License: Creative Commons Attribution-NoDerivs 3.0 Unported license (CC BY-ND 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2010.2468
URN: urn:nbn:de:0030-drops-24685
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2010/2468/
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Esparza, Javier ; Gaiser, Andreas ; Kiefer, Stefan

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},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2010/2468},
  URN =		{urn:nbn:de:0030-drops-24685},
  doi =		{10.4230/LIPIcs.STACS.2010.2468},
  annote =	{Keywords: Computing fixed points, numerical approximation, stochastic models, branching processes}
}

Keywords: Computing fixed points, numerical approximation, stochastic models, branching processes
Collection: 27th International Symposium on Theoretical Aspects of Computer Science
Issue Date: 2010
Date of publication: 09.03.2010

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