Upper bounds for Newton's method on monotone polynomial systems, and P-time model checking of probabilistic one-counter automata

TL;DR

This paper establishes worst-case upper bounds on the convergence rate of Newton's method for arbitrary monotone polynomial systems, enabling P-time model checking of probabilistic automata with desired precision.

Contribution

It provides the first P-time algorithm for quantitative model checking of probabilistic one-counter automata using bounds on Newton's method convergence.

Findings

Derived worst-case upper bounds on Newton's method iterations

Bounded convergence rate as a function of input size and epsilon

Enabled P-time model checking for probabilistic automata

Abstract

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 {\em termination probabilities}, and computing these probabilities in turn boils down to computing the {\em least fixed point} (LFP) solution of a corresponding {\em monotone polynomial system} (MPS) of equations, denoted x=P(x). It was shown by Etessami & Yannakakis 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, & Luttenberger 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. 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 epsilon > 0, on the number of iterations required for decomposed Newton's method (even with rounding) to converge within additive error epsilon > 0 of q^*, for any MPS with LFP solution q^*. Our upper bounds are essentially optimal in terms of several important parameters. 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 desired precision, of discrete-time QBDs and (equivalently) probabilistic 1-counter automata, with respect to any (fixed) omega-regular or LTL property.