Programming in metric temporal logic

Abstract

We present a fragment of metric temporal logic called bounded universal Horn formulae as a theoretical basis for temporal reasoning in logic programming. We characterize its semantics in terms of fixed points and canonical models, and present an efficient proof method as operational semantics based on SLD-resolution with constraints. Although the complexity of real-time logics is very high in general — the validity problem for most of them is Π 1 l -complete already for propositional fragments in case of dense time structures — we show that the class of bounded universal Horn formulae admits complete and efficient proof methods exploiting uniform proofs and linear time complexity of basic steps of the proof method. The results obtained heavily rely on the fragment investigated and make it necessary to establish some basic results like compactness and approximation of the least model by at most ω-steps of the corresponding fixed point operator directly without recourse to standard methods (in dense case). The fragment itself is sufficiently expressive for a variety of applications ranging from real-time systems, temporal (deductive) data bases, and sequence evaluation purposes. We show that the fragment is the greatest of the metric temporal logic — in discrete and dense case — having the properties classically desired for logic programming languages.