Beyond $\omega$BS-regular Languages: $\omega$T-regular Expressions and Counter-Check Automata

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 ωBand 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.