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Algorithmic Verification of Recursive Probabilistic State Machines

  • Kousha Etessami
  • Mihalis Yannakakis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3440)

Abstract

Recursive Markov Chains (RMCs) ([EY05]) are a natural abstract model of procedural probabilistic programs and related systems involving recursion and probability. They succinctly define a class of denumerable Markov chains that generalize multi-type branching (stochastic) processes. In this paper, we study the problem of model checking an RMC against a given ω-regular specification. Namely, given an RMC A and a Büchi automaton B, we wish to know the probability that an execution of A is accepted by B. We establish a number of strong upper bounds, as well as lower bounds, both for qualitative problems (is the probability = 1, or = 0?), and for quantitative problems (is the probability ≥ p ?, or, approximate the probability to within a desired precision). Among these, we show that qualitative model checking for general RMCs can be decided in PSPACE in |A| and EXPTIME in |B|, and when A is either a single-exit RMC or when the total number of entries and exits in A is bounded, it can be decided in polynomial time in |A|. We then show that quantitative model checking can also be done in PSPACE in |A|, and in EXPSPACE in |B|. When B is deterministic, all our complexities in |B| come down by one exponential.

For lower bounds, we show that the qualitative model checking problem, even for a fixed RMC, is EXPTIME-complete. On the other hand, even for reachability analysis, we showed in [EY05] that our PSPACE upper bounds in A can not be improved upon without a breakthrough on a well-known open problem in the complexity of numerical computation.

Keywords

Markov Chain Model Check Reachability Analysis Strongly Connect Component Model Check Problem 
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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Kousha Etessami
    • 1
  • Mihalis Yannakakis
    • 2
  1. 1.School of InformaticsUniversity of Edinburgh 
  2. 2.Department of Computer ScienceColumbia University 

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