17 Automata-theoretic techniques for temporal reasoning

Abstract

This chapter describes an automata-theoretic approach to temporal reasoning. The basic idea underlying this approach is that for any temporal formula, one can construct a finite state automaton that accepts the computations that satisfy the formula. For linear temporal logics, the automaton runs on infinite words while for branching temporal logics the automaton runs on infinite trees. The simple combinatorial structures that emerge from the automata-theoretic approach decouple the logical and algorithmic components of temporal reasoning and yield clear and asymptotically optimal algorithms. Many modal and temporal logics can be viewed as fragments of monadic second-order logic over trees in a suitable signature, so there is a clear theoretical link between modal logic and automata theory. This link turns out to have practical repercussions for computational applications. Two types of temporal logics can be distinguished: linear and branching. By viewing temporal formulas as giving rise to “alternating automata”, a theoretically transparent and practical perspective on both validity and model checking can be gained, one of the most significant applications of contemporary modal logic.