Robust Synchronization in Markov Decision Processes

  • Laurent Doyen
  • Thierry Massart
  • Mahsa Shirmohammadi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8704)


We consider synchronizing properties of Markov decision processes (MDP), viewed as generators of sequences of probability distributions over states. A probability distribution is p-synchronizing if the probability mass is at least p in some state, and a sequence of probability distributions is weakly p-synchronizing, or strongly p-synchronizing if respectively infinitely many, or all but finitely many distributions in the sequence are p-synchronizing.

For each synchronizing mode, an MDP can be (i) sure winning if there is a strategy that produces a 1-synchronizing sequence; (ii) almost-sure winning if there is a strategy that produces a sequence that is, for all ε > 0, a (1-ε)-synchronizing sequence; (iii) limit-sure winning if for all ε > 0, there is a strategy that produces a (1-ε)-synchronizing sequence.

For each synchronizing and winning mode, we consider the problem of deciding whether an MDP is winning, and we establish matching upper and lower complexity bounds of the problems, as well as the optimal memory requirement for winning strategies: (a) for all winning modes, we show that the problems are PSPACE-complete for weak synchronization, and PTIME-complete for strong synchronization; (b) we show that for weak synchronization, exponential memory is sufficient and may be necessary for sure winning, and infinite memory is necessary for almost-sure winning; for strong synchronization, linear-size memory is sufficient and may be necessary in all modes; (c) we show a robustness result that the almost-sure and limit-sure winning modes coincide for both weak and strong synchronization.


Markov Decision Process Probability Mass Winning Strategy Membership Problem Strongly Connect Component 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    ArXiv CoRR (2014), (Full version)
  2. 2.
    Agrawal, M., Akshay, S., Genest, B., Thiagarajan, P.S.: Approximate verification of the symbolic dynamics of Markov chains. In: LICS, pp. 55–64. IEEE (2012)Google Scholar
  3. 3.
    de Alfaro, L., Henzinger, T.A.: Concurrent omega-regular games. In: Proc. of LICS, pp. 141–154 (2000)Google Scholar
  4. 4.
    de Alfaro, L., Henzinger, T.A., Kupferman, O.: Concurrent reachability games. Theor. Comput. Sci. 386(3), 188–217 (2007)CrossRefzbMATHGoogle Scholar
  5. 5.
    Baier, C., Bertrand, N., Größer, M.: On decision problems for probabilistic Büchi automata. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 287–301. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Baldoni, R., Bonnet, F., Milani, A., Raynal, M.: On the solvability of anonymous partial grids exploration by mobile robots. In: Baker, T.P., Bui, A., Tixeuil, S. (eds.) OPODIS 2008. LNCS, vol. 5401, pp. 428–445. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Burkhard, H.D.: Zum längenproblem homogener experimente an determinierten und nicht-deterministischen automaten. Elektronische Informationsverarbeitung und Kybernetik 12(6), 301–306 (1976)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Cerný, J.: Poznámka k. homogénnym experimentom s konecnymi automatmi. Matematicko-fyzikálny Časopis 14(3), 208–216 (1964)zbMATHGoogle Scholar
  9. 9.
    Chadha, R., Korthikanti, V.A., Viswanathan, M., Agha, G., Kwon, Y.: Model checking MDPs with a unique compact invariant set of distributions. In: QEST, pp. 121–130. IEEE (2011)Google Scholar
  10. 10.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. J. ACM 42(4), 857–907 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Doyen, L., Massart, T., Shirmohammadi, M.: Infinite synchronizing words for probabilistic automata. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 278–289. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Doyen, L., Massart, T., Shirmohammadi, M.: Synchronizing objectives for Markov decision processes. In: Proc. of iWIGP. EPTCS, vol. 50, pp. 61–75 (2011)Google Scholar
  13. 13.
    Doyen, L., Massart, T., Shirmohammadi, M.: Infinite synchronizing words for probabilistic automata (Erratum). CoRR abs/1206.0995 (2012)Google Scholar
  14. 14.
    Doyen, L., Massart, T., Shirmohammadi, M.: Limit synchronization in Markov decision processes. In: Muscholl, A. (ed.) FOSSACS 2014. LNCS, vol. 8412, pp. 58–72. Springer, Heidelberg (2014)Google Scholar
  15. 15.
    Filar, J., Vrieze, K.: Competitive Markov Decision Processes. Springer (1997)Google Scholar
  16. 16.
    Gimbert, H., Oualhadj, Y.: Probabilistic automata on finite words: Decidable and undecidable problems. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010, Part II. LNCS, vol. 6199, pp. 527–538. Springer, Heidelberg (2010)Google Scholar
  17. 17.
    Henzinger, T.A., Mateescu, M., Wolf, V.: Sliding window abstraction for infinite Markov chains. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 337–352. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  18. 18.
    Imreh, B., Steinby, M.: Directable nondeterministic automata. Acta Cybern. 14(1), 105–115 (1999)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Kfoury, D.: Synchronizing sequences for probabilistic automata. Studies in Applied Mathematics 29, 101–103 (1970)MathSciNetGoogle Scholar
  20. 20.
    Korthikanti, V.A., Viswanathan, M., Agha, G., Kwon, Y.: Reasoning about MDPs as transformers of probability distributions. In: QEST, pp. 199–208. IEEE (2010)Google Scholar
  21. 21.
    Martyugin, P.: Computational complexity of certain problems related to carefully synchronizing words for partial automata and directing words for nondeterministic automata. Theory Comput. Syst. 54(2), 293–304 (2014)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Vardi, M.Y.: Automatic verification of probabilistic concurrent finite-state programs. In: Proc. of FOCS, pp. 327–338. IEEE Computer Society (1985)Google Scholar
  23. 23.
    Volkov, M.V.: Synchronizing automata and the Cerny conjecture. In: Martín-Vide, C., Otto, F., Fernau, H. (eds.) LATA 2008. LNCS, vol. 5196, pp. 11–27. Springer, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Laurent Doyen
    • 1
  • Thierry Massart
    • 2
  • Mahsa Shirmohammadi
    • 1
    • 2
  1. 1.LSV, ENS Cachan & CNRSFrance
  2. 2.Université Libre de BruxellesBelgium

Personalised recommendations