Confluence of Conditional Rewriting Modulo
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908801255342080 |
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| author | Lucas, Salvador |
| author_facet | Lucas, Salvador |
| contents | Sets of equations E play an important computational role in rewriting-based systems R by defining an equivalence relation =E inducing a partition of terms into E-equivalence classes on which rewriting computations, denoted ->R/E and called *rewriting modulo E*, are issued. This paper investigates *confluence of ->R/E*, usually called *E-confluence*, for *conditional* rewriting-based systems, where rewriting steps are determined by conditional rules. We rely on Jouannaud and Kirchner's framework to investigate confluence of an abstract relation R modulo an abstract equivalence relation E on a set A. We show how to particularize the framework to be used with conditional systems. Then, we show how to define appropriate finite sets of *conditional pairs* to prove and disprove E-confluence. In particular, we introduce *Logic-based Conditional Critical Pairs* which do not require the use of (often infinitely many) E-unifiers to provide a finite representation of the *local peaks* considered in the abstract framework. We also introduce *parametric Conditional Variable Pairs* which are essential to deal with conditional rules in the analysis of E-confluence. Our results apply to well-known classes of rewriting-based systems. In particular, to *Equational (Conditional) Term Rewriting Systems*. In this realm, our E-confluence results strictly subsume previous approaches by Huet and Jouannaud and Kirchner. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01847 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Confluence of Conditional Rewriting Modulo Lucas, Salvador Logic in Computer Science Programming Languages Symbolic Computation Sets of equations E play an important computational role in rewriting-based systems R by defining an equivalence relation =E inducing a partition of terms into E-equivalence classes on which rewriting computations, denoted ->R/E and called *rewriting modulo E*, are issued. This paper investigates *confluence of ->R/E*, usually called *E-confluence*, for *conditional* rewriting-based systems, where rewriting steps are determined by conditional rules. We rely on Jouannaud and Kirchner's framework to investigate confluence of an abstract relation R modulo an abstract equivalence relation E on a set A. We show how to particularize the framework to be used with conditional systems. Then, we show how to define appropriate finite sets of *conditional pairs* to prove and disprove E-confluence. In particular, we introduce *Logic-based Conditional Critical Pairs* which do not require the use of (often infinitely many) E-unifiers to provide a finite representation of the *local peaks* considered in the abstract framework. We also introduce *parametric Conditional Variable Pairs* which are essential to deal with conditional rules in the analysis of E-confluence. Our results apply to well-known classes of rewriting-based systems. In particular, to *Equational (Conditional) Term Rewriting Systems*. In this realm, our E-confluence results strictly subsume previous approaches by Huet and Jouannaud and Kirchner. |
| title | Confluence of Conditional Rewriting Modulo |
| topic | Logic in Computer Science Programming Languages Symbolic Computation |
| url | https://arxiv.org/abs/2504.01847 |