The equivalence of finite valued transducers (on HDT0L languages) is decidable

Karel Culik, Juhani Karhumäki
1986 Theoretical Computer Science  
We show a generalization of the Ehrenfeucht Conjecutre: for every language there exists a (finite) test set with respect to normalized k-valued finite transducers with bounded number of states. Further, we show that, for each HDTOL language, such a test set can be effectively found. As a corollary we solve an open problem by Gurari and Ibarra: the equivalence problem for finite valued finite transducers is decidable. This is the first time the equivalence problem is shown to be decidable for a
more » ... arger class of multivalued transducers. decidability of the equivalence problem for k-valued finite transducers. In order to put our theorem into perspective we briefly review the history of main results on the equivalence of finite transducers. Equivalence problems for various types of finite transducers (finite automata with outputs) have been extensively studied since the beginning of automata theory. In his famous paper [31] Moore showed that the equivalence problem is decidable for length-preserving deterministic sequential machines. This decidability result was then gradually extended as follows: for deterministic gsm's Jones and Laaser [25], cf. also Blattner and Head [7], for single-valued transducers Schiitzenberger [37] and independently Blattner and Head [6], for deterministic two-way transducers Gurad [18, 19] , and, finally, for single-valued two-way transducers the present authors [ 12] . A strictly larger class than that of deterministic gsm's, but incomparable with the other classes above, is the class of deterministic two-tape acceptors, for which the decidability of the equivalence problem was proved by Bird [5] . On the other hand, Fischer and Rosenberg [ 16] showed the undecidability of the equivalence problem for nondeterministic finite transducers. At the same time Grifliths [21] generalized this undecidability for e-free nondeterministic gsm's, and finally Ibarra [23] proved it for e-free gsm's with unary input (or output) alphabet. Here, we further narrow the gap between the decidable and undecidable equivalence problems for finite transducers. We show that the problem is decidable for finitevalued finite transducers. We give a detailed proof for the case of one-way finite transducers, but using the techniques from our previous paper [12] , the result can be straightforwardly extended to finite-valued two-way finite transducers, too. Actually, we prove a considerably stronger result, namely that, for every HDTOL language and two natural numbers n and k, there effectively exists a test set with respect to normalized k-valued finite transducers with at most n states. Clearly, this result implies that, given an HDTOL language L and two finite valued transducers, we can test whether they are equivalent on L. The decidability of testing the equivalence of mappings of certain types on languages from a family ~ has been considered by many authors. Most relevant results from the point of view of this paper can be listed as follows. Testing the equivalence of mappings which are realized by finite transducers on a regular language is a special case of the equivalence problem for finite transducers since a restriction of a finite transduction to a regular set is again a finite transduction. In [27] , it was shown that this problem is decidable for (multivalued) mappings of the form 'morphism followed by inverse morphism', while it is undecidable for the mappings of the form 'inverse morphism followed by morphism'. This latter undecidability result was generalized in [29] for inverses of finite substitutions. A lot of attention was given to the problem of deciding the equivalence of two morphisms on a given language. Indeed, in order to prove the decidability of the DOL-sequence equivalence problem in [9] , it was shown that the equivalence of two morphisms can be tested on the D0L language generated by one of them. Subsequently, the following cases were shown to be decidable. Morphisms on a
doi:10.1016/0304-3975(86)90134-9 fatcat:oo5w5wvy55chzl57d4537wromq