On the Combination of Congruence Closure and Completion

ABSTRACT We present a graph-based method for constructing a congruence closure of a given set of ground equalities that combines the key ideas of two well-known approaches, completion and abstract congruence closure, in a natural way by relying on a specialized and optimized version of the more general, but less efficient, SOUR graphs. This approach allows for efficient implementations and a visual presentation that better illuminates the basic ideas underlying the construction of congruence closures and clarifies the role of original and extended signatures and the impact of rewrite techniques for ordering equalities.