4 Search Results for "Gramlich, Bernhard"

Document

DOI: 10.4230/LIPIcs.FSCD.2024.27

A Verified Algorithm for Deciding Pattern Completeness

Authors: René Thiemann and Akihisa Yamada

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Pattern completeness is the property that the left-hand sides of a functional program cover all cases w.r.t. pattern matching. In the context of term rewriting a related notion is quasi-reducibility, a prerequisite if one wants to perform ground confluence proofs by rewriting induction. In order to certify such confluence proofs, we develop a novel algorithm that decides pattern completeness and that can be used to ensure quasi-reducibility. One of the advantages of the proposed algorithm is its simple structure: it is similar to that of a regular matching algorithm and, unlike an existing decision procedure for quasi-reducibility, it avoids enumerating all terms up to a given depth. Despite the simple structure, proving the correctness of the algorithm is not immediate. Therefore we formalize the algorithm and verify its correctness using the proof assistant Isabelle/HOL. To this end, we not only verify some auxiliary algorithms, but also design an Isabelle library on sorted term rewriting. Moreover, we export the verified code in Haskell and experimentally evaluate its performance. We observe that our algorithm significantly outperforms existing algorithms, even including the pattern completeness check of the GHC Haskell compiler.

Cite as

René Thiemann and Akihisa Yamada. A Verified Algorithm for Deciding Pattern Completeness. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{thiemann_et_al:LIPIcs.FSCD.2024.27,
  author =	{Thiemann, Ren\'{e} and Yamada, Akihisa},
  title =	{{A Verified Algorithm for Deciding Pattern Completeness}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{27:1--27:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.27},
  URN =		{urn:nbn:de:0030-drops-203566},
  doi =		{10.4230/LIPIcs.FSCD.2024.27},
  annote =	{Keywords: Isabelle/HOL, pattern matching, term rewriting}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.32

Termination of Generalized Term Rewriting Systems

Authors: Salvador Lucas

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

We investigate termination of Generalized Term Rewriting Systems (GTRS), which extend Conditional Term Rewriting Systems by considering replacement restrictions on selected arguments of function symbols, as in Context-Sensitive Rewriting, and conditional rewriting rules whose conditional part may include not only a mix of the usual (reachability, joinability,...) conditions, but also atoms defined by a set of definite Horn clauses. GTRS can be used to prove confluence and termination of Generalized Rewrite Theories and Maude programs. We have characterized confluence of terminating GTRS as the joinability of a finite set of conditional pairs. Since termination of GTRS is underexplored to date, this paper introduces a Dependency Pair Framework which is well-suited to automatically (dis)prove termination of GTRS.

Cite as

Salvador Lucas. Termination of Generalized Term Rewriting Systems. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{lucas:LIPIcs.FSCD.2024.32,
  author =	{Lucas, Salvador},
  title =	{{Termination of Generalized Term Rewriting Systems}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{32:1--32:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.32},
  URN =		{urn:nbn:de:0030-drops-203616},
  doi =		{10.4230/LIPIcs.FSCD.2024.32},
  annote =	{Keywords: Program Analysis, Reduction-Based Systems, Termination}
}

Document

DOI: 10.4230/LIPIcs.RTA.2012.193

On Soundness Conditions for Unraveling Deterministic Conditional Rewrite Systems

Authors: Karl Gmeiner, Bernhard Gramlich, and Felix Schernhammer

Published in: LIPIcs, Volume 15, 23rd International Conference on Rewriting Techniques and Applications (RTA'12) (2012)

Abstract

We study (un)soundness of transformations of conditional term rewriting systems (CTRSs) into unconditional term rewriting systems (TRSs). The focus here is on analyzing (un)soundness of so-called unravelings, the most basic and natural class of such transformations. We extend our previous analysis from normal 1-CTRSs to the more general class of deterministic CTRSs (DCTRSs) where extra variables in right-hand sides of rules are allowed to a certain extent. We prove that the previous soundness results based on weak left-linearity and on right-linearity can be extended from normal 1-CTRSs to DCTRSs. Counterexamples show that such an extension to DCTRSs does not work for the previous criteria which were based on confluence and on non-erasingness, not even for right-stable systems. Yet, we prove weaker versions of soundness criteria based on confluence and on non-erasingness. Finally, we compare our approach and results with other recently established soundness criteria for unraveling DCTRSs.

Cite as

Karl Gmeiner, Bernhard Gramlich, and Felix Schernhammer. On Soundness Conditions for Unraveling Deterministic Conditional Rewrite Systems. In 23rd International Conference on Rewriting Techniques and Applications (RTA'12). Leibniz International Proceedings in Informatics (LIPIcs), Volume 15, pp. 193-208, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{gmeiner_et_al:LIPIcs.RTA.2012.193,
  author =	{Gmeiner, Karl and Gramlich, Bernhard and Schernhammer, Felix},
  title =	{{On Soundness Conditions for Unraveling Deterministic Conditional Rewrite Systems}},
  booktitle =	{23rd International Conference on Rewriting Techniques and Applications (RTA'12)},
  pages =	{193--208},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-38-5},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{15},
  editor =	{Tiwari, Ashish},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2012.193},
  URN =		{urn:nbn:de:0030-drops-34936},
  doi =		{10.4230/LIPIcs.RTA.2012.193},
  annote =	{Keywords: Conditional term rewriting system (CTRS), deterministic CTRS, transformation, simulation, soundness}
}

Document

DOI: 10.4230/LIPIcs.RTA.2010.119

On (Un)Soundness of Unravelings

Authors: Karl Gmeiner, Bernhard Gramlich, and Felix Schernhammer

Published in: LIPIcs, Volume 6, Proceedings of the 21st International Conference on Rewriting Techniques and Applications (2010)

Abstract

We revisit (un)soundness of transformations of conditional into unconditional rewrite systems. The focus here is on so-called unravelings, the most simple and natural kind of such transformations, for the class of normal conditional systems without extra variables. By a systematic and thorough study of existing counterexamples and of the potential sources of unsoundness we obtain several new positive and negative results. In particular, we prove the following new results: Confluence, non-erasingness and weak left-linearity (of a given conditional system) each guarantee soundness of the unraveled version w.r.t. the original one. The latter result substantially extends the only known sufficient criterion for soundness, namely left-linearity. Furthermore, by means of counterexamples we refute various other tempting conjectures about sufficient conditions for soundness.

Cite as

Karl Gmeiner, Bernhard Gramlich, and Felix Schernhammer. On (Un)Soundness of Unravelings. In Proceedings of the 21st International Conference on Rewriting Techniques and Applications. Leibniz International Proceedings in Informatics (LIPIcs), Volume 6, pp. 119-134, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{gmeiner_et_al:LIPIcs.RTA.2010.119,
  author =	{Gmeiner, Karl and Gramlich, Bernhard and Schernhammer, Felix},
  title =	{{On (Un)Soundness of Unravelings}},
  booktitle =	{Proceedings of the 21st International Conference on Rewriting Techniques and Applications},
  pages =	{119--134},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-18-7},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{6},
  editor =	{Lynch, Christopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2010.119},
  URN =		{urn:nbn:de:0030-drops-26485},
  doi =		{10.4230/LIPIcs.RTA.2010.119},
  annote =	{Keywords: Conditional rewriting, transformation into unconditional systems, unsoundness, unraveling}
}

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2 Gmeiner, Karl
2 Gramlich, Bernhard
2 Schernhammer, Felix
1 Lucas, Salvador
1 Thiemann, René
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1 Theory of computation → Automated reasoning
1 Theory of computation → Equational logic and rewriting
1 Theory of computation → Higher order logic
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1 Theory of computation → Pattern matching
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1 Conditional rewriting
1 Conditional term rewriting system (CTRS)
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