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On the Decision Problem for Two-Variable First-Order Logic

Published online by Cambridge University Press:  15 January 2014

Erich Grädel ,
Phokion G. Kolaitis  and
Moshe Y. Vardi
Erich Grädel
Affiliation:
Lehrgebiet Mathematische Grundlagen Der Informatik Rwth Aachen, D-52056 Aachen, Germany, E-mail: [email protected], URL: http://www.informatik.rwth-aachen.de/WWW-math/index.html
Phokion G. Kolaitis
Affiliation:
Computer Science Department, University of California, Santa Cruz, CA 95064, USA, E-mail: [email protected]
Moshe Y. Vardi
Affiliation:
Department of Computer Science, Rice University, Houston, TX 77005-1892, USA, E-mail: [email protected], URL: http://www.cs.rice.edu/~vardi
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Abstract

We identify the computational complexity of the satisfiability problem for FO2, the fragment of first-order logic consisting of all relational first-order sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO2 has the finite-model property, which means that if an FO2-sentence is satisiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable FO2-sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO2-sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO2 is NEXPTIME-complete.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 3 , Issue 1 , March 1997 , pp. 53 - 69
Copyright
Copyright © Association for Symbolic Logic 1997

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