International Workshop on Logic, Rationality and Interaction

Logic, Rationality, and Interaction pp 255-267 | Cite as

Algebraic Semantics for Dynamic Dynamic Logic

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9394)

Abstract

Dynamic dynamic logic (DDL) is a generalisation of propositional dynamic logic PDL and dynamic epistemic logic. In this paper, we develop algebraic semantics for DDL without the constant program. We introduce inductive and continuous modal Kleene algebras for PDL and show the validity of reduction axioms in algebraic models and hence the algebraic completeness of DDL.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Danecki, R.: Nondeterministic propositional dynamic logic with intersection is decidable. In: Skowron, A. (ed.) SCT 1984. LNCS, vol. 208, pp. 34–53. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  2. 2.
    Enderton, H.B.: Computability Theory: An Introduction to Recursion Theory. Elsevier (2011)Google Scholar
  3. 3.
    Fischer, M.J., Ladner, R.E.: Propositional dynamic logic of regular programs. Journal of Computer and System Sciences 18(2), 194–211 (1979)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Harel, D.: Dynamic logic. In: Handbook of Philosophical Logic, vol. II, pp. 496–604. D. Reidel Publishers (1984)Google Scholar
  5. 5.
    Kozen, D.: A representation theorem for models of *-free pdl. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 351–362. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  6. 6.
    Kozen, D.: On induction vs.*-continuity. In: Kozen, D. (ed.) Logic of Programs 1981. LNCS, vol. 131, pp. 167–176. Springer, Heidelberg (1982)CrossRefGoogle Scholar
  7. 7.
    Kozen, D.: Kleene algebra with tests. ACM Transactions on Programming Languages and Systems 19(3), 427–443 (1997)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kurz, A., Palmigiano, A.: Epistemic updates on algebras. Logical Methods in Computer Science 9(4:17), 1–28 (2013)MathSciNetMATHGoogle Scholar
  9. 9.
    Lutz, C., Walther, D.: PDL with Negation of Atomic Programs. Journal of Applied Non-Classical Logic 15(2), 189–214 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lutz, C.: PDL with intersection and converse is decidable. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 413–427. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Ma, M., Sano, K.: How to update neighborhood models. In: Grossi, D., Roy, O., Huang, H. (eds.) LORI. LNCS, vol. 8196, pp. 204–217. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  12. 12.
    Ma, M., Palmigiano, A., Sadrzadeh, M.: Algebraic semantics and model completeness for intuitionistic public announcement logic. Annals of Pure and Applied Logic 165(4), 963–995 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Pratt, V.: Dynamic algebras: examples, constructions, applications. Studia Logica 50(3-4), 571–605 (1991)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Resende, P.: Lectures on étale groupoids, inverse semigroups and quantales (2006), http://www.math.ist.utl.pt/~pmr/poci55958/gncg51gamap-version2.pdf
  15. 15.
    Segerberg, K.: A completeness theorem in the modal logic of programs. Banach Center Publications 9(1), 31–46 (1982)MathSciNetMATHGoogle Scholar
  16. 16.
    Girard, P., Seligman, J., Liu, F.: General dynamic dynamic logic. In: Ghilardi, S., Bolander, T., Moss, L. (eds.) Advances in Modal Logic, vol. 9, pp. 239–260. Colledge Publications (2012)Google Scholar
  17. 17.
    Tiuryn, J., Harel, D., Kozen, D.: Dynamic Logic. MIT Press (2000)Google Scholar
  18. 18.
    Plaza, L.: Logics of public communications. Synthese 158(2), 165–179 (2007)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Vakarelov, D.: Filtration theorem for dynamic algebras with tests and inverse operator. In: Salwicki, A. (ed.) Logic of Programs 1980. LNCS, vol. 148, pp. 314–324. Springer, Heidelberg (1983)CrossRefGoogle Scholar
  20. 20.
    van Benthem, J., Pacuit, E.: Dynamic logics of evidence-based beliefs. Studia Logica 99(1), 61–92 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    van Ditmarsch, H., van der Hoek, W., Kooi, B.P.: Dynamic epistemic logic. Springer Science & Business Media (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute for Logic and IntelligenceSouthwest UniversityChongqingChina
  2. 2.Department of PhilosophyUniversity of AucklandAuklandNew Zealand

Personalised recommendations