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.

@article{abs-2310-05014,
  title = {Congruence Closure Modulo Groups},
  author = {Dohan Kim},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 21, Issue 1},
  year = {2025},
  doi = {10.46298/lmcs-21(1:20)2025}
}