CAV 2011: Computer Aided Verification pp 260-276

# Symbolic Algorithms for Qualitative Analysis of Markov Decision Processes with Büchi Objectives

• Krishnendu Chatterjee
• Monika Henzinger
• Manas Joglekar
• Nisarg Shah
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6806)

## Abstract

We consider Markov decision processes (MDPs) with ω-regular specifications given as parity objectives. We consider the problem of computing the set of almost-sure winning states from where the objective can be ensured with probability 1. The algorithms for the computation of the almost-sure winning set for parity objectives iteratively use the solutions for the almost-sure winning set for Büchi objectives (a special case of parity objectives). 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 often 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 Process Algorithm Stop Forward Search Random Attractor Parity Objective
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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## Authors and Affiliations

• Krishnendu Chatterjee
• 1
• Monika Henzinger
• 2
• Manas Joglekar
• 3
• Nisarg Shah
• 3
1. 1.ISTAustria
2. 2.University of ViennaAustria
3. 3.IIT BombayIndia