Confluence of Conditional Rewriting Modulo

2025-04-02

Annual Conference for Computer Science Logic

AI Summary

This paper addresses the decidability problem of confluence modulo an equational theory (E) (i.e., (E)-confluence) for conditional rewrite systems. To overcome the fundamental limitation of traditional approaches—which rely on enumerating infinitely many (E)-unifiers and thus fail to finitely characterize local peaks—we introduce *logical conditional critical pairs* and *parameterized conditional variable pairs*, thereby reducing (E)-confluence to a finite, decidable critical-pair analysis. Building upon the Jouannaud–Kirchner abstract rewriting framework, we integrate logical constraint solving, conditional term normalization, and equational reasoning to achieve, for the first time, a finite characterization of local peaks and a fully decidable verification procedure for (E)-confluence in conditional rewriting modulo theories. The resulting theory enables the construction of a finite critical pair set that uniformly covers major classes of conditional rewrite systems, providing a rigorous foundation for formal verification and automated theorem proving.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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*.

Problem

• Investigates confluence of conditional rewriting modulo equations
• Defines conditional pairs to prove/disprove E-confluence
• Introduces Logic-based Conditional Critical Pairs for finite analysis

Innovation

• Conditional rewriting modulo equivalence classes
• Logic-based Conditional Critical Pairs
• Parametric Conditional Variable Pairs