Minimal Proof Search for Modal Logic K Model Checking

  • Abdallah Saffidine
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7519)

Abstract

Most modal logics such as S5, LTL, or ATL are extensions of Modal Logic K. While the model checking problems for LTL and to a lesser extent ATL have been very active research areas for the past decades, the model checking problem for the more basic MMLK has important applications as a formal framework for perfect information multi-player games on its own.

We present MPS, an effort number based algorithm solving the model checking problem for MMLK. We prove two important properties for MPS beyond its correctness. The (dis)proof exhibited by MPS is of minimal cost for a general definition of cost, and MPS is an optimal algorithm for finding (dis)proofs of minimal cost. Optimality means that any comparable algorithm either needs to explore a bigger or equal state space than MPS, or is not guaranteed to find a (dis)proof of minimal cost on every input.

As such, our work relates to A* and AO* in heuristic search, to Proof Number Search and DFPN+ in two-player games, and to counterexample minimization in software model checking.

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References

  1. 1.
    Allis, L.V., van der Meulen, M., van den Herik, H.J.: Proof-Number Search. Artificial Intelligence 66(1), 91–124 (1994)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Blackburn, P., De Rijke, M., Venema, Y.: Modal Logic, vol. 53. Cambridge University Press (2001)Google Scholar
  3. 3.
    Clarke, E.M., Grumberg, O., McMillan, K.L., Zhao, X.: Efficient generation of counterexamples and witnesses in symbolic model checking. In: Proceedings of the 32nd Annual ACM/IEEE Design Automation Conference, pp. 427–432. ACM (1995)Google Scholar
  4. 4.
    Clarke, E.M., Grumberg, O., Peled, D.A.: Model checking. The MIT Press (1999)Google Scholar
  5. 5.
    Clarke, E.M., Jha, S., Lu, Y., Veith, H.: Tree-like counterexamples in model checking. In: Proceedings of the 17th Annual IEEE Symposium on Logic in Computer Science, pp. 19–29. IEEE (2002)Google Scholar
  6. 6.
    Cleaveland, R.: Tableau-based model checking in the propositional mu-calculus. Acta Informatica 27(8), 725–747 (1989)MathSciNetMATHGoogle Scholar
  7. 7.
    Groce, A., Visser, W.: What went wrong: Explaining counterexamples. In: Model Checking Software, pp. 121–136 (2003)Google Scholar
  8. 8.
    Kishimoto, A., Müller, M.: A solution to the GHI problem for depth-first proof-number search. Information Sciences 175(4), 296–314 (2005)CrossRefGoogle Scholar
  9. 9.
    Korf, R.E.: Depth-first iterative-deepening: an optimal admissible tree search. Artificial Intelligence 27(1), 97–109 (1985)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lange, M.: Model checking propositional dynamic logic with all extras. Journal of Applied Logic 4(1), 39–49 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lomuscio, A., Qu, H., Raimondi, F.: MCMAS: A Model Checker for the Verification of Multi-Agent Systems. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 682–688. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Müller, M.: Proof-Set Search. In: Schaeffer, J., Müller, M., Björnsson, Y. (eds.) CG 2002. LNCS, vol. 2883, pp. 88–107. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Nagai, A.: Df-pn algorithm for searching AND/OR trees and its applications. Ph.D. thesis, University of Tokyo (December 2001)Google Scholar
  14. 14.
    Pearl, J.: Heuristics: intelligent search strategies for computer problem solving. Addison Wesley Publishing Company (1984)Google Scholar
  15. 15.
    Schijf, M., Allis, L.V., Uiterwijk, J.W.: Proof-number search and transpositions. ICCA Journal 17(2), 63–74 (1994)Google Scholar
  16. 16.
    Shoham, Y., Leyton-Brown, K.: Multiagent systems: Algorithmic, game-theoretic, and logical foundations. Cambridge University Press (2009)Google Scholar
  17. 17.
    van der Hoek, W., Wooldridge, M.J.: Model checking knowledge and time. In: Bošnački, D., Leue, S. (eds.) SPIN 2002. LNCS, vol. 2318, pp. 95–111. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  18. 18.
    van Ditmarsch, H.P., van der Hoek, W., Kooi, B.P.: Concurrent dynamic epistemic logic for MAS. In: Proceedings of the Second International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 201–208. ACM (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Abdallah Saffidine
    • 1
  1. 1.LAMSADEUniversité Paris-DauphineFrance

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