On branching programs with bounded uncertainty

Abstract

We propose an information-theoretic approach to proving lower bounds on the size of branching programs (b.p.). The argument is based on Kraft-McMillan type inequalities for the average amount of uncertainty about (or entropy of) a given input during various stages of the computation. We first demonstrate the approach for read-once b.p. Then we introduce a strictly larger class of so-called 'gentle' b.p. and, using the suggested approach, prove that some explicit Boolean functions, including the Clique function and a particular Pointer function (which belongs to AC0), cannot be computed by gentle program of polynomial size. These lower bounds are new since explicit functions, which are known to be hard for all previously considered restricted classes of b.p. (like (1, +s)-b.p. or syntactic read-k-times b.p.) can be easily computed by gentle b.p. of polynomial size.