Separating Oblivious and Non-oblivious BPs

Abstract

This paper gives an exponential separation on the depth of branching programs (BPs) between oblivious and non-oblivious BPs. Namely, there is a difference just like the difference between sequential and NC computation: (i) There is a Boolean function f1 of N variables which can be computed by a polynomial-size, syntactic BP with a depth of 2 logN - log logN + 1 but cannot be computed by any oblivious BPs with a depth of (2-ε)N for some ε ∈ o(1). (ii) Similarly, there is an f 2 computed by a syntactic depth of log3 N but not by an oblivious depth of Ώ(N logN). (iii) We also show that any (unrestricted) BP of depth t can be simulated by an oblivious BP with a depth of N + ⌈(t - logN)/(log logN + C)⌉·N. The third result implies that f 1 cannot be computed by any BP with a depth less than logN +log logN and f 2 not with a depth of o(logN·log logN). Note that most bounds in this paper include factors and lower-degree terms.