Unbounded Fan-In Circuits and Associative Functions

Abstract The computation of finite semigroups using unbounded fan-in circuits are considered. There are constant-depth, polynomial size circuits for semigroup product iff the semigroup does not contain a nontrivial group as a subset. In the case that the semigroup in fact does not contain a group, then for any primitive recursive function f circuits of size O ( nf −1 ( n )) and constant depth exist for the semigroup product of n elements. The depth depends upon the choice of the primitive recursive function f . The circuits not only compute the semigroup product, but every prefix of the semigroup product. A consequence is that the same bounds apply for circuits computing the sum of two n-bit numbers.