Fast Left-Linear Semi-Unification

Semi-unification is a generalization of both unification and matching with applications in proof theory, term rewriting systems, polymorphic type inference, and natural language processing. It is the problem of solving a set of term inequalities M1 ≤ N1, . . . ,Mk ≤ Nk, where ≤ is interpreted as the subsumption preordering on (first-order) terms. Whereas the general problem has recently been shown to be undecidable, several special cases are decidable. Kfoury, Tiuryn, and Urzyczyn proved that left-linear semi-unification (LLSU) is decidable by giving an exponential time decision procedure. We improve their result as follows. 1. We present a generic polynomial-time algorithm L1 for LLSU, which shows that LLSU is in P. 2. We show that L1 can be implemented in time O(n) by using a fast dynamic transitive closure algorithm. 3. We prove that LLSU is P-complete under log-space reductions, thus giving evidence that there are no fast (NC-class) parallel algorithms for LLSU. ∗In Proc. Int’l Conf. on Computing and Information (ICCI), Niagara Falls, Canada, May 1990, Springer-Verlag Lecture Notes in Computer Science, Vol. 468, pp. 82-91