  In the last years, various extensions of {\omega}-regular languages have been proposed in the literature, including {\omega}B-regular ({\omega}-regular languages extended with boundedness), {\omega}S-regular ({\omega}-regular languages extended with strict unboundedness), and {\omega}BS-regular languages (the combination of {\omega}B- and {\omega}S-regular ones). While the first two classes satisfy a generalized closure property, namely, the complement of an {\omega}B-regular (resp., {\omega}S-regular) language is an {\omega}S-regular (resp., {\omega}B-regular) one, the last class is not closed under complementation. The existence of non-{\omega}BS-regular languages that are the complements of some {\omega}BS-regular ones and express fairly natural properties of reactive systems motivates the search for other well-behaved classes of extended {\omega}-regular languages. In this paper, we introduce the class of {\omega}T-regular languages, that includes meaningful languages which are not {\omega}BS-regular. We first define it in terms of {\omega}T-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 {\omega}T-regular languages (whether or not {\omega}T-regular languages are expressively complete with respect to counter-check automata is still an open problem). Finally, we provide an encoding of {\omega}T-regular expressions into S1S+U. 
Beyond $\omega$BS-regular Languages: $\omega$T-regular Expressions and   Counter-Check Automata