Decidable and undecidable fragments of first-order branching temporal logics

Ian Hodkinson Frank Wolter Michael Zakharyaschev

To appear at Symposium on Logic in Computer Science (LICS'2002), Copenhagen, Denmark, July 22nd - 25th, 2002


In this paper we analyze the decision problem for fragments of first-order extensions of branching time temporal logics such as computational tree logics CTL and CTL* or Prior's Ockhamist logic of historical necessity. On the one hand, we show that the one-variable fragments of logics like first-order CTL* - such as the product of propositional CTL* with simple propositional modal logic S5, or even the one-variable bundled first-order temporal logic with sole temporal operator `some time in the future' - are undecidable. On the other hand, it is proved that by restricting applications of first-order quantifiers to state (i.e., path-independent) formulas, and applications of temporal operators and path quantifiers to formulas with at most one free variable, we can obtain decidable fragments. The same arguments show decidability of `non-local' propositional CTL*, in which truth values of propositional atoms depend on the history as well as the current time. The positive decidability results can serve as a unifying framework for devising expressive and effective time-dependent knowledge representation formalisms, e.g., temporal description or spatio-temporal logics.

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