Matrix analysis
The complexity of probabilistic verification
Journal of the ACM (JACM)
Numerical Methods for Structured Markov Chains (Numerical Mathematics and Scientific Computation)
Numerical Methods for Structured Markov Chains (Numerical Mathematics and Scientific Computation)
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
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
Computing the Least Fixed Point of Positive Polynomial Systems
SIAM Journal on Computing
Exact algorithms for solving stochastic games: extended abstract
Proceedings of the forty-third annual ACM symposium on Theory of computing
Efficient analysis of probabilistic programs with an unbounded counter
CAV'11 Proceedings of the 23rd international conference on Computer aided verification
Model Checking of Recursive Probabilistic Systems
ACM Transactions on Computational Logic (TOCL)
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
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A central computational problem for analyzing and model checking various classes of infinite-state recursive probabilistic systems (including quasi-birth-death processes, multi-type branching processes, stochastic context-free grammars, probabilistic pushdown automata and recursive Markov chains) is the computation of termination probabilities, and computing these probabilities in turn boils down to computing the least fixed point (LFP) solution of a corresponding monotone polynomial system (MPS) of equations, denoted x=P(x). It was shown by Etessami and Yannakakis [11] that a decomposed variant of Newton's method converges monotonically to the LFP solution for any MPS that has a non-negative solution. Subsequently, Esparza, Kiefer, and Luttenberger [7] obtained upper bounds on the convergence rate of Newton's method for certain classes of MPSs. More recently, better upper bounds have been obtained for special classes of MPSs ([10, 9]). However, prior to this paper, for arbitrary (not necessarily strongly-connected) MPSs, no upper bounds at all were known on the convergence rate of Newton's method as a function of the encoding size |P| of the input MPS, x=P(x). In this paper we provide worst-case upper bounds, as a function of both the input encoding size |P|, and ε0, on the number of iterations required for decomposed Newton's method (even with rounding) to converge to within additive error ε0 of q*, for an arbitrary MPS with LFP solution q*. Our upper bounds are essentially optimal in terms of several important parameters of the problem. Using our upper bounds, and building on prior work, we obtain the first P-time algorithm (in the standard Turing model of computation) for quantitative model checking, to within arbitrary desired precision, of discrete-time QBDs and (equivalently) probabilistic 1-counter automata, with respect to any (fixed) ω-regular or LTL property.