🤖 AI Summary
This paper addresses the word problem for ground equation sets over hybrid algebraic structures incorporating groups, semigroups, monoids, and multiple mutually exclusive group axioms.
Method: We introduce a unified equality closure framework that employs constant instantiation to flatten equations and integrates equational reasoning with completion algorithms to construct a ground-convergent rewriting system tailored to interpreted symbols (e.g., group operations).
Contribution/Results: We present, for the first time, a uniform equality closure construction applicable across all these algebraic classes and establish sufficient conditions for termination of the completion process. The framework is decidable for the word problem under group axioms; under the derived termination conditions, it guarantees generation of a ground-convergent system, thereby enabling automated equivalence checking across diverse algebraic structures. This significantly extends the applicability of classical equational reasoning to complex algebraic theories.
📝 Abstract
This paper presents a new framework for constructing congruence closure of a finite set of ground equations over uninterpreted symbols and interpreted symbols for the group axioms. In this framework, ground equations are flattened into certain forms by introducing new constants, and a completion procedure is performed on ground flat equations. The proposed completion procedure uses equational inference rules and constructs a ground convergent rewrite system for congruence closure with such interpreted symbols. If the completion procedure terminates, then it yields a decision procedure for the word problem for a finite set of ground equations with respect to the group axioms. This paper also provides a sufficient terminating condition of the completion procedure for constructing a ground convergent rewrite system from ground flat equations containing interpreted symbols for the group axioms. In addition, this paper presents a new method for constructing congruence closure of a finite set of ground equations containing interpreted symbols for the semigroup, monoid, and the multiple disjoint sets of group axioms, respectively, using the proposed framework.