On the decision problem for two-variable first-order logic

Bulletin of Symbolic Logic 3 (1):53-69 (1997)
We identify the computational complexity of the satisfiability problem for FO 2 , 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 FO 2 has the finite-model property, which means that if an FO 2 -sentence is satisfiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable FO 2 -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 FO 2 -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 FO 2 is NEXPTIME-complete
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DOI 10.2307/421196
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References found in this work BETA
On Languages with Two Variables.Michael Mortimer - 1975 - Mathematical Logic Quarterly 21 (1):135-140.
The Unsolvability of the Gödel Class with Identity.Warren D. Goldfarb - 1984 - Journal of Symbolic Logic 49 (4):1237-1252.
Correction to a Note on the Entscheidungsproblem.Alonzo Church - 1936 - Journal of Symbolic Logic 1 (3):101-102.

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Logics for the Relational Syllogistic.Ian Pratt-hartmann & Lawrence S. Moss - 2009 - Review of Symbolic Logic 2 (4):647-683.
On the Restraining Power of Guards.Erich Grädel - 1999 - Journal of Symbolic Logic 64 (4):1719-1742.
Deciding Regular Grammar Logics with Converse Through First-Order Logic.Stéphane Demri & Hans De Nivelle - 2005 - Journal of Logic, Language and Information 14 (3):289-329.
On Preservation Theorems for Two-Variable Logic.Erich Gradel & Eric Rosen - 1999 - Mathematical Logic Quarterly 45 (3):315-325.

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