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Small substructures and decidability issues for first-order logic with two variables

Published online by Cambridge University Press:  12 March 2014

Emanuel Kieroński  and
Martin Otto
Emanuel Kieroński
Affiliation:
Institute of Computer Science, University of Wrocław, Joliot-Curie 15, PL-50-383 Wrocław, Poland, E-mail: kiero@cs.uni.wroc.pl
Martin Otto
Affiliation:
Department of Mathematics, Technische Universität Darmstadt, Schlossgartenstrasse 7, D-64289 Darmstadt, Germany, E-mail: otto@mathematik.tu-darmstsdt.de
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Abstract

We study first-order logic with two variables FO2 and establish a small substructure property. Similar to the small model property for FO2 we obtain an exponential size bound on embedded substructures, relative to a fixed surrounding structure that may be infinite. We apply this technique to analyse the satisfiability problem for FO2 under constraints that require several binary relations to be interpreted as equivalence relations. With a single equivalence relation, FO2 has the finite model property and is complete for non-deterministic exponential time, just as for plain FO2. With two equivalence relations, FO2 does not have the finite model property, but is shown to be decidable via a construction of regular models that admit finite descriptions even though they may necessarily be infinite. For three or more equivalence relations, FO2 is undecidable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 3 , September 2012 , pp. 729 - 765
Copyright
Copyright © Association for Symbolic Logic 2012

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