The complexity of probabilistic verification
Journal of the ACM (JACM)
Modeling and verification of randomized distributed real-time systems
Modeling and verification of randomized distributed real-time systems
Competitive Markov decision processes
Competitive Markov decision processes
Languages, automata, and logic
Handbook of formal languages, vol. 3
Computing strongly connected components in a linear number of symbolic steps
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
An Algorithm for Strongly Connected Component Analysis in n log n Symbolic Steps
FMCAD '00 Proceedings of the Third International Conference on Formal Methods in Computer-Aided Design
Model Checking of Probabalistic and Nondeterministic Systems
Proceedings of the 15th Conference on Foundations of Software Technology and Theoretical Computer Science
Quantitative stochastic parity games
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Verification of the randomized consensus algorithm of Aspnes and Herlihy: a case study
Distributed Computing
Code aware resource management
Proceedings of the 5th ACM international conference on Embedded software
Magnifying-lens abstraction for Markov decision processes
CAV'07 Proceedings of the 19th international conference on Computer aided verification
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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 ċ √m) symbolic steps as compared to the previous known algorithm that takes O(n2) 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 ċ √n) symbolic steps, as compared to the previous known O(n2) 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 ċ √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 5ċn symbolic steps, whereas our new algorithm takes 4ċn symbolic steps.