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].
@InProceedings{huschenbett_et_al:LIPIcs.STACS.2014.374, author = {Huschenbett, Martin and Kufleitner, Manfred}, title = {{Ehrenfeucht-Fra\"{i}ss\'{e} Games on Omega-Terms}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {374--385}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.374}, URN = {urn:nbn:de:0030-drops-44729}, doi = {10.4230/LIPIcs.STACS.2014.374}, annote = {Keywords: regular language, first-order logic, finite monoid, Ehrenfeucht-Fra\"{i}ss\'{e} games, pseudoidentity} }
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