Join Algorithms for the Theory of Uninterpreted Functions

The join of two sets of facts, E1 and E2, is defined as the set of all facts that are implied independently by both E1 and E2. Congruence closure is a widely used representation for sets of equational facts in the theory of uninterpreted function symbols (UFS). We present an optimal join algorithm for special classes of the theory of UFS using the abstract congruence closure framework. Several known join algorithms, which work on a strict subclass, can be cast as specific instantiations of our generic procedure. We demonstrate the limitations of any approach for computing joins that is based on the use of congruence closure. We also mention some interesting open problems in this area.