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Algebraic Foundations in Computer Science pp 112–145Cite as

Equational Weighted Tree Transformations with Discounting

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7020))

Abstract

We consider systems of equations of polynomial weighted tree transformations over the max-plus (or: arctic) semiring ℝ max  = ( ℝ  +  ∪ { − ∞ }, max , + , − ∞ ,0). We apply discounting with a parameter 0 ≤ d < 1 in order to guarantee the existence of the least solution, called least d-solution, of such systems. We compute least d-solutions under u-substitution mode, where u = [IO] or u = OI. We define a weighted relation over ℝ max to be u-d-equational, if it is a component of the least u-d-solution of such a system of equations in a pair of algebras. We mainly focus on u-d-equational weighted tree transformations which are equational relations obtained by considering the least u-d-solutions in pairs of term algebras. We also introduce u-d-equational weighted tree languages over ℝ max . We characterize u-d-equational weighted tree transformations in terms of weighted tree transformations defined by weighted d-bimorphisms, which are bimorphisms from d-recognizable weighted tree languages. Finally, we prove that a weighted relation is u-d-equational if and only if it is, roughly speaking, the morphic image of a weighted u-d-equational tree transformation.

This research was financially supported by the TÁMOP-4.2.1/B-09/1/KONV-2010-0005 program of the Hungarian National Development Agency.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Szeged, Árpád tér 2., H-6720, Szeged, Hungary

    Zoltán Fülöp

  2. Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

    George Rahonis

Authors
  1. Zoltán Fülöp
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  2. George Rahonis
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Editor information

Editors and Affiliations

  1. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Austria

    Werner Kuich

  2. Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

    George Rahonis

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Fülöp, Z., Rahonis, G. (2011). Equational Weighted Tree Transformations with Discounting. In: Kuich, W., Rahonis, G. (eds) Algebraic Foundations in Computer Science. Lecture Notes in Computer Science, vol 7020. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24897-9_6

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  • DOI: https://doi.org/10.1007/978-3-642-24897-9_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-24896-2

  • Online ISBN: 978-3-642-24897-9

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