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Revision #1 to TR99-004 | 6th November 1999 00:00

#### Almost k-Wise Independence and Hard Boolean Functions Revision of: TR99-004

Revision #1
Authors: Valentine Kabanets
Accepted on: 6th November 1999 00:00
Keywords:

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.

### Paper:

TR99-004 | 3rd February 1999 00:00

#### Almost \$k\$-Wise Independence and Boolean Functions Hard for Read-Once Branching Programs

TR99-004
Authors: Valentine Kabanets
Publication: 23rd February 1999 11:08
Keywords:

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.

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