N-axioms Parallel Unification

Abstract

MGU_{mon} and MGU_{k-rep} have a complementary role in unification in the complexity class NC. MGU_mon is the upper bound of the unification classes that fall in NC and whose inputs admit an unrestricted number of repeated variables. MGU_{k-rep} is the upper bound of the unification classes that still fall in NC but whose inputs admit an unrestricted arity on term constructors. No LogSpace reduction of the one to the other class is known. Moreover, very fast algorithms that solve the two separately are well known but no one is able to compute with both in polylog PRAM-Time. N-axioms unification extends the structure of unification inputs and brings out the notion of interleaving variable as a special repeated variable which serializes independet computations. Based on it, we define the unification class AMGU^k_{p/h} whose inputs have a fixed number of interleaving variables but admit unrestricted number of repeated variables and, at the same time, unrestricted arity for term constructors. Constructively, we prove that AMGU^k_{p/h} is in NC by introducing a new unification algorithm that works on graph contractions and solves AMGU^k_{p/h} in a polylog PRAM time of the input size. Finally, we prove that MGU_{mon}, MGU_{k-rep} , and MGU_{linear} all are LogSpace reducible to AMGU^k_{p/h}. Hence, AMGU^k_{p/h} becomes the upper bound of the unification classes that are proved to be in NC.