Computing Least Fixed Points of Probabilistic Systems of Polynomials

Authors Javier Esparza, Andreas Gaiser, Stefan Kiefer

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Javier Esparza
Andreas Gaiser
Stefan Kiefer

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Javier Esparza, Andreas Gaiser, and Stefan Kiefer. Computing Least Fixed Points of Probabilistic Systems of Polynomials. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 359-370, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)


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.
  • Computing fixed points
  • numerical approximation
  • stochastic models
  • branching processes


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