Circuits and Expressions with Non-associative Gates (extended Abstract) ?

We consider circuits and expressions whose gates carry out multiplication in a non-associative groupoid such as loop. We deene a class we call the polyabelian groupoids, formed by iterated quasidirect products of Abelian groups. We show that a loop can express arbitrary Boolean functions if and only if it is not polyabelian, in which case its Expression Evaluation and Circuit Value problems are NC 1-complete and P-complete respectively. This is not true for groupoids in general, and we give a counterexample. We show that Expression Evaluation is also NC 1-complete if the groupoid has a non-solvable multiplication semigroup, but is in TC 0 if the groupoid is both polyabelian and has a solvable multiplication semigroup. Thus, in the non-associative case, earlier results about the role of solvability in circuit complexity generalize in several diierent ways.