Superlinear Lower Bounds for Bounded-Width Branching Programs

We use algebraic techniques to obtain superlinear lower bounds on the size of bounded-width branching programs to solve a number of problems. In particular, we show that any bounded-width branching program computing a nonconstant threshold function has length (n log log n); improving on the previous lower bounds known to apply to all such threshold functions. We also show that any program over a nite solvable monoid computing products in a nonsolvable group has length (n log log n): This result is a step toward proving the conjecture that the circuit complexity class ACC 0 is properly contained in NC 1 : A preliminary version of this paper appeared in the Proceedings of the 1991 Structure in Complexity Theory Symposium. 1. The Main Results In this paper we describe a general algebraic technique for obtaining superlinear lower bounds on the length of bounded-width branching programs to solve certain problems. Our method is based on the interpretation, due to Barrington and Th erien 5] of these programs as generalizations of ordinary nite automata. We are thus able to apply the well-established connection between nite automata and nite monoids to branching programs. More precisely, we start from the simple algebraic fact that a homomorphism from a free monoid into a nite monoid N cannot simulate