Boundedness in languages of infinite words

Abstract

We define a new class of languages of $\omega$-words, strictly extending$\omega$-regular languages. One way to present this new class is by a type of regular expressions. Thenew expressions are an extension of $\omega$-regular expressions where two newvariants of the Kleene star $L^*$ are added: $L^B$ and $L^S$. These newexponents are used to say that parts of the input word have bounded size, andthat parts of the input can have arbitrarily large sizes, respectively. Forinstance, the expression $(a^Bb)^\omega$ represents the language of infinitewords over the letters $a,b$ where there is a common bound on the number ofconsecutive letters $a$. The expression $(a^Sb)^\omega$ represents a similarlanguage, but this time the distance between consecutive $b$'s is required totend toward the infinite. We develop a theory for these languages, with a focus on decidability andclosure. We define an equivalent automaton model, extending B\"uchi automata.The main technical result is a complementation lemma that works for languageswhere only one type of exponent---either $L^B$ or $L^S$---is used. We use the closure and decidability results to obtain partial decidabilityresults for the logic MSOLB, a logic obtained by extending monadic second-orderlogic with new quantifiers that speak about the size of sets