19 Search Results for "Middeldorp, Aart"


Document
Short Paper
Formalizing Almost Development Closed Critical Pairs (Short Paper)

Authors: Christina Kohl and Aart Middeldorp

Published in: LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023)


Abstract
We report on the first formalization of the almost-development closedness criterion for confluence of left-linear term rewrite systems and illustrate a problem we encountered while trying to formalize the original paper proof by van Oostrom.

Cite as

Christina Kohl and Aart Middeldorp. Formalizing Almost Development Closed Critical Pairs (Short Paper). In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 38:1-38:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{kohl_et_al:LIPIcs.ITP.2023.38,
  author =	{Kohl, Christina and Middeldorp, Aart},
  title =	{{Formalizing Almost Development Closed Critical Pairs}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{38:1--38:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-284-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{268},
  editor =	{Naumowicz, Adam and Thiemann, Ren\'{e}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.38},
  URN =		{urn:nbn:de:0030-drops-184130},
  doi =		{10.4230/LIPIcs.ITP.2023.38},
  annote =	{Keywords: Term rewriting, confluence, certification}
}
Document
Hydra Battles and AC Termination

Authors: Nao Hirokawa and Aart Middeldorp

Published in: LIPIcs, Volume 260, 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)


Abstract
We present a new encoding of the Battle of Hercules and Hydra as a rewrite system with AC symbols. Unlike earlier term rewriting encodings, it faithfully models any strategy of Hercules to beat Hydra. To prove the termination of our encoding, we employ type introduction in connection with many-sorted semantic labeling for AC rewriting and AC-RPO.

Cite as

Nao Hirokawa and Aart Middeldorp. Hydra Battles and AC Termination. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{hirokawa_et_al:LIPIcs.FSCD.2023.12,
  author =	{Hirokawa, Nao and Middeldorp, Aart},
  title =	{{Hydra Battles and AC Termination}},
  booktitle =	{8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)},
  pages =	{12:1--12:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-277-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{260},
  editor =	{Gaboardi, Marco and van Raamsdonk, Femke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.12},
  URN =		{urn:nbn:de:0030-drops-179965},
  doi =		{10.4230/LIPIcs.FSCD.2023.12},
  annote =	{Keywords: battle of Hercules and Hydra, term rewriting, termination}
}
Document
Polynomial Termination Over ℕ Is Undecidable

Authors: Fabian Mitterwallner and Aart Middeldorp

Published in: LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)


Abstract
In this paper we prove via a reduction from Hilbert’s 10th problem that the problem whether the termination of a given rewrite system can be shown by a polynomial interpretation in the natural numbers is undecidable, even for rewrite systems that are incrementally polynomially terminating. We also prove that incremental polynomial termination is an undecidable property of terminating rewrite systems.

Cite as

Fabian Mitterwallner and Aart Middeldorp. Polynomial Termination Over ℕ Is Undecidable. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{mitterwallner_et_al:LIPIcs.FSCD.2022.27,
  author =	{Mitterwallner, Fabian and Middeldorp, Aart},
  title =	{{Polynomial Termination Over \mathbb{N} Is Undecidable}},
  booktitle =	{7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)},
  pages =	{27:1--27:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-233-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{228},
  editor =	{Felty, Amy P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.27},
  URN =		{urn:nbn:de:0030-drops-163081},
  doi =		{10.4230/LIPIcs.FSCD.2022.27},
  annote =	{Keywords: Term Rewriting, Polynomial Termination, Undecidability}
}
Document
Completion for Logically Constrained Rewriting

Authors: Sarah Winkler and Aart Middeldorp

Published in: LIPIcs, Volume 108, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)


Abstract
We propose an abstract completion procedure for logically constrained term rewrite systems (LCTRSs). This procedure can be instantiated to both standard Knuth-Bendix completion and ordered completion for LCTRSs, and we present a succinct and uniform correctness proof. A prototype implementation illustrates the viability of the new completion approach.

Cite as

Sarah Winkler and Aart Middeldorp. Completion for Logically Constrained Rewriting. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 30:1-30:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{winkler_et_al:LIPIcs.FSCD.2018.30,
  author =	{Winkler, Sarah and Middeldorp, Aart},
  title =	{{Completion for Logically Constrained Rewriting}},
  booktitle =	{3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)},
  pages =	{30:1--30:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-077-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{108},
  editor =	{Kirchner, H\'{e}l\`{e}ne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2018.30},
  URN =		{urn:nbn:de:0030-drops-92001},
  doi =		{10.4230/LIPIcs.FSCD.2018.30},
  annote =	{Keywords: Constrained rewriting, completion, automation, theorem proving}
}
Document
System Description
ProTeM: A Proof Term Manipulator (System Description)

Authors: Christina Kohl and Aart Middeldorp

Published in: LIPIcs, Volume 108, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)


Abstract
Proof terms are a useful concept for reasoning about computations in term rewriting. Human calculation with proof terms is tedious and error-prone. We present ProTeM, a new tool that offers support for manipulating proof terms that represent multisteps in left-linear rewrite systems.

Cite as

Christina Kohl and Aart Middeldorp. ProTeM: A Proof Term Manipulator (System Description). In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 31:1-31:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{kohl_et_al:LIPIcs.FSCD.2018.31,
  author =	{Kohl, Christina and Middeldorp, Aart},
  title =	{{ProTeM: A Proof Term Manipulator}},
  booktitle =	{3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)},
  pages =	{31:1--31:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-077-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{108},
  editor =	{Kirchner, H\'{e}l\`{e}ne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2018.31},
  URN =		{urn:nbn:de:0030-drops-92013},
  doi =		{10.4230/LIPIcs.FSCD.2018.31},
  annote =	{Keywords: Proof terms, term rewriting, interactive tool}
}
Document
Confluence Competition 2018

Authors: Takahito Aoto, Makoto Hamana, Nao Hirokawa, Aart Middeldorp, Julian Nagele, Naoki Nishida, Kiraku Shintani, and Harald Zankl

Published in: LIPIcs, Volume 108, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)


Abstract
We report on the 2018 edition of the Confluence Competition, a competition of software tools that aim to (dis)prove confluence and related properties of rewrite systems automatically.

Cite as

Takahito Aoto, Makoto Hamana, Nao Hirokawa, Aart Middeldorp, Julian Nagele, Naoki Nishida, Kiraku Shintani, and Harald Zankl. Confluence Competition 2018. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 32:1-32:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{aoto_et_al:LIPIcs.FSCD.2018.32,
  author =	{Aoto, Takahito and Hamana, Makoto and Hirokawa, Nao and Middeldorp, Aart and Nagele, Julian and Nishida, Naoki and Shintani, Kiraku and Zankl, Harald},
  title =	{{Confluence Competition 2018}},
  booktitle =	{3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)},
  pages =	{32:1--32:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-077-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{108},
  editor =	{Kirchner, H\'{e}l\`{e}ne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2018.32},
  URN =		{urn:nbn:de:0030-drops-92023},
  doi =		{10.4230/LIPIcs.FSCD.2018.32},
  annote =	{Keywords: Confluence, competition, rewrite systems}
}
Document
Infinite Runs in Abstract Completion

Authors: Nao Hirokawa, Aart Middeldorp, Christian Sternagel, and Sarah Winkler

Published in: LIPIcs, Volume 84, 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)


Abstract
Completion is one of the first and most studied techniques in term rewriting and fundamental to automated reasoning with equalities. In an earlier paper we presented a new and formalized correctness proof of abstract completion for finite runs. In this paper we extend our analysis and our formalization to infinite runs, resulting in a new proof that fair infinite runs produce complete presentations of the initial equations. We further consider ordered completion - an important extension of completion that aims to produce ground-complete presentations of the initial equations. Moreover, we revisit and extend results of Métivier concerning canonicity of rewrite systems. All proofs presented in the paper have been formalized in Isabelle/HOL.

Cite as

Nao Hirokawa, Aart Middeldorp, Christian Sternagel, and Sarah Winkler. Infinite Runs in Abstract Completion. In 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 84, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{hirokawa_et_al:LIPIcs.FSCD.2017.19,
  author =	{Hirokawa, Nao and Middeldorp, Aart and Sternagel, Christian and Winkler, Sarah},
  title =	{{Infinite Runs in Abstract Completion}},
  booktitle =	{2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)},
  pages =	{19:1--19:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-047-7},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{84},
  editor =	{Miller, Dale},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2017.19},
  URN =		{urn:nbn:de:0030-drops-77252},
  doi =		{10.4230/LIPIcs.FSCD.2017.19},
  annote =	{Keywords: term rewriting, abstract completion, ordered completion, canonicity, Isabelle/HOL}
}
Document
Automating the First-Order Theory of Rewriting for Left-Linear Right-Ground Rewrite Systems

Authors: Franziska Rapp and Aart Middeldorp

Published in: LIPIcs, Volume 52, 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)


Abstract
The first-order theory of rewriting is decidable for finite left-linear right-ground rewrite systems. We present a new tool that implements the decision procedure for this theory. It is based on tree automata techniques. The tool offers the possibility to synthesize rewrite systems that satisfy properties that are expressible in the first-order theory of rewriting.

Cite as

Franziska Rapp and Aart Middeldorp. Automating the First-Order Theory of Rewriting for Left-Linear Right-Ground Rewrite Systems. In 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 52, pp. 36:1-36:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{rapp_et_al:LIPIcs.FSCD.2016.36,
  author =	{Rapp, Franziska and Middeldorp, Aart},
  title =	{{Automating the First-Order Theory of Rewriting for Left-Linear Right-Ground Rewrite Systems}},
  booktitle =	{1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)},
  pages =	{36:1--36:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-010-1},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{52},
  editor =	{Kesner, Delia and Pientka, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2016.36},
  URN =		{urn:nbn:de:0030-drops-60018},
  doi =		{10.4230/LIPIcs.FSCD.2016.36},
  annote =	{Keywords: first-order theory, ground rewrite systems, automation, synthesis}
}
Document
Leftmost Outermost Revisited

Authors: Nao Hirokawa, Aart Middeldorp, and Georg Moser

Published in: LIPIcs, Volume 36, 26th International Conference on Rewriting Techniques and Applications (RTA 2015)


Abstract
We present an elementary proof of the classical result that the leftmost outermost strategy is normalizing for left-normal orthogonal rewrite systems. Our proof is local and extends to hyper-normalization and weakly orthogonal systems. Based on the new proof, we study basic normalization, i.e., we study normalization if the set of considered starting terms is restricted to basic terms. This allows us to weaken the left-normality restriction. We show that the leftmost outermost strategy is hyper-normalizing for basically left-normal orthogonal rewrite systems. This shift of focus greatly extends the applicability of the classical result, as evidenced by the experimental data provided.

Cite as

Nao Hirokawa, Aart Middeldorp, and Georg Moser. Leftmost Outermost Revisited. In 26th International Conference on Rewriting Techniques and Applications (RTA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 36, pp. 209-222, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{hirokawa_et_al:LIPIcs.RTA.2015.209,
  author =	{Hirokawa, Nao and Middeldorp, Aart and Moser, Georg},
  title =	{{Leftmost Outermost Revisited}},
  booktitle =	{26th International Conference on Rewriting Techniques and Applications (RTA 2015)},
  pages =	{209--222},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-85-9},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{36},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2015.209},
  URN =		{urn:nbn:de:0030-drops-51986},
  doi =		{10.4230/LIPIcs.RTA.2015.209},
  annote =	{Keywords: term rewriting, strategies, normalization}
}
Document
Conditional Complexity

Authors: Cynthia Kop, Aart Middeldorp, and Thomas Sternagel

Published in: LIPIcs, Volume 36, 26th International Conference on Rewriting Techniques and Applications (RTA 2015)


Abstract
We propose a notion of complexity for oriented conditional term rewrite systems. This notion is realistic in the sense that it measures not only successful computations but also partial computations that result in a failed rule application. A transformation to unconditional context-sensitive rewrite systems is presented which reflects this complexity notion, as well as a technique to derive runtime and derivational complexity bounds for the latter.

Cite as

Cynthia Kop, Aart Middeldorp, and Thomas Sternagel. Conditional Complexity. In 26th International Conference on Rewriting Techniques and Applications (RTA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 36, pp. 223-240, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{kop_et_al:LIPIcs.RTA.2015.223,
  author =	{Kop, Cynthia and Middeldorp, Aart and Sternagel, Thomas},
  title =	{{Conditional Complexity}},
  booktitle =	{26th International Conference on Rewriting Techniques and Applications (RTA 2015)},
  pages =	{223--240},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-85-9},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{36},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2015.223},
  URN =		{urn:nbn:de:0030-drops-51999},
  doi =		{10.4230/LIPIcs.RTA.2015.223},
  annote =	{Keywords: conditional term rewriting, complexity}
}
Document
Improving Automatic Confluence Analysis of Rewrite Systems by Redundant Rules

Authors: Julian Nagele, Bertram Felgenhauer, and Aart Middeldorp

Published in: LIPIcs, Volume 36, 26th International Conference on Rewriting Techniques and Applications (RTA 2015)


Abstract
We describe how to utilize redundant rewrite rules, i.e., rules that can be simulated by other rules, when (dis)proving confluence of term rewrite systems. We demonstrate how automatic confluence provers benefit from the addition as well as the removal of redundant rules. Due to their simplicity, our transformations were easy to formalize in a proof assistant and are thus amenable to certification. Experimental results show the surprising gain in power.

Cite as

Julian Nagele, Bertram Felgenhauer, and Aart Middeldorp. Improving Automatic Confluence Analysis of Rewrite Systems by Redundant Rules. In 26th International Conference on Rewriting Techniques and Applications (RTA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 36, pp. 257-268, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{nagele_et_al:LIPIcs.RTA.2015.257,
  author =	{Nagele, Julian and Felgenhauer, Bertram and Middeldorp, Aart},
  title =	{{Improving Automatic Confluence Analysis of Rewrite Systems by Redundant Rules}},
  booktitle =	{26th International Conference on Rewriting Techniques and Applications (RTA 2015)},
  pages =	{257--268},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-85-9},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{36},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2015.257},
  URN =		{urn:nbn:de:0030-drops-52010},
  doi =		{10.4230/LIPIcs.RTA.2015.257},
  annote =	{Keywords: term rewriting, confluence, automation, transformation}
}
Document
Normalized Completion Revisited

Authors: Sarah Winkler and Aart Middeldorp

Published in: LIPIcs, Volume 21, 24th International Conference on Rewriting Techniques and Applications (RTA 2013)


Abstract
Normalized completion (Marché 1996) is a widely applicable and efficient technique for com- pletion modulo theories. If successful, a normalized completion procedure computes a rewrite system that allows to decide the validity problem using normalized rewriting. In this paper we consider a slightly simplified inference system for finite normalized completion runs. We prove correctness, show faithfulness of critical pair criteria in our setting, and propose a different notion of normalizing pairs. We then show how normalized completion procedures can benefit from AC- termination tools instead of relying on a fixed AC-compatible reduction order. We outline our implementation of this approach in the completion tool mkbtt and present experimental results, including new completions.

Cite as

Sarah Winkler and Aart Middeldorp. Normalized Completion Revisited. In 24th International Conference on Rewriting Techniques and Applications (RTA 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 21, pp. 319-334, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


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@InProceedings{winkler_et_al:LIPIcs.RTA.2013.319,
  author =	{Winkler, Sarah and Middeldorp, Aart},
  title =	{{Normalized Completion Revisited}},
  booktitle =	{24th International Conference on Rewriting Techniques and Applications (RTA 2013)},
  pages =	{319--334},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-53-8},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{21},
  editor =	{van Raamsdonk, Femke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2013.319},
  URN =		{urn:nbn:de:0030-drops-40702},
  doi =		{10.4230/LIPIcs.RTA.2013.319},
  annote =	{Keywords: term rewriting, completion}
}
Document
Beyond Peano Arithmetic – Automatically Proving Termination of the Goodstein Sequence

Authors: Sarah Winkler, Harald Zankl, and Aart Middeldorp

Published in: LIPIcs, Volume 21, 24th International Conference on Rewriting Techniques and Applications (RTA 2013)


Abstract
Kirby and Paris (1982) proved in a celebrated paper that a theorem of Goodstein (1944) cannot be established in Peano (1889) arithmetic. We present an encoding of Goodstein's theorem as a termination problem of a finite rewrite system. Using a novel implementation of ordinal interpretations, we are able to automatically prove termination of this system, resulting in the first automatic termination proof for a system whose derivational complexity is not multiple recursive. Our method can also cope with the encoding by Touzet (1998) of the battle of Hercules and Hydra, yet another system which has been out of reach for automated tools, until now.

Cite as

Sarah Winkler, Harald Zankl, and Aart Middeldorp. Beyond Peano Arithmetic – Automatically Proving Termination of the Goodstein Sequence. In 24th International Conference on Rewriting Techniques and Applications (RTA 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 21, pp. 335-351, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


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@InProceedings{winkler_et_al:LIPIcs.RTA.2013.335,
  author =	{Winkler, Sarah and Zankl, Harald and Middeldorp, Aart},
  title =	{{Beyond Peano Arithmetic – Automatically Proving Termination of the Goodstein Sequence}},
  booktitle =	{24th International Conference on Rewriting Techniques and Applications (RTA 2013)},
  pages =	{335--351},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-53-8},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{21},
  editor =	{van Raamsdonk, Femke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2013.335},
  URN =		{urn:nbn:de:0030-drops-40718},
  doi =		{10.4230/LIPIcs.RTA.2013.335},
  annote =	{Keywords: term rewriting, termination, automation, ordinals}
}
Document
Layer Systems for Proving Confluence

Authors: Bertram Felgenhauer, Harald Zankl, and Aart Middeldorp

Published in: LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)


Abstract
We introduce layer systems for proving generalizations of the modularity of confluence for first-order rewrite systems. Layer systems specify how terms can be divided into layers. We establish structural conditions on those systems that imply confluence. Our abstract framework covers known results like many-sorted persistence, layer-preservation and currying. We present a counterexample to an extension of the former to order-sorted rewriting and derive new sufficient conditions for the extension to hold.

Cite as

Bertram Felgenhauer, Harald Zankl, and Aart Middeldorp. Layer Systems for Proving Confluence. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 288-299, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)


Copy BibTex To Clipboard

@InProceedings{felgenhauer_et_al:LIPIcs.FSTTCS.2011.288,
  author =	{Felgenhauer, Bertram and Zankl, Harald and Middeldorp, Aart},
  title =	{{Layer Systems for Proving Confluence}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
  pages =	{288--299},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-34-7},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{13},
  editor =	{Chakraborty, Supratik and Kumar, Amit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.288},
  URN =		{urn:nbn:de:0030-drops-33265},
  doi =		{10.4230/LIPIcs.FSTTCS.2011.288},
  annote =	{Keywords: Term rewriting, Confluence, Modularity, Persistence}
}
Document
Revisiting Matrix Interpretations for Proving Termination of Term Rewriting

Authors: Friedrich Neurauter and Aart Middeldorp

Published in: LIPIcs, Volume 10, 22nd International Conference on Rewriting Techniques and Applications (RTA'11) (2011)


Abstract
Matrix interpretations are a powerful technique for proving termination of term rewrite systems, which is based on the well-known paradigm of interpreting terms into a domain equipped with a suitable well-founded order, such that every rewrite step causes a strict decrease. Traditionally, one uses vectors of non-negative numbers as domain, where two vectors are in the order relation if there is a strict decrease in the respective first components and a weak decrease in all other components. In this paper, we study various alternative well-founded orders on vectors of non-negative numbers based on vector norms and compare the resulting variants of matrix interpretations to each other and to the traditional approach. These comparisons are mainly theoretical in nature. We do, however, also identify one of these variants as a proper generalization of traditional matrix interpretations as a stand-alone termination method, which has the additional advantage that it gives rise to a more powerful implementation.

Cite as

Friedrich Neurauter and Aart Middeldorp. Revisiting Matrix Interpretations for Proving Termination of Term Rewriting. In 22nd International Conference on Rewriting Techniques and Applications (RTA'11). Leibniz International Proceedings in Informatics (LIPIcs), Volume 10, pp. 251-266, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)


Copy BibTex To Clipboard

@InProceedings{neurauter_et_al:LIPIcs.RTA.2011.251,
  author =	{Neurauter, Friedrich and Middeldorp, Aart},
  title =	{{Revisiting Matrix Interpretations for Proving Termination of Term Rewriting}},
  booktitle =	{22nd International Conference on Rewriting Techniques and Applications (RTA'11)},
  pages =	{251--266},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-30-9},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{10},
  editor =	{Schmidt-Schauss, Manfred},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2011.251},
  URN =		{urn:nbn:de:0030-drops-31222},
  doi =		{10.4230/LIPIcs.RTA.2011.251},
  annote =	{Keywords: term rewriting, termination, matrix interpretations}
}
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