Equivalence of Probabilistic \(\mu \)-Calculus and p-Automata

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10329)


An important characteristic of Kozen’s \(\mu \)-calculus is its strong connection with parity alternating tree automata. Here, we show that the probabilistic \(\mu \)-calculus \(\mu ^p\)-calculus and p-automata (parity alternating Markov chain automata) have an equally strong connection. Namely, for every \(\mu ^p\)-calculus formula we can construct a p-automaton that accepts exactly those Markov chains that satisfy the formula. For every p-automaton we can construct a \(\mu ^p\)-calculus formula satisfied in exactly those Markov chains that are accepted by the automaton. The translation in one direction relies on a normal form of the calculus and in the other direction on the usage of vectorial \(\mu ^p\)-calculus. The proofs use the game semantics of \(\mu ^p\)-calculus and automata to show that our translations are correct.


Markov Chain Atomic Proposition Tree Automaton Modal Formula Semantic Game 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Arnold, A., Niwiński, D.: Rudiments of \(\mu \)-calculus. Studies in Logic and the Foundations of Mathematics, vol. 146. Elsevier, New York (2001)Google Scholar
  2. 2.
    Baier, C., Katoen, J.P.: Principles of Model Checking. The MIT Press, Cambridge (2008)zbMATHGoogle Scholar
  3. 3.
    Blackburn, P., Benthem, J., Wolter, F.: Handbook of Modal Logic. Studies in Logic and Practical Reasoning. Elsevier, New York (2007)zbMATHGoogle Scholar
  4. 4.
    Bradfield, J., Stirling, C.: Modal \(\mu \)-calculi. In: Handbook of Modal Logic, pp. 721–756. Elsevier (2007)Google Scholar
  5. 5.
    Bradfield, J., Walukiewicz, I.: The \(\mu \)-calculus and model-checking. In: Handbook of Model Checking. Springer (2015)Google Scholar
  6. 6.
    Bruse, F., Friedmann, O., Lange, M.: On guarded transformation in the modal \(\mu \)-calculus. Logic J. IGPL 23(2), 194–216 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cleaveland, R., Purushothaman Iyer, S., Narasimha, M.: Probabilistic temporal logics via the modal \(\mu \)-calculus. Theor. Comput. Sci. 342(2–3), 316–350 (2005)Google Scholar
  8. 8.
    Castro, P., Kilmurray, C., Piterman, N.: Tractable probabilistic \(\mu \)-calculus that expresses probabilistic temporal logics. In: 32nd Symposium on Theoretical Aspects of Computer Science. Schloss Dagstuhl (2015)Google Scholar
  9. 9.
    Chatterjee, K., Piterman, N.: Obligation Blackwell games and p-Automata. CoRR, abs/1206.5174 (2013)Google Scholar
  10. 10.
    Emerson, E.A., Jutla S.: Tree automata, \(\mu \)-calculus and determinacy. In: Proceedings of 32nd Annual Symposium on Foundations of Computer Science, pp. 368–377. IEEE (1991)Google Scholar
  11. 11.
    Emerson, E.A., Lei, C.-L.: Efficient model checking in fragments of the propositional \(\mu \)-calculus. In: Proceedings of the First Annual IEEE Symposium on Logic in Computer Science, LICS, pp. 267–278 (1986)Google Scholar
  12. 12.
    Grädel, E., Thomas, W., Wilke, T.: Automata Logics, and Infinite Games: A Guide to Current Research. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  13. 13.
    Huth, M., Kwiatkowska, M.: Quantitative analysis and model checking. In: Proceedings of 12th Annual IEEE Symposium on Logic in Computer Science, Warsaw, Poland, 29 June– 2 July, pp. 111–122 (1997)Google Scholar
  14. 14.
    Janin, D., Walukiewicz, I.: Automata for the modal \(\mu \)-calculus and related results. Mathematical Foundations of Computer Science, pp. 552–562 (1995)Google Scholar
  15. 15.
    Kozen, D.: Results on the propositional \(\mu \)-calculus. Theor. Comput. Sci. 27(3), 333–354 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kupferman, O., Vardi, M.Y., Wolper, P.: An automata-theoretic approach to branching-time model checking. J. ACM 47(2), 312–360 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mio, M.: Game semantics for probabilistic modal \(\mu \)-calculi. Ph.D. thesis, University of Edinburgh (2012)Google Scholar
  18. 18.
    Mio, M.: Probabilistic modal \(\mu \)-calculus with independent product. Logical Methods Comput. Sci. 8(4), 1–36 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mio, M., Simpson, A.K.: Łukasiewicz \(\mu \)-calculus. CoRR, abs/1510.00797 (2015)Google Scholar
  20. 20.
    Niwiński, D.: Fixed points vs. infinite generation. In: Proceedings of the Third Annual IEEE Symposium on Logic in Computer Science, LICS, pp. 402–409 (1988)Google Scholar
  21. 21.
    Niwiński, D.: Fixed point characterization of infinite behavior of finite-state systems. Theor. Comput. Sci. 189(1–2), 1–69 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schneider, K.: Verification of Reactive Systems: Formal Methods and Algorithms. Texts in Theoretical Computer Science. Springer, Heidelberg (2004)CrossRefzbMATHGoogle Scholar
  23. 23.
    Wilke, T.: Alternating tree automata, parity games, and modal \(\mu \)-calculus. Bull. Soc. Math. Belg. 8(2), 359–391 (2001)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of LeicesterLeicesterUK

Personalised recommendations