Polynomials over the reals in proofs of termination : from theory to practice

Salvador Lucas

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 39, Issue: 3, page 547-586
  • ISSN: 0988-3754

Abstract

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This paper provides a framework to address termination problems in term rewriting by using orderings induced by algebras over the reals. The generation of such orderings is parameterized by concrete monotonicity requirements which are connected with different classes of termination problems: termination of rewriting, termination of rewriting by using dependency pairs, termination of innermost rewriting, top-termination of infinitary rewriting, termination of context-sensitive rewriting, etc. We show how to define term orderings based on algebraic interpretations over the real numbers which can be used for these purposes. From a practical point of view, we show how to automatically generate polynomial algebras over the reals by using constraint-solving systems to obtain the coefficients of a polynomial in the domain of the real or rational numbers. Moreover, as a consequence of our work, we argue that software systems which are able to generate constraints for obtaining polynomial interpretations over the naturals which prove termination of rewriting (e.g., AProVE, CiME, and TTT), are potentially able to obtain suitable interpretations over the reals by just solving the constraints in the domain of the real or rational numbers.

How to cite

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Lucas, Salvador. "Polynomials over the reals in proofs of termination : from theory to practice." RAIRO - Theoretical Informatics and Applications 39.3 (2010): 547-586. <http://eudml.org/doc/92778>.

@article{Lucas2010,
abstract = { This paper provides a framework to address termination problems in term rewriting by using orderings induced by algebras over the reals. The generation of such orderings is parameterized by concrete monotonicity requirements which are connected with different classes of termination problems: termination of rewriting, termination of rewriting by using dependency pairs, termination of innermost rewriting, top-termination of infinitary rewriting, termination of context-sensitive rewriting, etc. We show how to define term orderings based on algebraic interpretations over the real numbers which can be used for these purposes. From a practical point of view, we show how to automatically generate polynomial algebras over the reals by using constraint-solving systems to obtain the coefficients of a polynomial in the domain of the real or rational numbers. Moreover, as a consequence of our work, we argue that software systems which are able to generate constraints for obtaining polynomial interpretations over the naturals which prove termination of rewriting (e.g., AProVE, CiME, and TTT), are potentially able to obtain suitable interpretations over the reals by just solving the constraints in the domain of the real or rational numbers. },
author = {Lucas, Salvador},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Algebraic interpretations; polynomial orderings; rewriting; termination.; term rewriting; polynomial algebras; constraint-solving systems},
language = {eng},
month = {3},
number = {3},
pages = {547-586},
publisher = {EDP Sciences},
title = {Polynomials over the reals in proofs of termination : from theory to practice},
url = {http://eudml.org/doc/92778},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Lucas, Salvador
TI - Polynomials over the reals in proofs of termination : from theory to practice
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 3
SP - 547
EP - 586
AB - This paper provides a framework to address termination problems in term rewriting by using orderings induced by algebras over the reals. The generation of such orderings is parameterized by concrete monotonicity requirements which are connected with different classes of termination problems: termination of rewriting, termination of rewriting by using dependency pairs, termination of innermost rewriting, top-termination of infinitary rewriting, termination of context-sensitive rewriting, etc. We show how to define term orderings based on algebraic interpretations over the real numbers which can be used for these purposes. From a practical point of view, we show how to automatically generate polynomial algebras over the reals by using constraint-solving systems to obtain the coefficients of a polynomial in the domain of the real or rational numbers. Moreover, as a consequence of our work, we argue that software systems which are able to generate constraints for obtaining polynomial interpretations over the naturals which prove termination of rewriting (e.g., AProVE, CiME, and TTT), are potentially able to obtain suitable interpretations over the reals by just solving the constraints in the domain of the real or rational numbers.
LA - eng
KW - Algebraic interpretations; polynomial orderings; rewriting; termination.; term rewriting; polynomial algebras; constraint-solving systems
UR - http://eudml.org/doc/92778
ER -

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