The key insights of the proposed approach are:
Only certain ground equations entailed by the group axioms are added for the completion procedure, while taking only ground flat equations into account during its entire completion procedure. This keeps the completion procedure from interacting with the (nonground) convergent rewrite system for groups directly.
The arguments of each term headed by an associativity symbol are represented by a string, and a completion procedure for groups using their monoid presentations is extended to the proposed approach.
The paper consists of three main phases:
Phase I: Flatten the input set of ground equations into certain forms by introducing new constants.
Phase II: Add certain ground flat equations entailed by the group axioms using the flattened equations from Phase I.
Phase III: Apply a ground completion procedure on the set of ground flat equations obtained from Phase II.
If the proposed completion procedure terminates, it yields a decision procedure for the word problem for a finite set of ground equations with respect to the group axioms.
The paper also discusses a sufficient terminating condition of the proposed completion procedure by attempting to associate the set of ground flat equations with a monoid presentation of a group.
Additionally, the paper presents a new approach to constructing a rewriting-based congruence closure of a finite set of ground equations with respect to the semigroup, monoid, and the multiple disjoint sets of group axioms, respectively, using the same proposed ground completion procedure.
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