## Bulletin of Symbolic Logic

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

#### Abstract

We identify the computational complexity of the satisfiability problem for $\text{FO}^{2}$, 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 $\text{FO}^{2}$ has the finite-model property, which means that if an $\text{FO}^{2}$-sentence is satisfiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable $\text{FO}^{2}$-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 $\text{FO}^{2}$-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 $\text{FO}^{2}$ is NEXPTIME-complete.

#### Article information

Source
Bull. Symbolic Logic, Volume 3, Number 1 (1997), 53-69.

Dates
First available in Project Euclid: 20 June 2007

https://projecteuclid.org/euclid.bsl/1182353488

Mathematical Reviews number (MathSciNet)
MR1444914

Zentralblatt MATH identifier
0873.03009

JSTOR