Circuits, matrices, and nonassociative computation

It is shown that the formula and circuit evaluation problems in the nonassociative context capture natural complexity classes up to NP, thus extending the known result that the word problem over a groupoid is LOGCFL-complete. The problem of multiplying together matrices whose elements are taken from an algebraic structure more general than a semiring is defined and studied. It is shown that natural variants of this problem are complete for complexity classes such as NL, NC^k , AC^k, SC^k, and NP. In particular, the iterated multiplication problems involving O(log^k n) matrices over a structure (S; +,.) in which (S ; +) is a monoid or an aperiodic monoid are complete for NC^k+1 and for AC^k respectively, and an iterated multiplication problem variant involving matrices of size O(log ^k n) is complete for SC^k