Beyond ωBS-regular Languages: ωT-regular Expressions and Counter-Check Automata

Abstract

In the last years, various extensions of ω-regular languages have been proposed in the literature, including ωB-regular (ω-regular languages extended with boundedness), ωS-regular (ω-regular languages extended with strict unboundedness), and ωBS-regular languages (the combination of ωB-and ωS-regular ones). While the first two classes satisfy a generalized closure property, namely, the complement of an ωB-regular (resp., ωS-regular) language is an ωS-regular (resp., ωB-regular) one, the last class is not closed under complementation. The existence of non-ωBS-regular languages that are the complements of some ωBS-regular ones and express fairly natural properties of reactive systems motivates the search for other well-behaved classes of extended ω-regular languages. In this paper, we introduce the class of ωT -regular languages, that includes meaningful languages which are not ωBS-regular. We first define it in terms of ωT -regular expressions. Then, we introduce a new class of automata (counter-check automata) and we prove that (i) their emptiness problem is decidable in PTIME and (ii) they are expressive enough to capture ωT -regular languages (whether or not ωT -regular languages are expressively complete with respect to counter-check automata is still an open problem). Finally, we provide an encoding of ωT -regular expressions into S1S+U.

Beyond ω-regular languages: ωT-regular expressions and their automata and logic counterparts

“…Equivalent characterisations of extended ω-regular languages are given in [7,8] in terms of automata (ωB-, ωS-, and ωBS-automata) and classical logic (fragments of WS1S+U, i.e., the extension of WS1S with the unbounding quantifier U [14], that allows one to express properties which are satisfied by finite sets of arbitrarily large size). 1 In [8], the authors also show that the complement of an ωB-regular language is an ωS-regular one and vice versa, and that ωBS-regular languages, featuring both B-and S-constructors, strictly extend ωB-and ωS-regular languages and are not closed under complementation.…”

“…This paper is an extended and merged version of [1] and [2]. A preliminary partial account of the work on the T (resp., T s ) operator appeared in [1] (resp., [2,16]).…”

“…This paper is an extended and merged version of [1] and [2]. A preliminary partial account of the work on the T (resp., T s ) operator appeared in [1] (resp., [2,16]). Besides detailed proofs of all the results and many other specific improvements, which were missing in the conference papers [1,2], we added several results in Section 2 (among others, the interleaving semantics of the shuffle operator + and the analysis of its relationships with the original mix one are completely new) and the whole Section 7 (comparison between T /CCA and T s /CQA).…”

“…It is worth pointing out that the expression e in the statement is a standard regular expression, that is, e does not feature any occurrence of the constructors B, S, and T , as the formulation of the statement given in[1] (Proposition 1) was incorrect.…”