Combining Uni cation- and Disuni cation Algorithms|Tractable and Intractable Instances

We consider the problem of combining procedures that decide solvability of (dis)uniication problems over disjoint equational theories. Partial answers to the following questions are given: Which properties of the component theories imply intractability in the sense that there cannot be a polynomial combination algorithm, assuming P 6 = NP? Which general properties of the component theories guarantee tractability of the combination problem in the sense that there exists a deterministic and polynomial combination algorithm? A criterion is given that characterizes a large class K of equational theories E where general E-uniication is always NP-hard. We show that all regular equational theories E that contain a commutative or an associative function symbol belong to K. Other examples of equational theories in K concern non-regular cases as well. The combination algorithm described in BS92] can be used to reduce solvability of general E-uniication algorithms to solvability of E-and free (Robinson) uniication problems with linear constant restrictions. We show that for E 2 K there exists no polynomial optimization of this combination algorithm for deciding solvability of general E-uniication problems, unless P = NP. This supports the conjecture that for E 2 K there is no polynomial algorithm for combining E-uniication with constants with free uniication. In the second part of the paper we characterize a class of equational theories where disuniication algorithms can be combined deterministically and in polynomial time. All unitary, regular and collapse-free equational theories with polynomial uniication algorithms belong to this class.