# Congruence closure with ACI function symbols

## Abstract

Congruence closure is the following well known reasoning problem: given a premise set of equations between ground terms over uninterpreted function symbols, does a given query equation follow using the axioms of equality? Several methods have been provided for polynomial-time answers to this question. Here we consider this same setting, but where some of the function symbols are known to be associative, commutative, and idempotent (ACI). Given these additional axioms, does the query equation follow from the premise equations? We provide a sound and complete cubic-time procedure correctly answering such questions. The problem requires exponential space when adding only AC function symbols [14], but requiring idempotence restores tractability . Our procedure is defined by providing a sound and complete "local" rule set for the problem [9]. A "local formula" is a formula mentioning only terms appearing in the premises or query. A local rule set is one for which any derivable local formula has a derivation using only local intermediate formulas. Closures under local rule sets can immediately be constructed in polynomial time by refusing to infer non-local formulas. Finally, we present results on the integration of ACI function symbols and equality inference rules into more general local rule sets.

## Date of this Version

3-29-2013