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2-Exp Time lower bounds for propositional dynamic logics with intersection

  • Martin Lange (a1) and Carsten Lutz (a2)
Abstract

In 1984. Danecki proved that satisfiability in IPDL, i.e., Propositional Dynamic Logic (PDL) extended with an intersection operator on programs, is decidabie in deterministic double exponential time. Since then, the exact complexity of IPDL has remained an open problem: the best known lower bound was the ExpTime one stemming from plain PDL until, in 2004. the first author established ExpSpace-hardness. In this paper, we finally close the gap and prove that IPDL is hard for 2-ExpTime. thus 2-ExpTime-complete. We then sharpen our lower bound, showing that it even applies to IPDL without the test operator interpreted on tree structures.

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[1]Abrahamson, K., Decidability and expressiveness of logics of processes, Ph.D. thesis, University of Washington, Seattle, 1980.
[2]Aeanasiev, L., Blackburn, P., Dimitriou, I., Gaiffe, B., Goris, E., Maarx, M.. and de Rijke, M., PDL for ordered trees, Journal of Applied Non-Classical Logic, vol. 15 (2005), no. 2, pp. 115135.
[3]Alechina, N., Demri, S., and de Rijke, M., A modal perspective on path constraints, Journal of Logic and Computation, vol. 13 (2003), no. 6, pp. 939956.
[4]Baader, F., Augmenting concept languages by transitive closure of roles: An alternative to terminological cycles, Proceedings of the twelfth international joint conference on artificial intelligence (IJCAI-91), Sydney, Australia, 1991, pp. 446451.
[5]Baader, F., McGuiness, D. L., Nardi, D., and Patel-Schneider, P., The description logic handbook: Theory, implementation and applications, Cambridge University Press, 2003.
[6]Calvanese, D., de Giacomo, G., and Lenzerini, M., On the decidability of query containment under constraints, Proceedings of the 17th ACM SIGACT SIGMOD SIGART Symposium on Principles of Database Systems (PODS'98), 1998, pp. 149158.
[7]Chandra, A. K., Kozen, D. C., and Stockmeyer, L. J., Alternation, Journal of the ACM, vol. 28 (1981), no. 1, pp. 114133.
[8]Danecki, R., Nondeterministic propositional dynamic logic with intersection is decidable, Proceedings of the Fifth Symposium on Computation Theory (Zaborów, Poland) (Skowron, A., editor), LNCS, vol. 208, Springer, 1984, pp. 3453.
[9]Donini, F. M., Lenzerini, M., Nardi, D., and Nutt, W., The complexity of concept languages, Information and Computation, vol. 134 (1997), no. 1, pp. 158.
[10]Fischer, M. J. and Ladner, R. E., Propositional dynamic logic of regular programs, Journal of Computer and System Sciences, vol. 18 (1979), no. 2, pp. 194211.
[11]de Giacomo, G. and Lenzerini, M., Boosting the correspondence between description logics and propositional dynamic logics, Proceedings of the Twelfth National Conference on Artificial Intelligence (AAAI'94). Volume 1, AAAI Press, 1994, pp. 205212.
[12]Goldblatt, R., Parallel action: Concurrent dynamic logic with independent modalities, Studia Logica, vol. 51 (1992), no. 3-4, pp. 551578.
[13]Harel, D., Dynamic logic, Handbook of philosophical logic, volume II (Gabbay, D. M. and Guenthner, F., editors), D. Reidel Publishers, 1984, pp. 496604.
[14]Harel, D., Kozen, D., and Tiuryn, J., Dynamic logic, MIT Press, 2000.
[15]Harel, D., Rosner, R., and Vardi, M., On the power of bounded concurrency III: Reasoning about programs, Proceedings of the 5th Annual IEEE Symposium on Logic in Computer Science (Mitchell, J. C., editor),IEEE Computer Society Press, 1990, pp. 478488.
[16]Harel, D. and Sherman, R., Looping versus repeating in dynamic logic, Information and control, vol. 55 (1982), pp. 175192.
[17]Johannsen, J. and Lange, M., CTL+ is complete for double exponential time, ICALP: Annual International Colloquium on Automata, Languages and Programming (Baeten, J. C. M.et al., editors), LNCS, vol. 2719, Springer, 2003, pp. 767775.
[18]Kozen, D. C. and Tiuryn, J., Logics of programs, Handbook of theoretical computer science (van Leewen, J., editor), vol. B: Formal Models and Semantics, The MIT Press, 1990, pp. 789840.
[19]Lange, M., A lower complexity bound for propositional dynamic logic with intersection, Advances in modal logic volume 5 (Schmidt, R. A., Pratt-Hartmann, I., Reynolds, M., and Wansing, H., editors), King's College Publications, 2005, pp. 133147.
[20]Lutz, C. and Sattler, U., Mary likes all cats, Proceedings of the 2000 International Workshop in Description Logics (DL2000) (Baader, F. and Sattler, U., editors), CEUR-WS (http://ceur-ws.org/). no. 33, 2000, pp. 213226.
[21]Lutz, C. and Walther, D., PDL with negation of atomic programs, Journal of Applied Non-Classical Logic, vol. 15 (2005), no. 2, pp. 189214.
[22]Massacci, F., Decision procedures for expressive description logics with role intersection, composition and converse, Proceedings of the seventeenth International Conference on Artificial Intelligence (IJCAI-01) (Nebel, B., editor), Morgan Kaufmann Publishers, Inc., San Francisco, CA, 2001, pp. 193198.
[23]Mayer, A. J. and Stockmeyer, L. J., The complexity of PDL with interleaving, Theoretical Computer Science, vol. 161 (1996), no. 1-2, pp. 109122.
[24]Peleg, D., Concurrent dynamic logic, Proceedings of the seventeenth annual ACM Symposium on Theory of Computing, Providence, Rhode Island, May 6–8, 1985, ACM Press, 1985, pp. 232239.
[25]Pratt, V., Considerations on Floyd-Hoare logic, IEEE Symposium on Foundations of Computer Science (FOCS), 1976, pp. 109121.
[26]Schild, K. D., A correspondence theory for terminological logics: Preliminary report, Proceedings of the Twelfth International Joint Conference on Artificial Intelligence (IJCA1-91) (Mylopoulos, John and Reiter, Ray, editors), Morgan Kaufmann, 1991, pp. 466471.
[27]Schild, K. D., Combining terminological logics with tense logic, Progress in artificial intelligence – 6th Portuguese Conference on Artificial Intelligence, EPIA'93 (Filgueiras, M. and Damas, L., editors), Lecture Notes in Artificial Intelligence, vol. 727, Springer-Verlag, 1993, pp. 105120.
[28]Streett, R. S., Propositional dynamic logic of looping and converse is elementarily decidable, Information and Control, vol. 54 (1982), no. 1-2, pp. 121141.
[29]Vardi, M. Y., The taming of converse: Reasoning about two-way computations, Proceedings of the Conference on Logic of Programs (Parikh, Rohit, editor), LNCS, vol. 193, Springer, 1985, pp. 413424.
[30]Vardi, M. Y. and Stockmeyer, L., Improvedupper and lower bounds for modal logics of programs, Proceedings of the seventeenth annual ACM Symposium on Theory of Computing, Providence, Rhode Island, May 6–8, 1985, ACM Press, 1985, pp. 240251.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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