2 research outputs found
Upper Bounds for Newton's Method on Monotone Polynomial Systems, and P-Time Model Checking of Probabilistic One-Counter Automata
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