Ehrenfeucht-Fraïssé Games on Omega-Terms

Authors Martin Huschenbett, Manfred Kufleitner

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Martin Huschenbett
Manfred Kufleitner

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Martin Huschenbett and Manfred Kufleitner. Ehrenfeucht-Fraïssé Games on Omega-Terms. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 374-385, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


Fragments of first-order logic over words can often be characterized in terms of finite monoids or finite semigroups. Usually these algebraic descriptions yield decidability of the question whether a given regular language is definable in a particular fragment. An effective algebraic characterization can be obtained from identities of so-called omega-terms. In order to show that a given fragment satisfies some identity of omega-terms, one can use Ehrenfeucht-Fraisse games on word instances of the omega-terms. The resulting proofs often require a significant amount of book-keeping with respect to the constants involved. In this paper we introduce Ehrenfeucht-Fraisse games on omega-terms. To this end we assign a labeled linear order to every omega-term. Our main theorem shows that a given fragment satisfies some identity of omega-terms if and only if Duplicator has a winning strategy for the game on the resulting linear orders. This allows to avoid the book-keeping. As an application of our main result, we show that one can decide in exponential time whether all aperiodic monoids satisfy some given identity of omega-terms, thereby improving a result of [McCammond, Int. J. Algebra Comput. 2001].
  • regular language
  • first-order logic
  • finite monoid
  • Ehrenfeucht-Fraïssé games
  • pseudoidentity


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