, a, L) ≥ c} is the set of a’s preceding activities, determined by threshold c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' – Afol c (a, L) = {afol ∈ UA|caus(a, afol, L) ≥ c} is the set of a’s following activities, determined by threshold c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' – As(L) = {σ(1) | σ ∈ L ∧ σ ̸= ⟨⟩} is the set of start activities in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' – Ae(L) = {σ(|σ|) | σ ∈ L ∧ σ ̸= ⟨⟩} is the set of end activities in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Definition 3 (Petri Net, Labeled Petri Net).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' A Petri net N = (P, T, F) is a tuple, where P is the set of places, T is the set of transitions, P ∩ T = ∅, and F ⊆ (P × T) ∪ (T × P) is the set of arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' A labeled Petri net N = (P, T, F, l) is a Petri net (P, T, F) with a labeling function l ∈ T ↛ UA that maps a subset of transitions to activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' A t ∈ T is called invisible if t is not in the domain of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' For any x ∈ P ∪ T, N•x = {y|(y, x) ∈ F} denotes the set of input nodes and x N• = {y|(x, y) ∈ F} denotes the set of output nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The superscript N is dropped if it is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The notation can be generalized to set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' For any X ⊆ P ∪ T, •X = {y|∃x∈X(y, x) ∈ F} and X• = {y|∃x∈X(x, y) ∈ F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Definition 4 (Free-choice Net).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Let N = (P, T, F) be a Petri net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' N is a free-choice net if for any t1, t2 ∈ T : •t1 = •t2 or •t1 ∩ •t2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Definition 5 (Workflow Net (WF-net) [1,11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Let N = (P, T, F, l) be a labeled Petri net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' W = (P, T, F, l, ps, pe, ⊤, ⊥) is a WF-net iff (1) it has a dedi- cated source place ps ∈ P: •ps = ∅ and a dedicated sink place pe ∈ P: pe• = ∅ (2) ⊤ ∈ T: •⊤ = {ps}∧ps• = {⊤} and ⊥ ∈ T: ⊥• = {pe}∧•pe = {⊥} (3) every node x is on some path from ps to pe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=', ∀x∈P ∪T (ps, x) ∈ F ∗ ∧ (x, pe) ∈ F ∗, where F ∗ is the reflexive transitive closure of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Definition 6 (Short-circuited WF-net [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Let W = (P, T, F, l, ps, pe, ⊤, ⊥) be a WF-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The short-circuited WF-net of W, denoted by SC(W), is con- structed by SC(W)=(P, T ∪{t′}, F ∪{(⊥, t′), (t′, ⊤)}, l, ps, pe, ⊤, ⊥), where t′ /∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Discovering Sound Free-choice Workflow Nets With Non-block Structures 5 Definition 7 (Paths, Elementary Paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' A path of a Petri net N = (P, T, F) is a non-empty sequence of nodes ρ = ⟨x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=', xn⟩ such that (xi, xi+1) ∈ F for 1 ≤ i < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' ρ is an elementary path if xi ̸= xj for 1 ≤ i < j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Definition 8 (Incidence Matrix [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Let N = (P, T, F) be a Petri net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The incidence matrix N : (P × T) → {−1, 0, 1} of N is defined as N(p, t) = � � � � � 0 if ((p, t) /∈ F ∧ (t, p) /∈ F) ∨ ((p, t) ∈ F ∧ (t, p) ∈ F) −1 if (p, t) ∈ F ∧ (t, p) /∈ F 1 if (p, t) /∈ F ∧ (t, p) ∈ F For a Petri net N = (P, T, F) and its corresponding incidence matrix N, we use N(p) to denote the row vector of the corresponding p ∈ P and N(t) to denote the column vector of the corresponding t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Definition 9 (Linearly Dependent Nodes [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Let N = (P, T, F) be a Petri net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Q is the set of rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' A place p is linearly dependent if there exists a row vector ⃗v : P → Q such that ⃗v(p) = 0 and ⃗v · N = N(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' A transition t is linearly dependent if there exists a column vector ⃗v : T → Q such that ⃗v(t) = 0 and ⃗v · N = N(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Definition 10 (Synthesis Rules [10,11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Let W and W ′ be two free-choice workflow nets, and let SC(W) = (P, T, F, l, ps, pe, ⊤, ⊥) and SC(W ′) = (P ′, T ′, F ′, l′, ps, pe, ⊤, ⊥) be the corresponding short-circuited WF-nets: – Linear Dependent Place Rule ψP : W ′ is derived from W using ψP , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=', (W, W ′) ∈ ψP if (1) T ′ = T, P ′ = P ∪ {p} and p /∈ P is linear dependent in SC(W ′), F ′ = F ∪ �F where �F ⊆ (({p} × T) ∪ (T × {p})) (2) Every siphon in SC(W ′) contains ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' – Linear Dependent Transition Rule ψT : W ′ is derived from W using ψT , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=', (W, W ′) ∈ ψT if P ′ = P, T ′ = T ∪ {t} and t /∈ T is linear dependent in SC(W ′) and F ′ = F ∪ �F where �F ⊆ ((P ×{t})∪({t}×P)), and ∀t∈T ∩T ′l(t) = l′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' – Abstraction Rule ψA: (W, W ′) ∈ ψA if (1) there exists a set of transitions R ⊆ T and a set of places S ⊆ P such that (R × S ⊆ F) ∧ (R × S ̸= ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' (2) SC(W ′) is constructed by adding an additional place p /∈ P and a transition t /∈ T such that P ′ = P ∪ {p}, T ′ = T ∪ {t}, F ′ = (F\\(R × S)) ∪ ((R × {p}) ∪ ({p} × {t}) ∪ ({t} × S)), and ∀t∈T ∩T ′l(t) = l′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Applying the three synthesis rules (ψP , ψT , ψA) to derive W ′ from a sound free-choice workflow net W ensures that W ′ is also sound [13,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Three proper- ties need to be hold for a WF-net to be sound (1) safeness: places cannot hold multiple tokens at the same time (2) option to complete: it is always possible to reach the marking in which only the sink place is marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' (3) no dead transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Next, we introduce the initial net [11] and show some examples of synthesis rules applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Huang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' van der Aalst 𝑝𝑠 𝑝𝑒 ⊤ ⊥ 𝑝1 (a) h 𝑝𝑠 𝑝𝑒 ⊤ ⊥ 𝑝1 𝑡1 𝑝2 (b) g h 𝑝𝑠 𝑝𝑒 ⊤ ⊥ 𝑝1 𝑝2 𝑝3 𝑡2 𝑡1 (c) g h 𝑝𝑠 𝑝𝑒 ⊤ ⊥ 𝑝1 𝑝2 𝑝3 𝑡2 𝑡1 𝑝4 (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 2: Examples of synthesis rules applications starting from (a) The initial net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' (b) Using ψA, p2 and t1 are added to the initial net with R = {⊤} and S = {p1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' (c) Using ψA, p3 and t2 are added to previous net with R = {⊤} and S = {p2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' (d) p4 is added using ψp as p4 is a linear combination of p3 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Definition 11 (Initial Net [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Let W = (P, T, F, l, ps, pe, ⊤, ⊥) be a free- choice WF-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' W is an initial net if P = {ps, p1, pe}, T = {⊤, ⊥}, F = {(ps, ⊤), (⊤, p1), (p1, ⊥), (⊥, pe)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The initial net is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Clearly, it is a sound free-choice workflow net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Starting from the initial net, one can incrementally add additional nodes according to the synthesis rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 2 shows example applications of synthesis rules starting from the initial net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 4 Approach With the necessary concepts introduced, we are now ready to introduce the ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' We start by showing the basic idea of the approach with the help of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 3 before diving into each step in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Internally, the approach incremen- tally adds a new activity to an existing net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The figure shows a single iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' In each iteration, we have an existing model from the previous iteration and a log projected on the already added activities so far and the to-be-added one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' We start by locating the most likely position to add the new activity deter- mined by log heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The result of this step is a subset of nodes of the existing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The set of nodes will then be used to prune the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Then, the predefined patterns are applied to the existing net to get a set of candidate nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Lastly, we select the best net (next existing net) out of the candidates in terms of fitness and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Note that the existing net in the first iteration is initiated by the initial net (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' As a running example, consider the correspond- ing log that is used to discover the Petri net in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='1a by our approach: Ls = [⟨a, b, c, d, f, g, h⟩22, ⟨a, b, c, f, d, g, h⟩14, ⟨a, e, b, c, d, f, g, h⟩13, ⟨a, e, b, c, f, d, g, h⟩13, ⟨a, e, b, c, f, g, d, h⟩10, ⟨a, b, c, f, g, d, h⟩10, ⟨a, b, e, c, d, f, g, h⟩6, ⟨a, b, e, c, f, g, d, h⟩3, ⟨a, b, e, c, f, d, g, h⟩3, ⟨a, b, c, d, e, f, g, h⟩2, ⟨a, b, c, e, d, f, g, h⟩2, ⟨a, b, c, e, f, g, d, h⟩1, ⟨a, b, c, e, f, d, g, h⟩1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The instances provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 3 shows the 3rd iteration for the running example Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' In the following subsections, we introduce the details of each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Discovering Sound Free-choice Workflow Nets With Non-block Structures 7 (1) Pruning search space using log heuristics (2) Add new activity to the existing net with pre-defined patterns [ 𝑑, 𝑔, ℎ 76, 𝑔, 𝑑, ℎ 24] Projected Log 𝐿𝑖 Existing Net 𝑊𝑖 = (𝑃𝑖, 𝑇𝑖, 𝐹𝑖, 𝑙𝑖, 𝑝𝑠, 𝑝𝑒, ⊤, ⊥) (3) Select the best net for the next iteration Next Existing Net 𝑊𝑖+1 = (𝑃𝑖+1, 𝑇𝑖+1, 𝐹𝑖+1, 𝑙𝑖+1, 𝑝𝑠, 𝑝��, ⊤, ⊥) To-be-added Activity 𝛾(𝑖) 𝑉𝑖 ⊆ 𝑃𝑖 ∪ 𝑇𝑖 Set of Candidate Nets 𝐶𝑖 skip loop … … Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 3: An example of a single iteration of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content='1 Ordering Strategies for Adding Activities Before starting any iteration, we need to come up with an order for adding ac- tivities based on a given log L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' It is important as the quality of the discovered models often depends on the order of adding activities [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Moreover, in combi- nation with the search space pruning, it can influence the computation time for each iteration significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' In this paper, we introduce two ordering strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The first one is relatively straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The activities in L are simply ordered by their frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Definition 12 (Activities-Adding Order, Frequency-Based Ordering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' Let L ∈ B(U∗ A) and A = � σ∈L{a ∈ σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' γ ∈ A∗ is an activities-adding order for L if {a ∈ γ} = A and |γ| = |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf'} +page_content=' The frequency-based ordering is orderfreq(L) = γ such that γ is an activities-adding order and ∀1≤i