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

• Published in: The bulletin of symbolic logic Vol. 3; no. 1; pp. 53 - 69
• Main Authors: Grädel, Erich, Kolaitis, Phokion G., Vardi, Moshe Y.
• Format: Journal Article
• Language: English
• Published: New York, USA Cambridge University Press 01-03-1997; Association for Symbolic Logic
• Subjects: Cardinality; Communications; Computational complexity; Decidability; Finite model property; Grammatical conjunctions; Mathematical logic; Modal logic; Polynomials; Predicate logic; Satisfiability
• ISSN: 1079-8986; 1943-5894
• DOI: 10.2307/421196

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 satisfiable, 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.