Some algebraic and geometric computations in PSPACE
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
A problem that is easier to solve on the unit-cost algebraic RAM
Journal of Complexity
Information and Computation
The complexity of probabilistic verification
Journal of the ACM (JACM)
Foundations of statistical natural language processing
Foundations of statistical natural language processing
Random walks with “back buttons” (extended abstract)
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Reachability Analysis of Pushdown Automata: Application to Model-Checking
CONCUR '97 Proceedings of the 8th International Conference on Concurrency Theory
Model checking for probability and time: from theory to practice
LICS '03 Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science
Some NP-complete geometric problems
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
Model Checking Probabilistic Pushdown Automata
LICS '04 Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science
Analysis of recursive state machines
ACM Transactions on Programming Languages and Systems (TOPLAS)
Checking LTL Properties of Recursive Markov Chains
QEST '05 Proceedings of the Second International Conference on the Quantitative Evaluation of Systems
The temporal logic of programs
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
Automatic verification of probabilistic concurrent finite state programs
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations
Journal of the ACM (JACM)
On the Complexity of Numerical Analysis
SIAM Journal on Computing
Recursive markov decision processes and recursive stochastic games
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
On the decidability of temporal properties of probabilistic pushdown automata
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
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
Polynomial time algorithms for multi-type branching processesand stochastic context-free grammars
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Model checking stochastic branching processes
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
Stochastic context-free grammars, regular languages, and newton's method
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part II
CAV'13 Proceedings of the 25th international conference on Computer Aided Verification
Branching-time model-checking of probabilistic pushdown automata
Journal of Computer and System Sciences
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Recursive Markov Chains (RMCs) 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 several other stochastic models, and they are equivalent in a precise sense to probabilistic Pushdown Systems. In this article, we study the problem of model checking an RMC against an ω-regular specification, given in terms of a Büchi automaton or a Linear Temporal Logic (LTL) formula. Namely, given an RMC A and a property, we wish to know the probability that an execution of A satisfies the property. 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). The complexity upper bounds we obtain for automata and LTL properties are similar, although the algorithms are different. We present algorithms for the qualitative model checking problem that run in polynomial space in the size |A| of the RMC and exponential time in the size of the property (the automaton or the LTL formula). For several classes of RMCs, including single-exit RMCs (a class that encompasses some well-studied stochastic models, for instance, stochastic context-free grammars) the algorithm runs in polynomial time in |A|. For the quantitative model checking problem, we present algorithms that run in polynomial space in the RMC and exponential space in the property. For the class of linearly recursive RMCs we can compute the exact probability in time polynomial in the RMC and exponential in the property. For deterministic automata specifications, all our complexities in the specification come down by one exponential. For lower bounds, we show that the qualitative model checking problem, even for a fixed RMC, is already EXPTIME-complete. On the other hand, even for simple reachability analysis, we know from our prior work that our PSPACE upper bounds in A can not be improved substantially without a breakthrough on a well-known open problem in the complexity of numerical computation.