Level Two of the quantifier alternation hierarchy over infinite words

© 2017 Springer Science+Business Media, LLC 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 Σ 2 over infinite words. Here, Σ 2 consists of the first-order formulas with two blocks of quantifiers, starting with an existential quantifier. Its Boolean closure is (Formula presented.). Our first main result is an effective characterization of the Boolean closure of the alphabetic topology, that is, given an ω-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 (Formula presented.) from finite to infinite words.