First-Order Logic with Two Variables and Unary Temporal Logic

We investigate the power of first-order logic with only two variables over omega-words and finite words, a logic denoted by FO. We prove that FO can 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 FO to unary-TL converts every FO 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 transition is essentially optimal. While satisfiability for full linear temporal logic, as well as for unary-TL is known to be PSPACE-complete, we prove that satisfiability for FO is NEXP-complete, in sharp contrast to the fact that satisfiability for FO has non-elementary computational complexity. Our NEXP upper bound for FO satisfiability has the advantage of being in terms of the quantifier depth of the input formula. It is obtained using a small model property for FO of independent interest, namely: a satisfiable FO formula has a model whose "size" is at most exponential in the quantifier depth of the formula. Using our translation from FO 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 for unary-TL types.