Level Two of the Quantifier Alternation Hierarchy over Infinite Words

  • Manfred Kufleitner
  • Tobias WalterEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)


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 \(\varSigma _2\) over infinite words. Here, \(\varSigma _2\) consists of the first-order formulas with two blocks of quantifiers, starting with an existential quantifier. Its Boolean closure is \(\mathbb {B}\varSigma _2\). 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 \(\mathbb {B}\varSigma _2\) from finite to infinite words.


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.FMIUniversität StuttgartStuttgartGermany

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