Formal Methods in System Design

, Volume 42, Issue 3, pp 301–327

# Symbolic algorithms for qualitative analysis of Markov decision processes with Büchi objectives

• Krishnendu Chatterjee
• Monika Henzinger
• Manas Joglekar
• Nisarg Shah
Article

## Abstract

We consider Markov decision processes (MDPs) with Büchi (liveness) objectives. We consider the problem of computing the set of almost-sure winning states from where the objective can be ensured with probability 1. Our contributions are as follows: First, we present the first subquadratic symbolic algorithm to compute the almost-sure winning set for MDPs with Büchi objectives; our algorithm takes $$O(n \cdot\sqrt{m})$$ symbolic steps as compared to the previous known algorithm that takes O(n 2) symbolic steps, where n is the number of states and m is the number of edges of the MDP. In practice MDPs have constant out-degree, and then our symbolic algorithm takes $$O(n \cdot\sqrt{n})$$ symbolic steps, as compared to the previous known O(n 2) symbolic steps algorithm. Second, we present a new algorithm, namely win-lose algorithm, with the following two properties: (a) the algorithm iteratively computes subsets of the almost-sure winning set and its complement, as compared to all previous algorithms that discover the almost-sure winning set upon termination; and (b) requires $$O(n \cdot\sqrt{K})$$ symbolic steps, where K is the maximal number of edges of strongly connected components (scc’s) of the MDP. The win-lose algorithm requires symbolic computation of scc’s. Third, we improve the algorithm for symbolic scc computation; the previous known algorithm takes linear symbolic steps, and our new algorithm improves the constants associated with the linear number of steps. In the worst case the previous known algorithm takes 5⋅n symbolic steps, whereas our new algorithm takes 4⋅n symbolic steps.

### Keywords

Markov decision processes Probabilistic verification Büchi objectives Symbolic algorithms

## Notes

### Acknowledgements

We thank Fabio Somenzi for sharing the facts about the performance comparison of the algorithm of [3] and the algorithm of [12]. We thank anonymous reviewers for many helpful comments that improved the presentation of the paper.

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© Springer Science+Business Media New York 2012

## Authors and Affiliations

• Krishnendu Chatterjee
• 1
• Monika Henzinger
• 2
• Manas Joglekar
• 3
• Nisarg Shah
• 4
1. 1.IST AustriaKlosterneuburgAustria
2. 2.University of ViennaViennaAustria
3. 3.Stanford UniversityPalo AltoUSA
4. 4.Carnegie Mellon UniversityPittsburghUSA