Decidable fragments of rst-order temporal logics

In this paper, we introduce a new fragment of the rst-order temporal language, called the monodic fragment, in which all formulas beginning with a temporal operator (Since or Until) have at most one free variable. We show that the satissability problem for monodic formulas in various linear time structures can be reduced to the satissability problem for a certain fragment of classical rst-order logic. This reduction is then used to single out a number of decidable fragments of rst-order temporal logics and of two-sorted rst-order logics in which one sort is intended for temporal reasoning. Besides standard rst-order time structures, we consider also those that have only nite rst-order domains, and extend the results mentioned above to temporal logics of nite domains. We prove decidability in three diierent ways: using decidability of monadic second-order logic over the intended ows of time, by an explicit analysis of structures with natural numbers time, and by a composition method that builds a model from pieces in nitely many steps.