Almost k-wise independence and hard boolean functions

Abstract

Andreev et al. [3] gave constructions of Boolean functions (computable by polynomial-size circuits) with large lower bounds for read-once branching program (1-b.p.'s): a function in P with the lower bound 2n- polylog(n), a function in quasipolynomial time with the lower bound 2n-O(logn), and a function in LINSPACE with the lower bound 2 n-logn-O(1). We point out alternative, much simpler constructions of such Boolean functions by applying the idea of almost k-wise independence more directly, without the use of discrepancy set generators for large affine subspaces; our constructions are obtained by derandomizing the probabilistic proofs of existence of the corresponding combinatorial objects. The simplicity of our new constructions also allows us to observe that there exists a Boolean function in AC0 [2] (computable by a depth 3, polynomial-size circuit over the basis {∧,⊕,1}) with the optimal lower bound 2 n-log n-O(1) for 1-b.p.'s. © Springer-Verlag Berlin Heidelberg 2000.