Flow Faster: Efficient Decision Algorithms for Probabilistic Simulations

  • Lijun Zhang
  • Holger Hermanns
  • Friedrich Eisenbrand
  • David N. Jansen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4424)


Abstraction techniques based on simulation relations have become an important and effective proof technique to avoid the infamous state space explosion problem. In the context of Markov chains, strong and weak simulation relations have been proposed [17,6], together with corresponding decision algorithms [3,5], but it is as yet unclear whether they can be used as effectively as their non-stochastic counterparts. This paper presents drastically improved algorithms to decide whether one (discrete- or continuous-time) Markov chain strongly or weakly simulates another. The key innovation is the use of parametric maximum flow techniques to amortize computations.


  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: theory, algorithms, and applications. Prentice-Hall, Englewood Cliffs (1993)Google Scholar
  2. 2.
    Aziz, A., et al.: Verifying Continuous Time Markov Chains. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 269–276. Springer, Heidelberg (1996)Google Scholar
  3. 3.
    Baier, C., Engelen, B., Majster-Cederbaum, M.E.: Deciding Bisimilarity and Similarity for Probabilistic Processes. J. Comput. Syst. Sci. 60(1), 187–231 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Baier, C., et al.: Model-Checking Algorithms for Continuous-Time Markov Chains. IEEE Trans. Software Eng. 29(6), 524–541 (2003)CrossRefGoogle Scholar
  5. 5.
    Baier, C., Hermanns, H., Katoen, J.-P.: Probabilistic weak simulation is decidable in polynomial time. Inf. Process. Lett. 89(3), 123–130 (2004)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Katoen, J.-P., et al.: Simulation for Continuous-Time Markov Chains. In: Brim, L., et al. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 338–354. Springer, Heidelberg (2002)Google Scholar
  7. 7.
    Baier, C., et al.: Comparative branching-time semantics for Markov chains. Inf. Comput 200(2), 149–214 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cheriyan, J., Hagerup, T., Mehlhorn, K.: Can a Maximum Flow be Computed in \(\mathcal{O}(nm)\) Time? In: Paterson, M. (ed.) ICALP 1990. LNCS, vol. 443, pp. 235–248. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  9. 9.
    Clarke, E.M., Grumberg, O., Long, D.E.: Model Checking and Abstraction. ACM Transactions on Programming Languages and Systems 16(5), 1512–1542 (1994)CrossRefGoogle Scholar
  10. 10.
    Gallo, G., Grigoriadis, M.D., Tarjan, R.E.: A fast parametric maximum flow algorithm and applications. SIAM J. Comput. 18(1), 30–55 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gentilini, R., Piazza, C., Policriti, A.: From Bisimulation to Simulation: Coarsest Partition Problems. J. Autom. Reasoning 31(1), 73–103 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Goldberg, A.V.: Recent Developments in Maximum Flow Algorithms. In: Arnborg, S. (ed.) SWAT 1998. LNCS, vol. 1432, pp. 1–10. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  13. 13.
    Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. J. ACM 35(4), 921–940 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hansson, H., Jonsson, B.: A Logic for Reasoning about Time and Reliability. Formal Asp. Comput. 6(5), 512–535 (1994)zbMATHCrossRefGoogle Scholar
  15. 15.
    Henzinger, M.R., Henzinger, T.A., Kopke, P.W.: Computing Simulations on Finite and Infinite Graphs. In: FOCS, pp. 453–462 (1995)Google Scholar
  16. 16.
    Jonsson, B.: Simulations Between Specifications of Distributed Systems. In: Groote, J.F., Baeten, J.C.M. (eds.) CONCUR 1991. LNCS, vol. 527, pp. 346–360. Springer, Heidelberg (1991)Google Scholar
  17. 17.
    Jonsson, B., Larsen, K.G.: Specification and Refinement of Probabilistic Processes. In: LICS, pp. 266–277 (1991)Google Scholar
  18. 18.
    Milner, R.: Communication and Concurrency. Prentice-Hall, Englewood Cliffs (1989)zbMATHGoogle Scholar
  19. 19.
    Parker, D.: Implementation of Symbolic Model Checking for Probabilistic Systems. University of Birmingham (2002)Google Scholar
  20. 20.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986)zbMATHGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Lijun Zhang
    • 1
  • Holger Hermanns
    • 1
  • Friedrich Eisenbrand
    • 2
  • David N. Jansen
    • 3
    • 4
  1. 1.Department of Computer Science, Saarland University, SaarbrückenGermany
  2. 2.Department of Mathematics, University of PaderbornGermany
  3. 3.Department of Computer Science, University of Twente, EnschedeThe Netherlands
  4. 4.Software Modeling and Verification Group, RWTH AachenGermany

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