Verification, Induction, Termination Analysis pp 117-128

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6463) | Cite as

Verifying the Modal Logic Cube Is an Easy Task (For Higher-Order Automated Reasoners)

  • Christoph Benzmüller

Abstract

Prominent logics, including quantified multimodal logics, can be elegantly embedded in simple type theory (classical higher-order logic). Furthermore, off-the-shelf reasoning systems for simple type type theory exist that can be uniformly employed for reasoning within and about embedded logics. In this paper we focus on reasoning about modal logics and exploit our framework for the automated verification of inclusion and equivalence relations between them. Related work has applied first-order automated theorem provers for the task. Our solution achieves significant improvements, most notably, with respect to elegance and simplicity of the problem encodings as well as with respect to automation performance.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andrews, P.B., Brown, C.: TPS: A Hybrid Automatic-Interactive System for Developing Proofs. Journal of Applied Logic 4(4), 367–395 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Andrews, P.B.: General models and extensionality. Journal of Symbolic Logic 37, 395–397 (1972)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Andrews, P.B.: General models, descriptions, and choice in type theory. Journal of Symbolic Logic 37, 385–394 (1972)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Andrews, P.B.: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, 2nd edn. Kluwer Academic Publishers, Dordrecht (2002)CrossRefMATHGoogle Scholar
  5. 5.
    Backes, J., Brown, C.E.: Analytic tableaux for higher-order logic with choice. In: Giesl, J., Hähnle, R. (eds.) Automated Reasoning. LNCS (LNAI), vol. 6173, pp. 76–90. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Benzmüller, C.: Combining logics in simple type theory. In: The 11th International Workshop on Computational Logic in Multi-Agent Systems, Lisbon, Portugal. Lecture Notes in Artifical Intelligence, Springer, Heidelberg (2010)Google Scholar
  7. 7.
    Benzmüller, C., Brown, C.E., Kohlhase, M.: Higher order semantics and extensionality. Journal of Symbolic Logic 69, 1027–1088 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    C. Benzmüller, L.C. Paulson. Quantified Multimodal Logics in Simple Type Theory. SEKI Report SR–2009–02 (ISSN 1437-4447). SEKI Publications, DFKI Bremen GmbH, Safe and Secure Cognitive Systems, Cartesium, Enrique Schmidt Str. 5, D–28359 Bremen, Germany (2009), http://arxiv.org/abs/0905.2435
  9. 9.
    Benzmüller, C., Paulson, L.C.: Multimodal and intuitionistic logics in simple type theory. The Logic Journal of the IGPL (2010)Google Scholar
  10. 10.
    Benzmüller, C., Paulson, L.C., Theiss, F., Fietzke, A.: LEO-II — A Cooperative Automatic Theorem Prover for Higher-Order Logic. In: Baumgartner, P., Armando, A., Gilles, D. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 162–170. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Benzmüller, C., Rabe, F., Sutcliffe, G.: THF0 — The Core TPTP Language for Classical Higher-Order Logic. In: Baumgartner, P., Armando, A., Gilles, D. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 491–506. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Blackburn, P., Marx, M.: Tableaux for quantified hybrid logic. In: Egly, U., Fermüller, C.G. (eds.) TABLEAUX 2002. LNCS (LNAI), vol. 2381, pp. 38–52. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Blanchette, J.C., Nipkow, T.: Nitpick: A counterexample generator for higher-order logic based on a relational model finder. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 131–146. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Braüner, T.: Natural deduction for first-order hybrid logic. Journal of Logic, Language and Information 14(2), 173–198 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Church, A.: A formulation of the simple theory of types. Journal of Symbolic Logic 5, 56–68 (1940)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fitting, M.: Interpolation for first order S5. Journal of Symbolic Logic 67(2), 621–634 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Garson, J.: Modal logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, Winter 2009 edition (2009)Google Scholar
  18. 18.
    Goldblatt, R.: Logics of Time and Computation, 2nd edn. Lecture Notes, vol. 7. Center for the Study of Language and Information, Stanford (1992)MATHGoogle Scholar
  19. 19.
    Halleck, J.: John Halleck’s Logic Systems, http://www.cc.utah.edu/~nahaj/logic/structures/systems/index.html
  20. 20.
    Henkin, L.: Completeness in the theory of types. Journal of Symbolic Logic 15, 81–91 (1950)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kaminski, M., Smolka, G.: Terminating tableau systems for hybrid logic with difference and converse. Journal of Logic, Language and Information 18(4), 437–464 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL - A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002)Google Scholar
  23. 23.
    Rabe, F., Pudlak, P., Sutcliffe, G., Shen, W.: Solving the $100 Modal Logic Challenge. Journal of Applied Logic (2008) (page to appear)Google Scholar
  24. 24.
    Schulz, S.: E: A Brainiac Theorem Prover. AI Communications 15(2-3), 111–126 (2002)MATHGoogle Scholar
  25. 25.
    Sutcliffe, G.: TPTP, TSTP, CASC, etc. In: Diekert, V., Volkov, M., Voronkov, A. (eds.) CSR 2007. LNCS, vol. 4649, pp. 7–23. Springer, Heidelberg (2007)Google Scholar
  26. 26.
    Sutcliffe, G.: The TPTP problem library and associated infrastructure. J. Autom. Reasoning 43(4), 337–362 (2009)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Sutcliffe, G., Benzmüller, C.: Automated reasoning in higher-order logic using the TPTP THF infrastructure. Journal of Formalized Reasoning 3(1), 1–27 (2010)MathSciNetMATHGoogle Scholar
  28. 28.
    Weber, T.: SAT-Based Finite Model Generation for Higher-Order Logic. Ph.D. thesis, Dept. of Informatics, T.U. München (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Christoph Benzmüller
    • 1
  1. 1.Articulate SoftwareAngwinU.S.

Personalised recommendations