Circuits, Matrices, and Nonassociative Computation

Abstract

We consider the complexity of various computational problems over nonassociative algebraic structures. Specifically, we look at the problem of evaluating circuits, formulas, and words, over both nonassociative structures themselves and over matrices with elements in these structures. Extending past work, we show that such problems can characterize a wide variety of complexity classes up to and including NP. As an example, the word (i.e., iterated multiplication) problems involving a sequence of O(logkn) matrices over a structure ( S; +, ·) in which (S; +) is a monoid or an aperiodic monoid are complete for NCk+1 and for ACk, respectively, and a word problem variant involving matrices of size O(logkn) is complete for SCk.