Equational Theories and Validity for Logically Constrained Term Rewriting

Authors Takahito Aoto , Naoki Nishida , Jonas Schöpf



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Author Details

Takahito Aoto
  • Niigata University, Japan
Naoki Nishida
  • Nagoya University, Japan
Jonas Schöpf
  • University of Innsbruck, Austria

Acknowledgements

We thank the anonymous reviewers for their valuable feedback, which improved the paper.

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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)
https://doi.org/10.4230/LIPIcs.FSCD.2024.31

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Equational logic and rewriting
Keywords
  • constrained equation
  • constrained equational theory
  • logically constrained term rewriting
  • algebraic semantics
  • consistency

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