On recognizable trace languages

Traces are a model for concurrency, whose formal definition is due to Mazurkiewicz [9]. In short, an alphabet A is fixed, with an independence relation on A. Traces can be defined as equivalence classes of words on A, under the relation that allows commutation of consecutive independent letters; and they are uniquely represented by certain A-labeled posets. A trace language is a set of traces on a fixed independence alphabet. The notion of regularity for trace languages is derived naturally from the standard framework of formal language theory (the theory of languages of finite words): a trace language is regular if the set of linearizations of its elements is a regular (word) language. The robustness of this notion is established by two fundamental results: regular trace languages are exactly the sets of traces that are defined by monadic second-order sentences [19]; they are exactly the sets of traces that are accepted by the so-called Zielonka automata [20]. These automata capture, in their structure, the intrinsically distributed nature of the alphabet (with its independence relation) and of the language. A difference with classical language theory arises when rational expressions are considered: if letters a and b are independent, then (ab)∗ is not a regular trace language (the set of its linearizations is the archetypal context-free language of all words with an equal number of a’s and b’s), but a result by Ochmański [12] gives a restriction of rational expressions that describe exactly the regular trace languages. On the basis of these encouraging results, it was natural to think that the algebraic point of view – that is so successful in automata theory – would also help classifying regular trace languages. Since the set of all traces is naturally endowed with a monoid structure, it makes sense to consider the recognizable trace languages, that is, those that are accepted by a morphism into a finite monoid, or equivalently, those whose syntactic monoid is finite. It turns out (and it is not difficult to establish) that those are exactly the regular trace languages. It was shown (Guaiana, Restivo, Salemi [6] and Ebinger, Muscholl [4]) that star-free expressions are as expressive as first-order formulas, and that the corresponding trace languages are exactly those whose syntactic monoid is aperiodic. The same statement also holds for word languages (Schützenberger [15] and McNaughton, Papert [11]) and it was one of the results that led to Eilenberg’s intuition of the theory of varieties. This theory is based on the idea that certain combinatorial properties of recognizable languages are reflected in the algebraic properties of their syntactic monoids. More precisely, Eilenberg introduced varieties of languages: these are classes of recognizable languages closed under Boolean operations, left and right residuals and inverse morphisms. He also considered the classical notion of pseudovarieties of monoids (classes of finite monoids that are closed under taking submonoids, quotients and finite direct product) and he showed that (a) the languages whose syntactic monoid sits in a given pseudovariety V form a variety of languages V , and (b) that the resulting correspondenceV → V is one-to-one and onto between pseudovarieties