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Products of ‘transitive” modal logics

Published online by Cambridge University Press:  12 March 2014

D. Gabelaia ,
A. Kurucz ,
F. Wolter  and
M. Zakharyaschev
D. Gabelaia
Affiliation:
Department of Computer Science, King's College London, Strand, London WC2R 2LS, UKE-mail:, gabelaia@dcs.kcl.ac.uk
A. Kurucz
Affiliation:
Department of Computer Science, King's College London, Strand, London WC2R 2LS, UKE-mail:, kuag@dcs.kcl.ac.uk
F. Wolter
Affiliation:
Department of Computer Science, King's College London, Strand, London WC2R 2LS, UKE-mail:, mz@dcs.kcl.ac.uk
M. Zakharyaschev
Affiliation:
Department of Computer Science, University of Liverpool, Liverpool L69 7ZF, UKE-mail:, frank@csc.liv.ac.uk
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Abstract

We solve a major open problem concerning algorithmic properties of products of ‘transitive’ modal logics by showing that products and commutators of such standard logics as K4, S4, S4.1, K4.3, GL, or Grz are undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4, K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, if l1 and l2 are classes of transitive frames such that their depth cannot be bounded by any fixed n < ω, then the logic of the class {5ℑ1 × ℑ2 ∣ ℑ1 ∈ l1, ℑ2, ∈ l2} is undecidable. (On the contrary, the product of, say, K4 and the logic of all transitive Kripke frames of depth ≤ n, for some fixed n < ω, is decidable.) The complexity of these undecidable logics ranges from r.e. to co-r.e. and Π11-complete. As a consequence, we give the first known examples of Kripke incomplete commutators of Kripke complete logics.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 3 , September 2005 , pp. 993 - 1021
Copyright
Copyright © Association for Symbolic Logic 2005

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References

REFERENCES

[1]Artale, A. and Franconi, E., A survey of temporal extensions of description logics, Annals of Mathematics and Artificial Intelligence, vol. 30 (2001), pp. 171–210.CrossRefGoogle Scholar
[2]Artemov, S., Davoren, J., and Nerode, A., Modal logics and topological semantics for hybrid systems, Technical Report MSI 97-05, Cornell University, 1997.CrossRefGoogle Scholar
[3]Baader, F. and Ohlbach, H.J., A multi-dimensional terminological knowledge representation language, Journal of Applied Non-Classical Logics, vol. 5 (1995), pp. 153–197.CrossRefGoogle Scholar
[4]Berger, R., The undecidability of the domino problem, Memoirs of the American Mathematical Society, vol. 66 (1966).Google Scholar
[5]Chagrov, A. and Zakharyaschev, M., Modal logic, Oxford Logic Guides, vol. 35, Clarendon Press, Oxford, 1997.CrossRefGoogle Scholar
[6]Davoren, J. and Goré, R., Bimodal logics for reasoning about continuous dynamics, Advances in modal logic (Wolter, F., Wansing, H., de Rijke, M., and Zakharyaschev, M., editors), vol. 3, World Scientific, 2002, pp. 91–112.CrossRefGoogle Scholar
[7]Davoren, J. and Nerode, A., Logics for hybrid systems, Proceedings of the IEEE, vol. 88 (2000), pp. 985–1010.CrossRefGoogle Scholar
[8]Fagin, R., Halpern, J., Moses, Y., and Vardi, M., Reasoning about knowledge, MIT Press, 1995.Google Scholar
[9]Fine, K., Logics containing K4, Part I, this Journal, vol. 39 (1974), pp. 229–237.Google Scholar
[10]Fine, K., Logics containing K4, Part II, this Journal, vol. 50 (1985), pp. 619–651.Google Scholar
[11]Gabbay, D., Kurucz, A., Wolter, F., and Zakharyaschev, M., Many-dimensional modal logics: Theory and applications, Studies in Logic, vol. 148, Elsevier, 2003.Google Scholar
[12]Gabbay, D. and Shehtman, V., Products of modal logics. Part I, Logic Journal of the IGPL, vol. 6 (1998), pp. 73–146.CrossRefGoogle Scholar
[13]Gabbay, D. and Shehtman, V., Products of modal logics, Part III: Products of modal and temporal logics, Studio Logica, vol. 72 (2002), pp. 157–183.CrossRefGoogle Scholar
[14]Gabelaia, D., Topological semantics and two-dimensional combinations of modal logics, Ph.D. thesis, King's College London, 2005.Google Scholar
[15]Gabelaia, D., Kontchakov, R., Kurucz, A., Wolter, F., and Zakharyaschev, M., Combining spatial and temporal logics: expressiveness vs. complexity, Journal of Artificial Intelligence Research, vol. 23 (2005), pp. 167–243.CrossRefGoogle Scholar
[16]Gabelaia, D., Kurucz, A., Wolter, F., and Zakharyaschev, M., Non-primitive recursive decidability of products of modal logics with expanding domains. Submitted, available at http://dcs.kcl.ac.uk/staff/mz/expand.pdf, 2005.CrossRefGoogle Scholar
[17]Harel, D., A simple highly undecidable domino problem. Proceedings of the Conference on Logic and Computation (Clayton, Victoria, Australia), 01 1984.Google Scholar
[18]Harel, D., Recurring dominoes: Making the highly undecidable highly understandable. Annals of Discrete Mathematics, vol. 24 (1985), pp. 51–72.Google Scholar
[19]Harel, D., Pnueli, A., and Stavi, J., Propositional dynamic logic of nonregular programs. Journal of Computer and System Sciences, vol. 26 (1983), pp. 222–243.CrossRefGoogle Scholar
[20]Hirsch, R., Hodkinson, I., and Kurucz, A., On modal logics between K × K × K and S5 × S5 × S5, this Journal, vol. 67 (2002), pp. 221–234.Google Scholar
[21]Hopcroft, J.E., Motwani, R., and Ullman, J.D, Introduction to automata theory, languages, and computation, Addison-Wesley, 2001.Google Scholar
[22]Konev, B., Kontchakov, R., Wolter, F., and Zakharyaschev, M., On dynamic topological and metric logics, Proceedings of AiML-2004 (Manchester) (Schmidt, R., Pratt-Hartmann, I., Reynolds, M., and Wansing, H., editors), 09 2004, pp. 182–196.Google Scholar
[23]Konev, B., Wolter, F., and Zakharyaschev, M., Temporal logics over transitive states, Proceedings of the 20th International Conference on Automated Deduction (CADE-20), Lecture Notes in Computer Science, Springer, 2005 (In print).Google Scholar
[24]Kremer, P. and Mints, G., Dynamic topological logic, The Bulletin of Symbolic Logic, vol. 3 (1997), pp. 371–372.Google Scholar
[25]Kremer, P., Dynamic topological logic, Annals of Pure and Applied Logic, vol. 131 (2005), pp. 133–158.CrossRefGoogle Scholar
[26]Kurucz, A. and Zakharyaschev, M., A note on relativised products of modal logics, Advances in modal logic (Balbiani, P., Suzuki, N.-Y., Wolter, F., and Zakharyaschev, M., editors), vol. 4, King's College Publications, 2003, pp. 221–242.Google Scholar
[27]Marx, M. and Mikulás, Sz., An elementary construction for a non-elementary procedure, Studio Logica, vol. 72 (2002), pp. 253–263.CrossRefGoogle Scholar
[28]Marx, M. and Reynolds, M., Undecidability of compass logic, Journal of Logic and Computation, vol. 9 (1999), pp. 897–914.CrossRefGoogle Scholar
[29]Reif, J. and Sistla, A., A multiprocess network logic with temporal and spatial modalities, Journal of Computer and System Sciences, vol. 30 (1985), pp. 41–53.CrossRefGoogle Scholar
[30]Reynolds, M., A decidable temporal logic of parallelism, Notre Dame Journal of Formal Logic, vol. 38 (1997), pp. 419–436.CrossRefGoogle Scholar
[31]Reynolds, M. and Zakharyaschev, M., On the products of linear modal logics, Journal of Logic and Computation, vol. 11 (2001), pp. 909–931.CrossRefGoogle Scholar
[32]Segerberg, K., Two-dimensional modal logic, Journal of Philosophical Logic, vol. 2 (1973), pp. 77–96.CrossRefGoogle Scholar
[33]Shehtman, V., Two-dimensional modal logics, Mathematical Notices of the USSR Academy of Sciences, vol. 23 (1978), pp. 417–424, (Translated from Russian).Google Scholar
[34]Shehtman, V., A new version of the filtration method, Proceedings of AiML-2004 (Manchester) (Schmidt, R., Pratt-Hartmann, I., Reynolds, M., and Wansing, H., editors), 09 2004, pp. 344–356.Google Scholar
[35]Spaan, E., Complexity of modal logics, Ph.D. thesis, Department of Mathematics and Computer Science, University of Amsterdam, 1993.Google Scholar
[36]Thomason, S., Reduction of second-order logic to modal logic, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 107–114.CrossRefGoogle Scholar
[37]Wang, H., Dominos and the case of the decision problem, Mathematical theory of automata, Polytechnic Institute, Brooklyn, 1963, pp. 23–55.Google Scholar
[38]Wolter, F. and Zakharyaschev, M., Qualitative spatio-temporal representation and reasoning: a computational perspective. Exploring Artificial Intelligence in the New Millenium (Lakemeyer, G. and Nebel, B., editors), Morgan Kaufmann, 2002, pp. 175–216.Google Scholar
[39]Zakharyaschev, M., Canonical formulas for K4, Part I: Basic results, this Journal, vol. 57 (1992), pp. 1377–1402.Google Scholar
[40]Zakharyaschev, M. and Alekseev, A., All finitely axiomatizable normal extensions of K4.3 are decidable, Mathematical Logic Quarterly, vol. 41 (1995), pp. 15–23.CrossRefGoogle Scholar