Title: 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. The
new expressions are an extension of $\omega$-regular expressions where two new
variants of the Kleene star $L^*$ are added: $L^B$ and $L^S$. These new
exponents are used to say that parts of the input word have bounded size, and
that parts of the input can have arbitrarily large sizes, respectively. For
instance, the expression $(a^Bb)^\omega$ represents the language of infinite
words over the letters $a,b$ where there is a common bound on the number of
consecutive letters $a$. The expression $(a^Sb)^\omega$ represents a similar
language, but this time the distance between consecutive $b$'s is required to
tend toward the infinite.
We develop a theory for these languages, with a focus on decidability and
closure. We define an equivalent automaton model, extending Büchi automata.
The main technical result is a complementation lemma that works for languages
where only one type of exponent---either $L^B$ or $L^S$---is used.
We use the closure and decidability results to obtain partial decidability
results for the logic MSOLB, a logic obtained by extending monadic second-order
logic with new quantifiers that speak about the size of sets.