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Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations
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Information Processing Letters
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Automata, Probability, and Recursion
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Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations
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Discounted Properties of Probabilistic Pushdown Automata
LPAR '08 Proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning
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Sliding Window Abstraction for Infinite Markov Chains
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Methods for Quantitative Analysis of Probabilistic Pushdown Automata
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A note on the attractor-property of infinite-state Markov chains
Information Processing Letters
PReMo: an analyzer for probabilistic recursive models
TACAS'07 Proceedings of the 13th international conference on Tools and algorithms for the construction and analysis of systems
On models and query languages for probabilistic processes
ACM SIGMOD Record
Probabilistic XML via Markov Chains
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SIAM Journal on Computing
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Recursive concurrent stochastic games
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Recursive markov decision processes and recursive stochastic games
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ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
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FSTTCS '05 Proceedings of the 25th international conference on Foundations of Software Technology and Theoretical Computer Science
Model Checking of Recursive Probabilistic Systems
ACM Transactions on Computational Logic (TOCL)
Analysis of recursive probabilistic models
ATVA'06 Proceedings of the 4th international conference on Automated Technology for Verification and Analysis
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The VLDB Journal — The International Journal on Very Large Data Bases
Branching-time model-checking of probabilistic pushdown automata
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Analyzing probabilistic pushdown automata
Formal Methods in System Design
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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.