A Criterion for Intractability of E-uniication with Free Function Symbols and Its Relevance for Combination of Uniication Algorithms

All applications of equational uniication in the area of term rewriting and theorem proving require algorithms for general E-uniica-tion, i.e., E-uniication with free function symbols. On this background, the complexity of general E-uniication algorithms has been investigated for a large number of equational theories. For most of the relevant cases, the problem of deciding solvability of general E-uniication problems was found to be NP-hard. We ooer a partial explanation. 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.