A propositional dense time logic
This paper extends propositional linear time temporal logic (PTL) to propositional dense time logic (PDTL). While a PTL model is a single sequence of states, a PDTL model, called an omega-tree, consists of a nested sequence of states. Two new operators, called within and everywhere are introduced to access nested sequences. Besides its application in describing activities for Artificial Intelligence, PDTL can be used to represent more naturally procedural abstractions in control flow. PDTL is shown to be decidable by a tableau based method, and a complete axiomatization is given.
PDTL's omega tree models allow a dense mix of events. By imposing a stability condition on the propositions we get a subset of the omega tree models called ordinal trees which are free of dense mix. This logic called Propositional Ordinal Tree Logic (POTL) is also shown to be decidable in exponential time. Ordinal tree models though based on dense points, represent interval based information which maybe refined to any finite level. Hence POTL is a good bridge between point based and interval based temporal logics.
Ordinal trees can be easily embedded as a temporal data structure in a conventional logic programming language and thus provide a framework for temporal logic programming.
KeywordsTemporal logic dense time ordinal trees
- M. Abadi and Z. Manna. Temporal logic programming. In International Conference on Logic Programming San Fransisco, CA, pages 4–16, 1987.Google Scholar
- 2]M. Ahmed and G. Venkatesh. Dense time logic programming. In Second Symposium on Logical Formalizations of Commonsenae Reasoning, Austin, TX, 1993.Google Scholar
- R. Alur and T. Henzinger. Real time logics: Complexity and expressiveness. In Logic in Computer Science, pages 390–01, 1990.Google Scholar
- M. Baudinet. Temporal logic programming is complete and expressive. In 16th POPL, Austin, TX, pages 267–79, 1989.Google Scholar
- J. P. Burgess. Basic tense logic. In D. Gabbay and F. Guenther, editors, Handbook of Philosophical Logic, volume II, pages 89–133. D. Reidel, Dordrecht, Holland, 1984.Google Scholar
- D. Gabbay. Modal and temporal logic programming. In A. Galton, editor, Temporal Logics and their applications, pages 195–273. Academic Press, 1987.Google Scholar
- D. Gabbay. Modal and temporal logic programming-ii. In T. Dodd, editor, Logic Programming, pages 82–123. Intellect, Oxford, 1991.Google Scholar
- D. Gabbay, A. Pnueli, S. Shelah, and S. Stavi. The temporal analysis of fairness. In 7th POPL, Las Vegas, NE, pages 163–173, 1980.Google Scholar
- D. Harel. Propositional dynamic logic. In D. Gabbay and F. Guenther, editors, Handbook of Philosophical Logic, volume II, pages 507–544. D. Reidel, Dordrecht, Holland, 1984.Google Scholar
- L. Lamport. Temporal logic of actions. TR 57, Digital, 1990.Google Scholar
- O. Lichtenstein, A. Pnueli, and L. Zuck. The glory of the past. In Proc. Conf. Logics of Programs, pages 196–218. Springer Verlag, 1985. LNCS 193.Google Scholar
- B. C. Moszkowski. Executing Temporal Logic Programs. Cambridge University Press, Cambridge, 1986.Google Scholar
- M. O. Rabin. Decidability of second order theories and automata on infinite trees. Transactions of AMS, 141:1–35, July 1969.Google Scholar
- W. Thomas. Automata on infinite trees. In J. V. Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, pages 165–186. North Holland, Amsterdam, 1990.Google Scholar
- J. F. A. K. van Benthem. The Logic of Time. D. Reidel, Dordrecht, Holland, 1983.Google Scholar
- J. F. A. K. van Benthem. Time, logic and computation. In J. deBakker, W. deRoever, and G. Rozenberg, editors, Linear Time, Branching Time and Partial Order in Logics and Models of Concurrency. Springer Verlag, 1989. LNCS 354.Google Scholar