Unification Algorithms Cannot be Combined in Polynomial Time

We establish that there is no polynomial-time general combination algorithm for uniication in nitary equational theories, unless the complexity class #P of counting problems is contained in the class FP of function problems solvable in polynomial-time. The prevalent view in complexity theory is that such a collapse is extremely unlikely for a number of reasons, including the fact that the containment of #P in FP implies that P = NP. Our main result is obtained by establishing the intractrability of the counting problem for general AG-uniication, where AG is the equational theory of Abelian groups. Speciically, we show that computing the cardinality of a minimal complete set of uniiers for general AG-uniication is a #P-hard problem. In contrast, AG-uniication with constants is solvable in polynomial time. Since an algorithm for general AG-uniication can be obtained as a combination of a polynomial-time algorithm for AG-uniication with constants and a polynomial-time algorithm for syntactic uniication, it follows that no polynomial-time general combination algorithm exists, unless #P is contained in FP.