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URN: urn:nbn:de:0030-drops-24685
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### Computing Least Fixed Points of Probabilistic Systems of Polynomials

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### Abstract

We study systems of equations of the form $X_1 = f_1(X_1, \ldots, X_n), \ldots, X_n = f_n(X_1, \ldots, X_n)$ where each $f_i$ is a polynomial with nonnegative coefficients that add up to~$1$. The least nonnegative solution, say~$\mu$, of such equation systems is central to problems from various areas, like physics, biology, computational linguistics and probabilistic program verification. We give a simple and strongly polynomial algorithm to decide whether $\mu=(1,\ldots,1)$ holds. Furthermore, we present an algorithm that computes reliable sequences of lower and upper bounds on~$\mu$, converging linearly to~$\mu$. Our algorithm has these features despite using inexact arithmetic for efficiency. We report on experiments that show the performance of our algorithms.

### BibTeX - Entry

@InProceedings{esparza_et_al:LIPIcs:2010:2468,
author =	{Javier Esparza and Andreas Gaiser and Stefan Kiefer},
title =	{{Computing Least Fixed Points of  Probabilistic Systems of Polynomials}},
booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
pages =	{359--370},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-16-3},
ISSN =	{1868-8969},
year =	{2010},
volume =	{5},
editor =	{Jean-Yves Marion and Thomas Schwentick},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},