The "Last" Decision Problem for Rational Trace Languages

It is established here that it is decidable whether a rational set of a free partially commutative monoid (i.e. trace monoid) is recognizable or not if and only if the commutation relation is transitive (i.e. if the trace monoid is isomorphic to a free product of free commutative monoids). The bulk of the paper consists in a characterization of recognizable sets of free products via generalized finite automata. I n t r o d u c t i o n 1 Since the work of Mazurkiewiecz [12], trace monoids are currently recognized as a possible model for parallel or conct~rrent programs. This paper deals with the recognizability of rational sets in such monoids. In order to present the result and its interpretation in terms of programs, let us first recall the "standard" terrdnology and notations of the domain: A is an alphabet, 0 C_ A x A a symmetric relation on A is the commutation relation. The free partially commutative monoid, or trace monoid, M(A, 0) is the quotient of A* by [0] -where [0] is the congruence of the free monoid A* generated by the set of pairs {(ab, ha) I (a, b) E 0}. Elements of M(A, 0) are called traces, subsets of M(A, 0) trace languages. The family of rational (or regular) subsets of M(A, 0) is denoted by Rat M(A, O). Trace monoids are a model for the behaviour of a parallel program in the sense that one computation of such a program is interpreted as the set of sequences (i. e. elements of A*) of operations that can be obtained using all possible commutations between them. This set of equivalent computations is well represented by one element of the trace monoid. A rational subset of the trace monoid is a description of the set of computations which a parallel program performs (see [7,12]). We address here the problem of deciding whether a rational set of M(A, 0) is recognizable and we prove : This work has been supported in part by the "'Programme de recherches coordonndes ~ Mathdmatiques et Informatique of the Ministate de la Recherche et de la Technologie, by the ESPRIT-BRA Working Group 3166 ASMICS and by the BID Program of the Universidade de Sa5 Paulo