Proving Confluence in the Confluence Framework with CONFident

TL;DR

The paper introduces CONFident, a framework that automates confluence proofs for generalized term rewriting systems using a modular divide-and-conquer approach.

Contribution

It presents a novel framework and implementation for automatically proving and disproving confluence in complex rewriting systems with diverse rule types.

Findings

Successfully proves confluence for various rewriting system variants.

Automates complex confluence proofs using a modular strategy.

Handles conditional and context-sensitive rewriting systems.

Abstract

This article describes the *Confluence Framework*, a novel framework for proving and disproving confluence using a divide-and-conquer modular strategy, and its implementation in CONFident. Using this approach, we are able to automatically prove and disprove confluence of *Generalized Term Rewriting Systems*, where (i) only selected arguments of function symbols can be rewritten and (ii) a rather general class of conditional rules can be used. This includes, as particular cases, several variants of rewrite systems such as (context-sensitive) *term rewriting systems*, *string rewriting systems*, and (context-sensitive) *conditional term rewriting systems*. The divide-and-conquer modular strategy allows us to combine in a proof tree different techniques for proving confluence, including modular decompositions, checking joinability of (conditional) critical and variable pairs, transformations, etc., and auxiliary tasks required by them, e.g., joinability of terms, joinability of conditional pairs, etc.