A decision procedure for combinations of propositional temporal logic and other specialized theories

Abstract

We present a decision procedure for formulae of discrete linear time propositional temporal logic whose propositional part may include assertions in a specialized theory. The combined decision procedure may be viewed as an extension of known decision procedures for quantifier-free theories to theories including temporal logic connectives. A combined decision procedure given by Pratt restricted to linear time temporal logic, runs in polynomial space relative to an oracle for the underlying theory. Our procedure differs from this one in that it can handle assertions containing arbitrary mixtures of global variables, whose values cannot change with time, and state variables, whose values can change with time. This new procedure can also handle assertions containing functions and predicates whose interpretations do not change with time. However, it requires the computation of least and greatest fixed points and has a worse asymptotic running time than that of Pratt. This new procedure has been implemented and seems efficient enough to be practical on simple formulae, although an upper bound derived for the worst case running time is triple exponential in the length of the formula. The techniques used appear to apply to logics other than temporal logic which have decision procedures based on tableaux.