text: We offer a theorem that highlights the broad applicability of these modeling techniques .4 If f ( input , output ) is a weighted regular relation , then the following statements are equivalent : ( 1 ) f is a joint probabilistic relation ; ( 2 ) f can be computed by a Markovian FST that halts with probability 1 ; ( 3 ) f can be expressed as a probabilistic regexp , i.e. , a regexp built up from atomic expressions a : b ( for a E E U -LCB- E -RCB- , b E A U -LCB- E -RCB- ) using concatenation , probabilistic union + p , and probabilistic closure * p. For defining conditional relations , a good regexp language is unknown to us , but they can be defined in several other ways : ( 1 ) via FSTs as in Fig. 1c , ( 2 ) by compilation of weighted rewrite rules ( Mohri and Sproat , 1996 ) , ( 3 ) by compilation of decision trees ( Sproat and Riley , 1996 ) , ( 4 ) as a relation that performs contextual left-to-right replacement of input substrings by a smaller conditional relation ( Gerdemann and van Noord , 1999 ) ,5 ( 5 ) by conditionalization of a joint relation as discussed below .