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The FO2 alternation hierarchy is decidable Kufleitner, Manfred Weil, Pascal first-order logic regular language automata theory semigroup ranker We consider the two-variable fragment FO2[<] of first-order logic over finite words. Numerous characterizations of this class are known. Therien and Wilke have shown that it is decidable whether a given regular language is definable in FO2[<]. From a practical point of view, as shown by Weis, FO2[<] is interesting since its satisfiability problem is in NP. Restricting the number of quantifier alternations yields an infinite hierarchy inside the class of FO2[<]-definable languages. We show that each level of this hierarchy is decidable. For this purpose, we relate each level of the hierarchy with a decidable variety of finite monoids. Our result implies that there are many different ways of climbing up the FO2[<]-quantifier alternation hierarchy: deterministic and co-deterministic products, Mal'cev products with definite and reverse definite semigroups, iterated block products with J-trivial monoids, and some inductively defined omega-term identities. A combinatorial tool in the process of ascension is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle programs of Schwentick, Therien, and Vollmer. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Manfred Kufleitner and Pascal Weil 2012 Is Part Of LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CSL.2012.426 urn:nbn:de:0030-drops-36888 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.426 eng https://creativecommons.org/licenses/by-nc-nd/3.0/legalcode