Skip to main content Accessibility help

A stochastic interpretation of propositional dynamic logic: expressivity

  • Ernst-Erich Doberkat (a1)


We propose a probabilistic interpretation of Propositional Dynamic Logic (PDL). We show that logical and behavioral equivalence are equivalent over general measurable spaces. This is done first for the fragment of straight line programs and then extended to cater for the nondeterministic nature of choice and iteration, expanded to PDL as a whole. Bisimilarity is also discussed and shown to be equivalent to logical and behavioral equivalence, provided the base spaces are Polish spaces. We adapt techniques from coalgebraic stochastic logic and point out some connections to Souslin's operation from descriptive set theory. This leads to a discussion of complete stochastic Kripke models and model completion, which permits an adequate treatment of the test operator.



Hide All
[1] Blackburn, P., de Rijke, M., and Venema, Y., Modal logic, Cambridge Tracts in Theoretical Computer Science, vol. 53, Cambridge University Press, Cambridge, UK, 2001.
[2] Blackburn, P. and van Benthem, J., Modal logic: A semantic perspective, Handbook of modal logic (Blackburn, P. et al., editors), Elsevier, Amsterdam, 2007, pp. 184.
[3] Bonanno, G., Modal logic and game theory—two alternative approaches, Risk Decision and Policy, vol. 7 (2002), pp. 309324.
[4] Desharnais, J., Edalat, A., and Panangaden, P., Bisimulation of labelled Markov processes, Information and Computation, vol. 179 (2002), no. 2, pp. 163193.
[5] Doberkat, E.-E., An analysis of Floyd's algorithm for heapconstruction, Information and Control, vol. 61 (1984), pp. 114131.
[6] Doberkat, E.-E., Stochastic relations: congruences, hisimulations and the Hennessy–Milner theorem, SIAM Journal on Computing, vol. 35 (2006), no. 3, pp. 590626.
[7] Doberkat, E.-E., Kleisli morphisms and randomized congruences for the Giry monad, Journal of Pure and Applied Algebra, vol. 211 (2007), pp. 638664.
[8] Doberkat, E.-E., Stochastic relations. Foundations for Markov transition systems, Chapman & Hall/CRC Press, Boca Raton, New York, 2007.
[9] Doberkat, E.-E., Stochastic coalgebraic logic, EATCS Monographs in Theoretical Computer Science, Springer-Verlag, 2009.
[10] Doberkat, E.-E., A note on the coalgebraic interpretation of game logic, Rendiconti dell'Istituto di Matematica dell'Università di Trieste, vol. 42 (2010), pp. 191204.
[11] Doberkat, E.-E. and Kurz, A., Special issue on coalgebraic logic, Mathematical Structures in Computer Science, vol. 21 (2011 ).
[12] Doberkat, E.-E. and Schubert, Ch., Coalgebraic logic over general measurable spaces—a survey, Mathematical Structures in Computer Science, vol. 21 (2011), pp. 175234, special issue on coalgebraic logic.
[13] Doberkat, E.-E. and Srivastava, S. M., Measurable selections, transition probabilities and Kripke models, Technical Report 185. Chair for Software Technology, Technische Universität Dortmund, 05 2010.
[14] Giry, M., A categorical approach to probability theory, Categorical aspects of topology and analysis, Lecture Notes in Mathematical, vol. 915, Springer-Verlag, Berlin, 1981, pp. 6885.
[15] Halmos, P. R., Measure theory, Van Nostrand Reinhold, New York, 1950.
[16] Harrenstein, B. P., der Hoek, W. van, Meyer, J.-J. Ch., and Witteveen, C., A modal characterization of Nash equilibrium, Fundamenta Informaticae, vol. 57 (2003), pp. 281321.
[17] Hennessy, M. and Milner, R., On observing nondeterminism and concurrency, Proceedings of ICALP'80, Lecture Notes in Computer Science, vol. 85, Springer-Verlag, Berlin, 1980, pp. 395409.
[18] Kechris, A. S., Classical descriptive set theory, Graduate Texts in Mathematics, Springer-Verlag, Berlin, 1994.
[19] Knuth, D. E., The art of computer programming. Volume III, Sorting and searching, Addison-Wesley, Reading, MA, 1973.
[20] Kurz, A., Specifying coalgebras with modal logic, Theoretical Computer Science, vol. 260 (2001), pp. 119138.
[21] Lubin, A., Extensions of measures and the von Neumann selection theorem, Proceedings of the American Mathematical Society, vol. 43 (1974), no. 1, pp. 118122.
[22] Parikh, R., The logic of games and its applications, Topics in the theory of computation (Karpinski, M. and van Leeuwen, J., editors), vol. 24, Elsevier, 1985, pp. 111140.
[23] Pauly, M., Game logic far game theorists, Technical Report INS-R0017, CWI, Amsterdam, 2000.
[24] Pauly, M. and Parikh, R., Game logic—an overview, Studia Logica, vol. 75 (2003), no. 2, pp. 165182.
[25] Rutten, J. J. M. M., Universal coalgebra: a theory of systems, Theoretical Computer Science, vol. 249 (2000), no. 1, pp. 380, special issue on modern algebra and its applications.
[26] Schröder, L. and Pattinson, D., Modular algorithms for heterogeneous modal logics, Proceedings of ICALP, Lecture Notes in Computer Science, vol. 4596, 2007, pp. 459471.
[27] Srivastava, S. M., A course on Borei sets, Graduate Texts in Mathematics, Springer-Verlag, Berlin, 1998.
[28] Terraf, P. Sànchez, Unprovability of the logical characterization of bisimulation, Information and Computation, vol. 209 (2011), no. 7, pp. 10481056.
[29] der Hoek, W. van and Pauly, M., Modal logic for games and information, Handbook of modal logic (Blackburn, P., van Benthem, J., and Wolter, F., editors), Studies in Logic and Practical Reasoning, vol. 3, Elsevier, Amsterdam, 2007, pp. 10771148.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed