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

We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finite relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an (...) algebra is ω-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this approach are looked at, and include the step by step method. (shrink)

Spatio-temporal logic is a variant of branching temporal logic where one of the so-called causal relations on spacetime plays the role of a time flow. Allowing only rational numbers as space and time co-ordinates, we prove that a first-order spatio-temporal theory over this flow is recursively enumerable if and only if the dimension of spacetime does not exceed 2. The situation is somewhat different compared to the case of real co-ordinates, because we establish that even dimension 2 does not permit (...) recursive enumerability in this case. The proof of the result on rational spacetime involves a more deeper portion of spacetime geometry than the corresponding, more evident result for the real co-ordinates. (shrink)