On the relationship of congruence closureand unification

Abstract

Congruence closure is a fundamental operation for symbolic computation. Unification closureis defined as its directional dual, i.e., on the same inputs but top-down as opposed to bottom-up. Unifying terms is another fundamental operation for symbolic computation and is commonly computed using unification closure. We clarify the directional duality by reducing unification closure to a special form of congruence closure. This reduction reveals a correspondence between repeated variables in terms to be unified and equalities of monadic ground terms. We then show that: (1) single equality congruence closure on a directed acyclic graph, and (2) acyclic congruence closure of a fixed number of equalities, are in the parallel complexity class NC. The directional dual unification closures in these two cases are known to be log-space complete for PTIME. As a consequence of our reductions we show that if the number of repeated variables in the input terms is fixed, then term unification can be performed in NC; this extends the known parallelizable cases of term unification. Using parallel complexity we also clarify the relationship of unification closure and the testing of deterministic finite automata for equivalence.