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DOI: 10.4230/LIPIcs.FSCD.2024.31

Equational Theories and Validity for Logically Constrained Term Rewriting

Authors: Takahito Aoto, Naoki Nishida, and Jonas Schöpf

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

Abstract

Logically constrained term rewriting is a relatively new formalism where rules are equipped with constraints over some arbitrary theory. Although there are many recent advances with respect to rewriting induction, completion, complexity analysis and confluence analysis for logically constrained term rewriting, these works solely focus on the syntactic side of the formalism lacking detailed investigations on semantics. In this paper, we investigate a semantic side of logically constrained term rewriting. To this end, we first define constrained equations, constrained equational theories and validity of the former based on the latter. After presenting the relationship of validity and conversion of rewriting, we then construct a sound inference system to prove validity of constrained equations in constrained equational theories. Finally, we give an algebraic semantics, which enables one to establish invalidity of constrained equations in constrained equational theories. This algebraic semantics derives a new notion of consistency for constrained equational theories.

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Takahito Aoto, Naoki Nishida, and Jonas Schöpf. Equational Theories and Validity for Logically Constrained Term Rewriting. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 31:1-31:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{aoto_et_al:LIPIcs.FSCD.2024.31,
  author =	{Aoto, Takahito and Nishida, Naoki and Sch\"{o}pf, Jonas},
  title =	{{Equational Theories and Validity for Logically Constrained Term Rewriting}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{31:1--31:21},
  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.2024.31},
  URN =		{urn:nbn:de:0030-drops-203607},
  doi =		{10.4230/LIPIcs.FSCD.2024.31},
  annote =	{Keywords: constrained equation, constrained equational theory, logically constrained term rewriting, algebraic semantics, consistency}
}

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Invited Talk

DOI: 10.4230/LIPIcs.FSCD.2020.4

Certifying the Weighted Path Order (Invited Talk)

Authors: René Thiemann, Jonas Schöpf, Christian Sternagel, and Akihisa Yamada

Published in: LIPIcs, Volume 167, 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)

Abstract

The weighted path order (WPO) unifies and extends several termination proving techniques that are known in term rewriting. Consequently, the first tool implementing WPO could prove termination of rewrite systems for which all previous tools failed. However, we should not blindly trust such results, since there might be problems with the implementation or the paper proof of WPO. In this work, we increase the reliability of these automatically generated proofs. To this end, we first formally prove the properties of WPO in Isabelle/HOL, and then develop a verified algorithm to certify termination proofs that are generated by tools using WPO. We also include support for max-polynomial interpretations, an important ingredient in WPO. Here we establish a connection to an existing verified SMT solver. Moreover, we extend the termination tools NaTT and TTT2, so that they can now generate certifiable WPO proofs.

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René Thiemann, Jonas Schöpf, Christian Sternagel, and Akihisa Yamada. Certifying the Weighted Path Order (Invited Talk). In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{thiemann_et_al:LIPIcs.FSCD.2020.4,
  author =	{Thiemann, Ren\'{e} and Sch\"{o}pf, Jonas and Sternagel, Christian and Yamada, Akihisa},
  title =	{{Certifying the Weighted Path Order}},
  booktitle =	{5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)},
  pages =	{4:1--4:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-155-9},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{167},
  editor =	{Ariola, Zena M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2020.4},
  URN =		{urn:nbn:de:0030-drops-123263},
  doi =		{10.4230/LIPIcs.FSCD.2020.4},
  annote =	{Keywords: certification, Isabelle/HOL, reduction order, termination analysis}
}

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Equational Theories and Validity for Logically Constrained Term Rewriting

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Certifying the Weighted Path Order (Invited Talk)

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