Abstract:
Andreev et al.~\cite{ABCR97} 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(\log n)}, and a function in LINSPACE with the lower bound 2^{n-\log n -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 AC^0[2] (computable by a depth 3, polynomial-size circuit over the basis {\wedge,\oplus,1}) with the optimal lower bound 2^{n-\log n -O(1)} for 1-b.p.'s.

Abstract:
Andreev et al.~\cite{ABCR97} give constructions of Boolean functions (computable by polynomial-size circuits) that require large read-once branching program (1-b.p.'s): a function in P that requires 1-b.p. of size at least $2^{n-\polylog(n)}$, a function in quasipolynomial time that requires 1-b.p. of size at least $2^{n-O(\log n)}$, and a function in LINSPACE that requires 1-b.p. of size $2^{n-\log n -O(1)}$. 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.