First-Order Logic with Two Variables and Unary Temporal Logic

Abstract

We investigate the power of first-order logic with only two variables over<br />omega-words and finite words, a logic denoted by FO2. We prove that FO2 can<br />express precisely the same properties as linear temporal logic with only the unary temporal operators: “next”, “previously”, “sometime in the future”, and “sometime in the past”, a logic we denote by unary-TL. Moreover, our translation from FO2 to unary-TL converts every FO2 formula to an equivalent unary-TL formula that is at most exponentially larger, and whose operator depth is at most twice the quantifier depth of the first-order formula. We show that this translation is optimal.<br />While satisfiability for full linear temporal logic, as well as for<br />unary-TL, is known to be PSPACE-complete, we prove that satisfiability<br />for FO2 is NEXP-complete, in sharp contrast to the fact that satisfiability<br />for FO3 has non-elementary computational complexity. Our NEXP time<br />upper bound for FO2 satisfiability has the advantage of being in terms of<br />the quantifier depth of the input formula. It is obtained using a small model property for FO2 of independent interest, namely: a satisfiable FO2 formula has a model whose “size” is at most exponential in the quantifier depth of the formula. Using our translation from FO2 to unary-TL we derive this small model property from a corresponding small model property for unary-TL. Our proof of the small model property for unary-TL is based on an analysis of unary-TL types.