First-order logic with two variables and unary temporal logic

We investigate the power of first-order logic with only two variables over /spl omega/-words and finite words, a logic denoted by FO/sup 2/. We prove that FO/sup 2/ 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/sup 2/ to unary-TL converts every FO/sup 2/ 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. 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/sup 2/ is NEXP-complete, in sharp contrast to the fact that satisfiability for FO/sup 3/ has non-elementary computational complexity. Our NEXP time upper bound for FO/sup 2/ 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/sup 2/ of independent interest, namely: a satisfiable FO/sup 2/ formula has a model whose "size" is at most exponential in the quantifier depth of the formula. Using our translation from FO/sup 2/ 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.