Level Two of the Quantifier Alternation Hierarchy over Infinite Words

TL;DR

This paper characterizes the Boolean closure of the alphabetic topology over infinite words and uses this to decide if certain $ ext{omega}$-regular languages are definable within this fragment, extending prior finite-word results.

Contribution

It provides an effective characterization of the Boolean closure of the alphabetic topology over infinite words and applies this to decide definability in $ ext{B}\Sigma_2$ fragment.

Findings

Decidability of whether an $ ext{omega}$-regular language is a Boolean combination of open sets in the alphabetic topology.

Extension of Place and Zeitoun's decidability results from finite to infinite words.

Effective criteria for membership in the Boolean closure of the alphabetic topology.

Abstract

The study of various decision problems for logic fragments has a long history in computer science. This paper is on the membership problem for a fragment of first-order logic over infinite words; the membership problem asks for a given language whether it is definable in some fixed fragment. The alphabetic topology was introduced as part of an effective characterization of the fragment $\Sigma_2$ over infinite words. Here, $\Sigma_2$ consists of the first-order formulas with two blocks of quantifiers, starting with an existential quantifier. Its Boolean closure is $\mathbb{B}\Sigma_2$. Our first main result is an effective characterization of the Boolean closure of the alphabetic topology, that is, given an $\omega$-regular language $L$, it is decidable whether $L$ is a Boolean combination of open sets in the alphabetic topology. This is then used for transferring Place and Zeitoun's recent decidability result for $\mathbb{B}\Sigma_2$ from finite to infinite words.