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An extension of Kleene’s and Ochmański’s theorems to infinite traces. (English) Zbl 0795.68116

Summary: As was noted by Mazurkiewicz, traces constitute a convenient tool for describing finite behaviour of concurrent systems. Extending in a natural way Mazurkiewicz’s original definition, infinite traces have recently been introduced enabling one to deal with infinite behaviour of nonterminating concurrent systems. In this paper we examine the basic families of recognizable sets and of rational sets of infinite traces. The seminal Kleene characterization of recognizable subsets of the free monoid and its subsequent extensions to infinite words due to Büchi and to finite traces due to Ochmański are the cornerstones of the corresponding theories. The main result of our paper is an extension of these characterizations to the domain of infinite traces. Using recognizing and weakly recognizing morphisms, as well as a generalization of the Schützenberger product of monoids, we prove various closure properties of recognizable trace languages. Moreover, we establish normal-form representations for recognizable and rational sets of infinite traces.

MSC:

68Q45 Formal languages and automata
68Q10 Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.)
20M35 Semigroups in automata theory, linguistics, etc.
68Q55 Semantics in the theory of computing
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