Strong Completeness for Markovian Logics

  • Dexter Kozen
  • Radu Mardare
  • Prakash Panangaden
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8087)

Abstract

In this paper we present Hilbert-style axiomatizations for three logics for reasoning about continuous-space Markov processes (MPs): (i) a logic for MPs defined for probability distributions on measurable state spaces, (ii) a logic for MPs defined for sub-probability distributions and (iii) a logic defined for arbitrary distributions. These logics are not compact so one needs infinitary rules in order to obtain strong completeness results.

We propose a new infinitary rule that replaces the so-called Countable Additivity Rule (CAR) currently used in the literature to address the problem of proving strong completeness for these and similar logics. Unlike the CAR, our rule has a countable set of instances; consequently it allows us to apply the Rasiowa-Sikorski lemma for establishing strong completeness. Our proof method is novel and it can be used for other logics as well.

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References

  1. 1.
    Aumann, R.: Interactive epistemology I: knowledge. International Journal of Game Theory 28, 263–300 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aumann, R.: Interactive epistemology II: probability. International Journal of Game Theory 28, 301–314 (1999)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Billingsley, P.: Probability and Measure. Wiley-Interscience (1995)Google Scholar
  4. 4.
    Cardelli, L., Larsen, K.G., Mardare, R.: Continuous markovian logic - from complete axiomatization to the metric space of formulas. In: CSL, pp. 144–158 (2011)Google Scholar
  5. 5.
    Desharnais, J., Edalat, A., Panangaden, P.: A logical characterization of bisimulation for labelled Markov processes. In: Proceedings of the 13th IEEE Symposium On Logic In Computer Science, Indianapolis, pp. 478–489. IEEE Press (June 1998)Google Scholar
  6. 6.
    Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for labeled Markov processes. Information and Computation 179(2), 163–193 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Doberkat, E.-E.: Stochastic Relations. Foundations for Markov Transition Systems. Chapman and Hall, New York (2007)CrossRefMATHGoogle Scholar
  8. 8.
    Dudley, R.M.: Real Analysis and Probability. Wadsworth and Brookes/Cole (1989)Google Scholar
  9. 9.
    Fagin, R., Halpern, J.Y.: Reasoning about knowledge and probability. Journal of the ACM 41(2), 340–367 (1994)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Goldblatt, R.: On the role of the Baire category theorem in the foundations of logic. Journal of Symbolic Logic, 412–422 (1985)Google Scholar
  11. 11.
    Goldblatt, R.: Deduction systems for coalgebras over measurable spaces. Journal of Logic and Computation 20(5), 1069–1100 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Heifetz, A., Mongin, P.: Probability logic for type spaces. Games and Economic Behavior 35(1-2), 31–53 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kozen, D., Larsen, K.G., Mardare, R., Panangaden, P.: Stone duality for markov processes. In: Proceedings of the 28th Annual IEEE Symposium on Logic in Computer Science: LICS 2013. IEEE Computer Society (2013)Google Scholar
  14. 14.
    Larsen, K.G., Skou, A.: Bisimulation through probablistic testing. Information and Computation 94, 1–28 (1991)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Mardare, R., Cardelli, L., Larsen, K.G.: Continuous markovian logics - axiomatization and quantified metatheory. Logical Methods in Computer Science 8(4) (2012)Google Scholar
  16. 16.
    Panangaden, P.: Labelled Markov Processes. Imperial College Press (2009)Google Scholar
  17. 17.
    Rasiowa, H., Sikorski, R.: A proof of the completeness theorem of gödel. Fund. Math. 37, 193–200 (1950)MathSciNetMATHGoogle Scholar
  18. 18.
    Schröder, L., Pattinson, D.: Modular algorithms for heterogeneous modal logics. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 459–471. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Zhou, C.: A complete deductive system for probability logic with application to Harsanyi type spaces. PhD thesis, Indiana University (2007)Google Scholar
  20. 20.
    Zhou, C.: Probability logic of finitely additive beliefs. J. Logic, Language and Information 19(3), 247–282 (2010)CrossRefMATHGoogle Scholar
  21. 21.
    Zhou, C., Ying, M.: Approximating Markov processes through filtration. Theoretical Computer Science 446, 75–97 (2012)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Dexter Kozen
    • 1
  • Radu Mardare
    • 2
  • Prakash Panangaden
    • 3
  1. 1.Computer Science DepartmentCornell UniversityIthacaUSA
  2. 2.Department of Computer ScienceAalborg UniversityDenmark
  3. 3.School of Computer ScienceMcGill UniversityMontrealCanada

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