article

Max/plus tree automata for termination of term rewriting

Authors:

Adam Koprowski,

Johannes WaldmannAuthors Info & Claims

Acta Cybernetica, Volume 19, Issue 2

Pages 357 - 392

Published: 01 February 2009 Publication History

Abstract

We use weighted tree automata as certificates for termination of term rewriting systems. The weights are taken from the arctic semiring: natural numbers extended with -∞, with the operations "max" and "plus". In order to find and validate these certificates automatically, we restrict their transition functions to be representable by matrix operations in the semiring. The resulting class of weighted tree automata is called path-separated.

This extends the matrix method for term rewriting and the arctic matrix method for string rewriting. In combination with the dependency pair method, this allows for some conceptually simple termination proofs in cases where only much more involved proofs were known before. We further generalize to arctic numbers "below zero": integers extended with -∞. This allows to treat some termination problems with symbols that require a predecessor semantics.

Correctness of this approach has been formally verified in the Coq proof assistant and the formalization has been contributed to the CoLoR library of certified termination techniques. This allows formal verification of termination proofs using the arctic matrix method in combination with the dependency pair transformation. This contribution brought a substantial performance gain in the certified category of the 2008 edition of the termination competition.

The method has been implemented by leading termination provers. We report on experiments with its implementation in one such tool, Matchbox, developed by the second author.

We also show that our method can simulate a previous method of quasi-periodic interpretations, if restricted to interpretations of slope one on unary signatures.

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Published In

cover image Acta Cybernetica

Acta Cybernetica Volume 19, Issue 2

February 2009

311 pages

ISSN:0324-721X

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Publisher

Institute of Informatics, University of Szeged

Szeged, Hungary

Publication History

Published: 01 February 2009

Received: 01 August 2008

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Mitterwallner FMiddeldorp AThiemann RSobocinski PLago UEsparza J(2024)Linear Termination is UndecidableProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662081(1-12)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662081
Paul E(2024)Finite Sequentiality of Finitely Ambiguous Max-Plus Tree AutomataTheory of Computing Systems10.1007/s00224-021-10049-668:4(615-661)Online publication date: 1-Aug-2024
https://dl.acm.org/doi/10.1007/s00224-021-10049-6
Yolcu EAaronson SHeule M(2023)An Automated Approach to the Collatz ConjectureJournal of Automated Reasoning10.1007/s10817-022-09658-867:2Online publication date: 25-Apr-2023
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Yamada A(2022)Tuple Interpretations for Termination of Term RewritingJournal of Automated Reasoning10.1007/s10817-022-09640-466:4(667-688)Online publication date: 1-Nov-2022
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Yamada A(2021)Multi-Dimensional Interpretations for Termination of Term RewritingAutomated Deduction – CADE 2810.1007/978-3-030-79876-5_16(273-290)Online publication date: 12-Jul-2021
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Fuhs C(2019)Transforming Derivational Complexity of Term Rewriting to Runtime ComplexityFrontiers of Combining Systems10.1007/978-3-030-29007-8_20(348-364)Online publication date: 4-Sep-2019
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Giesl JAschermann CBrockschmidt MEmmes FFrohn FFuhs CHensel JOtto CPlücker MSchneider-Kamp PStröder TSwiderski SThiemann R(2017)Analyzing Program Termination and Complexity Automatically with AProVEJournal of Automated Reasoning10.1007/s10817-016-9388-y58:1(3-31)Online publication date: 1-Jan-2017
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