Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations
Journal of the ACM (JACM)
One-counter Markov decision processes
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Reachability games on extended vector addition systems with states
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming: Part II
Time-bounded reachability in tree-structured QBDs by abstraction
Performance Evaluation
Runtime analysis of probabilistic programs with unbounded recursion
ICALP'11 Proceedings of the 38th international conference on Automata, languages and programming - Volume Part II
Approximating the termination value of one-counter MDPS and stochastic games
ICALP'11 Proceedings of the 38th international conference on Automata, languages and programming - Volume Part II
Efficient analysis of probabilistic programs with an unbounded counter
CAV'11 Proceedings of the 23rd international conference on Computer aided verification
Survey: Equilibria, fixed points, and complexity classes
Computer Science Review
Approximating the termination value of one-counter MDPs and stochastic games
Information and Computation
Analyzing probabilistic pushdown automata
Formal Methods in System Design
Decidability of Weak Simulation on One-Counter Nets
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
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We begin by observing that (discrete-time) Quasi-Birth-Death Processes (QBDs) are equivalent, in a precise sense, to (discrete-time) probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs (TL-QBDs) and Tree-Structured QBDs (TS-QBDs) are equivalent to both probabilistic Pushdown Systems (pPDSs) and Recursive Markov Chains (RMCs).We then proceed to exploit these connections to obtain a number of new algorithmic upper and lower bounds for central computational problems about these models. Our main result is this: for an arbitrary QBD (even a null-recurrent one), we can approximate its termination probabilities (i.e., its G matrix) to within i bits of precision (i.e., within additive error 1/2^i), in time polynomial in both the encoding size of the QBD and in i, in the unit-cost rational arithmetic RAM model of computation. Specifically, we show that a decomposed Newton's method can be used to achieve this.We emphasize that this bound is very different from the well-known "linear/quadratic convergence" of numerical analysis, known for QBDs and TL-QBDs, which typically gives no constructive bound in terms of the encoding size of the system being solved. In fact, we observe (based on recent results for pPDSs) that for the more general TL-QBDs this bound fails badly. Specifically, in the worst case Newton's method "converges linearly" to the termination probabilities for TL-QBDs, but requires exponentially many iterations in the encoding size of the TL-QBD to approximate these probabilities within any non-trivial constant error c