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The Coinductive Formulation of Common Knowledge

  • Colm Baston
  • Venanzio CaprettaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10895)

Abstract

We study the coinductive formulation of common knowledge in type theory. We formalise both the traditional relational semantics and an operator semantics, similar in form to the epistemic system S5, but at the level of events on possible worlds rather than as a logical derivation system. We have two major new results. Firstly, the operator semantics is equivalent to the relational semantics: we discovered that this requires a new hypothesis of semantic entailment on operators, not known in previous literature. Secondly, the coinductive version of common knowledge is equivalent to the traditional transitive closure on the relational interpretation. All results are formalised in the proof assistants Agda and Coq.

References

  1. 1.
    Barwise, J.: Three views of common knowledge. In: Vardi, M.Y. (ed.) Proceedings of the 2nd Conference on Theoretical Aspects of Reasoning about Knowledge, Pacific Grove, CA, March 1988, pp. 365–379. Morgan Kaufmann (1988)Google Scholar
  2. 2.
    Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development. Coq’Art: The Calculus of Inductive Constructions. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-662-07964-5CrossRefzbMATHGoogle Scholar
  3. 3.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, New York (2001)CrossRefGoogle Scholar
  4. 4.
    Bucheli, S., Kuznets, R., Struder, T.: Two ways to common knowledge. Electron. Notes Theor. Comput. Sci. 262, 83–98 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Capretta, V.: General recursion via coinductive types. Log. Methods Comput. Sci. 1(2), 1–18 (2005).  https://doi.org/10.2168/LMCS-1(2:1)2005MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Capretta, V.: Common knowledge as a coinductive modality. In: Barendsen, E., Geuvers, H., Capretta, V., Niqui, M. (eds.) Reflections on Type Theory, Lambda Calculus, and the Mind, pp. 51–61. ICIS, Faculty of Science, Radbout University Nijmegen (2007). Essays Dedicated to Henk Barendregt on the Occasion of his 60th BirthdayGoogle Scholar
  7. 7.
    Coquand, T.: Infinite objects in type theory. In: Barendregt, H., Nipkow, T. (eds.) TYPES 1993. LNCS, vol. 806, pp. 62–78. Springer, Heidelberg (1994).  https://doi.org/10.1007/3-540-58085-9_72CrossRefGoogle Scholar
  8. 8.
    Fagin, R., Halpern, J.Y., Vardi, M.Y., Moses, Y.: Reasoning About Knowledge. MIT Press, Cambridge (1995)zbMATHGoogle Scholar
  9. 9.
    Gamow, G., Stern, M.: Puzzle Math. Viking Press, New York (1958)zbMATHGoogle Scholar
  10. 10.
    Garson, J.: Modal logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University (2016)Google Scholar
  11. 11.
    Giménez, E.: Codifying guarded definitions with recursive schemes. In: Dybjer, P., Nordström, B., Smith, J. (eds.) TYPES 1994. LNCS, vol. 996, pp. 39–59. Springer, Heidelberg (1995).  https://doi.org/10.1007/3-540-60579-7_3CrossRefGoogle Scholar
  12. 12.
    Hintikka, J.: Knowledge and Belief. Cornell University Press, Ithaca (1962)zbMATHGoogle Scholar
  13. 13.
    Keller, C., Werner, B.: Importing HOL light into Coq. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 307–322. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-14052-5_22CrossRefGoogle Scholar
  14. 14.
    Kripke, S.A.: A completeness theorem in modal logic. J. Symb. Logic 24(1), 1–14 (1959)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lescanne, P.: Common knowledge logic in a higher order proof assistant. In: Voronkov, A., Weidenbach, C. (eds.) Programming Logics. LNCS, vol. 7797, pp. 271–284. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-37651-1_11CrossRefGoogle Scholar
  16. 16.
    Lewis, C.I., Langford, C.H.: Symbolic Logic. The Century Co., New York (1932)zbMATHGoogle Scholar
  17. 17.
    Reynolds, J.C.: User-defined types and procedural data structures as complementary approaches to data abstraction. In: Gries, D. (ed.) Programming Methodology. MCS, pp. 309–317. Springer, New York (1978).  https://doi.org/10.1007/978-1-4612-6315-9_22CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Functional Programming Lab, School of Computer ScienceUniversity of NottinghamNottinghamUK

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