Circuits and Expressions with Non-Associative Gates

We consider circuits and expressions whose gates carry out multiplication in a non-associative algebra such as a quasigroup or loop. We define a class we call the polyabelian algebras, formed by iterated quasidirect products of Abelian groups. We show that a quasigroup can express arbitrary Boolean functions if and only if it is not polyabelian, in which case its EXPRESSION EVALUATION and CIRCUIT VALUE problems are NC1-complete and P-complete respectively. This is not true for algebras in general, and we give a counter-example. <p> We show that EXPRESSION EVALUATION is also NC1-complete if the TC0 if the algebra is both polyabelian and has a solvable multiplication semigroup, e.g., for a nilpotent loop or group. <p> Thus, in the non-associative case, earlier results about the role of solvability in circuit complexity generalize in several different ways.