This paper is devoted to the languages accepted by constant-depth, polynomial-size families of circuits in which every circuit element computes the sum of its input bits modulo a fixed period q. It has been conjectured that such a circuit family cannot compute the AND function of n inputs. Here it is shown that such circuit families are equivalent in power to polynomial-length programs over finite solvable groups; in particular, the conjecture implies that Barrington's result on the computational power of branching programs over nonsolvable groups cannot be extended to solvable groups. It is also shown that polynomial-length programs over dihedral groups cannot compute the AND function. Furthermore, it is shown that the conjecture is equivalent to a characterization, in terms of finite semigroups and formal logic, of the regular languages accepted by such circuit families. There is, moreover, considerable independent evidence for this characterization. This last result is established using new theorems, of independent interest, concerning the algebraic structure of finite categories.
The power of nondeterminism in polynomial- size bounded-width branching programs
