On the Decidability of the Equivalence for k-Valued Transducers (Extended Abstract) Rodrigo de Souza ⋆ TELECOM ParisTech, 46, rue Barrault, 75634 Paris Cedex 13 rsouza@enst.fr Abstract. We give a new proof for the decidability of the equivalence of two k-valued transducers, a result originally established by Culik and Karh¨ umaki and independently by Weber. Our proof relies on two con- structions we have recently introduced to decompose a k-valued trans- ducer and to decide whether a transducer is k-valued. As a result, our proof is entirely based on the structure of the transducers under inspec- tion, and the complexity it yields is of single exponential order on the number of states. This improves Weber’s result by one exponential. 1 Introduction This communication is the third part of a complete reworking of the theory of k-valued rational relations and transducers which makes it appear as a natural generalisation of the theory of rational functions and functional transducers, not only at the level of results but also at the level of proofs. In [1], we present a construction to decompose a k-valued transducer into a sum of k functional and unambiguous ones of single exponential size. The existence of such a decompo- sition has been settled by Weber in [2]; but this proof yields a bound of double exponential order. And in [3] we generalise a technique of [4] to a new algorithm to decide whether a transducer is k-valued, a result originally due to Gurari and Ibarra [5]. Here, we combine the techniques of [1] and [3] into a procedure to decide the equivalence of k-valued transducers in single exponential time. Equivalence of automata is a most fundamental problem in the field of au- tomata theory and has been studied for several formalisms. For transducers the equivalence reduces to the Post Correspondence Problem and is thus un- decidable [6]. This result has been subsequently sharpened by Griffiths [7] for generalised sequential machines with no empty words in the transitions and next by Ibarra [8] for the same device over a unary input (or output) alphabet. Things change for k-valued transducers. For the functional ones (k = 1), the decidability of the equivalence follows from the decidability of the functionality, a particular case of Gurari and Ibarra’s theorem which had been established by Sch¨ utzenberger [9] and independently by Blattner and Head [10]. The general ⋆ Research supported by CAPES Foundation (Brazilian government). M. Ito and M. Toyama (Eds.): DLT 2008, LNCS 5257, pp. 252–263, 2008. c  Springer-Verlag Berlin Heidelberg 2008