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Derivation Tree Analysis for Accelerated Fixed-Point Computation

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Developments in Language Theory (DLT 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5257))

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Abstract

We show that for several classes of idempotent semirings the least fixed-point of a polynomial system of equations is equal to the least fixed-point of a linear system obtained by “linearizing” the polynomials of in a certain way. Our proofs rely on derivation tree analysis, a proof principle that combines methods from algebra, calculus, and formal language theory, and was first used in [5] to show that Newton’s method over commutative and idempotent semirings converges in a linear number of steps. Our results lead to efficient generic algorithms for computing the least fixed-point. We use these algorithms to derive several consequences, including an O(N 3) algorithm for computing the throughput of a context-free grammar (obtained by speeding up the O(N 4) algorithm of [2]), and a generalization of Courcelle’s result stating that the downward-closed image of a context-free language is regular [3].

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Authors and Affiliations

  1. Institut für Informatik, Technische Universität München, 85748, Garching, Germany

    Javier Esparza, Stefan Kiefer & Michael Luttenberger

Authors
  1. Javier Esparza
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  2. Stefan Kiefer
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  3. Michael Luttenberger
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Masami Ito Masafumi Toyama

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Esparza, J., Kiefer, S., Luttenberger, M. (2008). Derivation Tree Analysis for Accelerated Fixed-Point Computation. In: Ito, M., Toyama, M. (eds) Developments in Language Theory. DLT 2008. Lecture Notes in Computer Science, vol 5257. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85780-8_24

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  • DOI: https://doi.org/10.1007/978-3-540-85780-8_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85779-2

  • Online ISBN: 978-3-540-85780-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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