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LIPIcs - Leibniz International Proceedings in Informatics. 25th International Symposium on Theoretical Aspects of Computer Science
Doi
Abstract
Monotone systems of polynomial equations (MSPEs) are systems of
fixed-point equations X1​=f1​(X1​,ldots,Xn​),ldots,Xn​=fn​(X1​,ldots,Xn​) where each fi​ is a polynomial with
positive real coefficients. The question of computing the least
non-negative solution of a given MSPE vecX=vecf(vecX)
arises naturally in the analysis of stochastic models such as
stochastic context-free grammars, probabilistic pushdown automata,
and back-button processes. Etessami and Yannakakis have recently
adapted Newton\u27s iterative method to MSPEs. In a previous paper we
have proved the existence of a threshold kvecf​ for strongly
connected MSPEs, such that after kvecf​ iterations of
Newton\u27s method each new iteration computes at least 1 new bit of
the solution. However, the proof was purely existential. In this
paper we give an upper bound for kvecf​ as a function of the
minimal component of the least fixed-point muvecf of vecf(vecX). Using this result we show that kvecf​ is at most
single exponential resp. linear for strongly connected MSPEs
derived from probabilistic pushdown automata resp. from
back-button processes. Further, we prove the existence of a
threshold for arbitrary MSPEs after which each new iteration
computes at least 1/w2h new bits of the solution, where w and
h are the width and height of the DAG of strongly connected
components
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