Some algebraic and geometric computations in PSPACE
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Foundations of statistical natural language processing
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Introduction To Automata Theory, Languages, And Computation
Introduction To Automata Theory, Languages, And Computation
Model Checking Probabilistic Pushdown Automata
LICS '04 Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science
Quantitative Analysis of Probabilistic Pushdown Automata: Expectations and Variances
LICS '05 Proceedings of the 20th Annual IEEE Symposium on Logic in Computer Science
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Checking LTL Properties of Recursive Markov Chains
QEST '05 Proceedings of the Second International Conference on the Quantitative Evaluation of Systems
Numerical Methods for Structured Markov Chains (Numerical Mathematics and Scientific Computation)
Numerical Methods for Structured Markov Chains (Numerical Mathematics and Scientific Computation)
On the convergence of Newton's method for monotone systems of polynomial equations
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Quasi-Birth-Death Processes, Tree-Like QBDs, Probabilistic 1-Counter Automata, and Pushdown Systems
QEST '08 Proceedings of the 2008 Fifth International Conference on Quantitative Evaluation of Systems
Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations
Journal of the ACM (JACM)
Discounted Properties of Probabilistic Pushdown Automata
LPAR '08 Proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning
Concentration of Measure for the Analysis of Randomized Algorithms
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One-counter Markov decision processes
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Computing the Least Fixed Point of Positive Polynomial Systems
SIAM Journal on Computing
Algorithmic verification of recursive probabilistic state machines
TACAS'05 Proceedings of the 11th international conference on Tools and Algorithms for the Construction and Analysis of Systems
Analyzing probabilistic pushdown automata
Formal Methods in System Design
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We study the runtime in probabilistic programs with unbounded recursion. As underlying formal model for such programs we use probabilistic pushdown automata (pPDA) which exactly correspond to recursive Markov chains. We show that every pPDA can be transformed into a stateless pPDA (called "pBPA") whose runtime and further properties are closely related to those of the original pPDA. This result substantially simplifies the analysis of runtime and other pPDA properties. We prove that for every pPDA the probability of performing a long run decreases exponentially in the length of the run, if and only if the expected runtime in the pPDA is finite. If the expectation is infinite, then the probability decreases "polynomially". We show that these bounds are asymptotically tight. Our tail bounds on the runtime are generic, i.e., applicable to any probabilistic program with unbounded recursion. An intuitive interpretation is that in pPDA the runtime is exponentially unlikely to deviate from its expected value.