Abstract
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.
Chapter PDF
Similar content being viewed by others
References
M. Abadi and Z. Manna. Temporal logic programming. In International Conference on Logic Programming San Fransisco, CA, pages 4–16, 1987.
M. Ahmed and G. Venkatesh. Dense time logic programming. In Second Symposium on Logical Formalizations of Commonsenae Reasoning, Austin, TX, 1993.
R. Alur and T. Henzinger. Real time logics: Complexity and expressiveness. In Logic in Computer Science, pages 390–01, 1990.
M. Baudinet. Temporal logic programming is complete and expressive. In 16th POPL, Austin, TX, pages 267–79, 1989.
M. Ben-Ari, J. Y. Halpern, and A. Pnueli. Deterministic propositional dynamic logic: Finite models, complexity and completeness. Journal of Computer and System Sciences, 25:402–417, 1982.
M. Ben-Ari, A. Pnueli, and Z. Manna. The temporal logic of branching time. Acta Informatica, 20:207–226, 1983.
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.
D. Gabbay. Modal and temporal logic programming. In A. Galton, editor, Temporal Logics and their applications, pages 195–273. Academic Press, 1987.
D. Gabbay. Modal and temporal logic programming-ii. In T. Dodd, editor, Logic Programming, pages 82–123. Intellect, Oxford, 1991.
D. Gabbay, A. Pnueli, S. Shelah, and S. Stavi. The temporal analysis of fairness. In 7th POPL, Las Vegas, NE, pages 163–173, 1980.
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.
L. Lamport. Temporal logic of actions. TR 57, Digital, 1990.
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.
B. C. Moszkowski. Executing Temporal Logic Programs. Cambridge University Press, Cambridge, 1986.
M. O. Rabin. Decidability of second order theories and automata on infinite trees. Transactions of AMS, 141:1–35, July 1969.
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.
J. F. A. K. van Benthem. The Logic of Time. D. Reidel, Dordrecht, Holland, 1983.
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.
P. Wolper. Temporal logic can be more expressive. Information and Control, 56:72–93, 1983.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ahmed, M., Venkatesh, G. (1993). A propositional dense time logic. In: Gaudel, M.C., Jouannaud, J.P. (eds) TAPSOFT'93: Theory and Practice of Software Development. CAAP 1993. Lecture Notes in Computer Science, vol 668. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56610-4_91
Download citation
DOI: https://doi.org/10.1007/3-540-56610-4_91
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56610-6
Online ISBN: 978-3-540-47598-9
eBook Packages: Springer Book Archive