First-Order Logic with Two Variables and Unary Temporal Logic
We investigate the power of first-order logic with only two variables over -words and finite words, a logic denoted by . We prove that 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 to unary-TL converts every 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 is NEXP-complete, in sharp contrast to the fact that satisfiability for has non-elementary computational complexity. Our NEXP time upper bound for 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 of independent interest, namely: a satisfiable formula has a model whose ``size'' is at most exponential in the quantifier depth of the formula. Using our translation from 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