Abstract
We consider the temporal logic with since and until modalities. This temporal logic is expressively equivalent over the class of ordinals to first-order logic thanks to Kamp’s theorem. We show that it has a pspace-complete satisfiability problem over the class of ordinals. Among the consequences of our proof, we show that given the code of some countable ordinal ordinal α and a formula, we can decide in pspace whether the formula has a model over ordinal α. In order to show these results, we introduce a class of simple ordinal automata, as expressive as Büchi ordinal automata. The pspace upper bound for the satisfiability problem of the temporal logic is obtained through a reduction to the nonemptiness problem for the simple ordinal automata.
Partially supported by an invited professorship from ENS de Cachan and project AutoMathA (ESF).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bruyère, V., Carton, O.: Automata on linear orderings. Journal of Computer and System Sciences 73, 1–24 (2007)
Büchi, J.R., Siefkes, D.: The monadic second order theory of all countable ordinals. Lecture Notes in Mathematics, vol. 328. Springer, Heidelberg (1973)
Cachat, T.: Controller synthesis and ordinal automata. In: Graf, S., Zhang, W. (eds.) ATVA 2006. LNCS, vol. 4218, pp. 215–228. Springer, Heidelberg (2006)
Demri, S., Nowak, D.: Reasoning about transfinite sequences. International Journal of Foundations of Computer Science 18(1), 87–112 (2007)
Gabbay, D., Hodkinson, I., Reynolds, M.: Temporal Logic - Mathematical Foundations and Computational Aspects, vol. 1. OUP, Oxford (1994)
Godefroid, P., Wolper, P.: A partial approach to model checking. I & C 110(2), 305–326 (1994)
Kamp, J.: Tense Logic and the theory of linear order. PhD thesis, UCLA, USA (1968)
Kupferman, O., Vardi, M.Y., Wolper, P.: An automata-theoretic approach to branching-time model checking. J. ACM 47(2), 312–360 (2000)
Reynolds, M.: The complexity of the temporal logic over the reals. Under submission
Reynolds, M.: The complexity of the temporal logic with until over general linear time. Journal of Computer and System Sciences 66(2), 393–426 (2003)
Rohde, S.: Alternating Automata and The Temporal Logic of Ordinals. PhD thesis, University of Illinois (1997)
Sistla, A., Clarke, E.: The complexity of propositional linear temporal logic. J. ACM 32(3), 733–749 (1985)
Venema, Y.: Completeness via completeness: Since and until. In: de Rijke, M. (ed.) Diamonds and Defaults, pp. 279–286. Kluwer Academic Publishers, Dordrecht (1993)
Vardi, M., Wolper, P.: Reasoning about infinite computations. I & C 115, 1–37 (1994)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Demri, S., Rabinovich, A. (2007). The Complexity of Temporal Logic with Until and Since over Ordinals . In: Dershowitz, N., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2007. Lecture Notes in Computer Science(), vol 4790. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75560-9_38
Download citation
DOI: https://doi.org/10.1007/978-3-540-75560-9_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-75558-6
Online ISBN: 978-3-540-75560-9
eBook Packages: Computer ScienceComputer Science (R0)