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 2^{n − − polylog( n)}, a function in quasipolynomial time with the lower bound 2^{n − −  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 − − logn − −  O(1)} for 1-b.p.’s.