Bounded-width polynomial-size branching programs recognize exactly those languages in NC1

摘要

We show that any language recognized by an NC1 circuit (fan-in 2, depth O(log n)) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC1 circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC1. Further, following Ruzzo (J. Comput. System Sci. 22 (1981), 365–383) and Cook (Inform. and Control 64 (1985) 2–22), if the branching programs are restricted to be ATIME(logn)-uniform, they recognize the same languages as do ATIME(log n)-uniform NC1 circuits, that is, those languages in ATIME(log n). We also extend the method of proof to investigate the complexity of the word problem for a fixed permutation group and show that polynomial size circuits of width 4 also recognize exactly nonuniform NC1.