Abstract
We investigate the computational behaviour of ‘two-dimensional’ propositional temporal logics over (ℕ, <) (with and without the next-time operator O) that are capable of reasoning about states with transitive relations. Such logics are known to be undecidable (even \(\Pi^{\rm 1}_{\rm 1}\)-complete) if the domains of states with those relations are assumed to be constant. Motivated by applications in the areas of temporal description logic and specification & verification of hybrid systems, in this paper we analyse the computational impact of allowing the domains of states to expand. We show that over finite expanding domains (with an arbitrary, tree-like, quasi-order, or linear transitive relation) the logics are recursively enumerable, but undecidable. If these finite domains eventually become constant then the resulting O-free logics are decidable (but not in primitive recursive time); on the other hand, when equipped with O they are not even recursively enumerable. Finally, we show that temporal logics over infinite expanding domains as above are undecidable even for the language with the sole temporal operator ‘eventually.’ The proofs are based on Kruskal’s tree theorem and reductions of reachability problems for lossy channel systems.
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References
Artemov, S., Davoren, J., Nerode, A.: Modal logics and topological semantics for hybrid systems. Technical Report MSI 97-05, Cornell University (1997)
Brand, D., Zafiropulo, P.: On communicating finite-state machines. Journal of the ACM 30, 323–342 (1983)
Davoren, J., Nerode, A.: Logics for hybrid systems. Proceedings of the IEEE 88, 985–1010 (2000)
Finkel, A.: Decidability of the termination problem for completely specified protocols. Distributed Computing 7, 129–135 (1994)
Fisher, M.: A resolution method for temporal logic. In: Myopoulos, J., Reiter, R. (eds.) Proceedings of IJCAI 1991, pp. 99–104. Morgan Kaufman, San Francisco (1991)
Fisher, M., Dixon, C., Peim, M.: Clausal temporal resolution. ACM Transactions on Computational Logic (TOCL) 2(1), 12–56 (2001)
Gabbay, D., Hodkinson, I., Reynolds, M.: Temporal Logic, vol. 1. Oxford University Press, Oxford (1994)
Gabbay, D., Kurucz, A., Wolter, F., Zakharyaschev, M.: Many-Dimensional Modal Logics: Theory and Applications. In: Studies in Logic, vol. 148, Elsevier, Amsterdam (2003)
Gabelaia, D., Kontchakov, R., Kurucz, A., Wolter, F., Zakharyaschev, M.: Combining spatial and temporal logics: expressiveness vs. complexity. Journal of Artificial Intelligence Research 23, 167–243 (2005)
Gabelaia, D., Kurucz, A., Wolter, F., Zakharyaschev, M.: Non-primitive recursive decidability of products of modal logics with expanding domains. Manuscript (2004), Available at http://www.dcs.kcl.ac.uk/staff/mz
Gabelaia, D., Kurucz, A., Wolter, F., Zakharyaschev, M.: Products of ‘transitive’ modal logics. Journal of Symbolic Logic (2005). (in print) Draft, available at http://www.dcs.kcl.ac.uk/staff/mz
Halpern, J., Vardi, M.: The complexity of reasoning about knowledge and time I: lower bounds. Journal of Computer and System Sciences 38, 195–237 (1989)
Higman, G.: Ordering by divisibility in abstract algebras. Proceedings of the London Mathematical Society 2, 326–336 (1952)
Hodkinson, I., Wolter, F., Zakharyaschev, M.: Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic 106, 85–134 (2000)
Hustadt, U., Konev, B.: TRP++ 2.0: A temporal resolution prover. In: Baader, F. (ed.) CADE 2003. LNCS (LNAI), vol. 2741, pp. 274–278. Springer, Heidelberg (2003)
Hustadt, U., Konev, B., Riazanov, A., Voronkov, A.: TeMP: A temporal monodic prover. In: Proceedings IJCAR 2004. LNCS (LNAI), vol. 3097, pp. 326–330. Springer, Heidelberg (2004)
Konev, B., Degtyarev, A., Dixon, C., Fisher, M., Hustadt, U.: Towards the implementation of first-order temporal resolution: the expanding domain case. Information and Computation. In: print. Available as Technical Report ULCS-03-005, The University of Liverpool, Department of Computer Science (2005)
Konev, B., Kontchakov, R., Wolter, F., Zakharyaschev, M.: On dynamic topological and metric logics. Manuscript (2005), Available at http://www.dcs.kcl.ac.uk/staff/mz
Kontchakov, R., Lutz, C., Wolter, F., Zakharyaschev, M.: Temporalising tableaux. Studia Logica 76, 91–134 (2004)
Kremer, P., Mints: Dynamic topological logic. Bulletin of Symbolic Logic 3, 371–372 (1997)
Kremer, P., Mints, G.: Dynamic topological logic. Annals of Pure and Applied Logic 131, 133–158 (2005)
Kruskal, J.B.: Well-quasi-orderings, the tree theorem, and Vázsonyi’s conjecture. Transactions of the American Mathematical Society 95, 210–225 (1960)
Lutz, C., Sattler, U., Wolter, F.: Modal logic and the two-variable fragment. In: Fribourg, L. (ed.) CSL 2001 and EACSL 2001. LNCS, vol. 2142, pp. 262–276. Springer, Heidelberg (2001)
Mayr, R.: Undecidable problems in unreliable computations. Theoretical Computer Science 297, 337–354 (2003)
Schild, K.: Combining terminological logics with tense logic. In: Proceedings of the 6th Portuguese Conference on Artificial Intelligence, Porto, pp. 105–120 (1993)
Schnoebelen, P.: Verifying lossy channel systems has nonprimitive recursive complexity. Information Processing Letters 83, 251–261 (2002)
Schwendimann, S.: Aspects of Computational Logic. PhD thesis, Universität Bern, Switzerland (1998)
Sistla, A., Clarke, E.: The complexity of propositional linear temporal logics. Journal of the Association for Computing Machinery 32, 733–749 (1985)
Sistla, A., German, S.: Reasoning with many processes. In: Proceedings of the Second IEEE Symposium on Logic in Computer Science, pp. 138–153 (1987)
Wolper, P.: The tableau method for temporal logic: An overview. Logique et Analyse 28, 119–152 (1985)
Wolter, F., Zakharyaschev, M.: Temporalizing description logics. In: Gabbay, D., de Rijke, M. (eds.) Frontiers of Combining Systems II, pp. 379–401. Studies Press/Wiley, Chichester (2000)
Wolter, F., Zakharyaschev, M.: Axiomatizing the monodic fragment of first-order temporal logic. Annals of Pure and Applied Logic 118, 133–145 (2002)
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Konev, B., Wolter, F., Zakharyaschev, M. (2005). Temporal Logics over Transitive States. In: Nieuwenhuis, R. (eds) Automated Deduction – CADE-20. CADE 2005. Lecture Notes in Computer Science(), vol 3632. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11532231_14
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