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

We identify the computational complexity of the satisfiability problem for FO, 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 has the finite-model property, which means that if an FO-sentence is satisfiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable FO-sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improveMortimer’s bound by one exponential and show that every satisfiable FO-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 is NEXPTIME-complete. §