Intractable unifiability problems and backtracking

Restrictions of the problem of finding all maximal unifiable or minimal nonunifiable subsets to those of certain sizes are shown to be NP-hard, and consequently inappropriate in general for reducing thrashing by intelligent backtracking in resolution theorem provers and logic program executions. We also show that existing backtrack methods based on the computation of all maximal unifiable subsets of assignments as a means to avoid thrashing are intractable because the solution length of these subsets can increase exponentially with the input length, and we give a corresponding result for minimal nonunifiability. The results apply not only to standard unification, but to a characterized collection of unification algorithms which includes unification without the occurs check. This now justifies the necessity for approximate or heuristic approaches to reduce thrashing in resolution theorem provers and the execution of logic programs.A version of this paper appears in Proceedings of the Third International Logic Programming Conference, London, Lecture Notes in Computer Science, Springer 225, 107–121 (1986).