5 research outputs found
A Nearly Optimal Lower Bound on the Approximate Degree of AC
The approximate degree of a Boolean function is the least degree of a real polynomial that
approximates pointwise to error at most . We introduce a generic
method for increasing the approximate degree of a given function, while
preserving its computability by constant-depth circuits.
Specifically, we show how to transform any Boolean function with
approximate degree into a function on variables with approximate degree at least . In particular, if , then
is polynomially larger than . Moreover, if is computed by a
polynomial-size Boolean circuit of constant depth, then so is .
By recursively applying our transformation, for any constant we
exhibit an AC function of approximate degree . This
improves over the best previous lower bound of due to
Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of
that holds for any function. Our lower bounds also apply to
(quasipolynomial-size) DNFs of polylogarithmic width.
We describe several applications of these results. We give:
* For any constant , an lower bound on the
quantum communication complexity of a function in AC.
* A Boolean function with approximate degree at least ,
where is the certificate complexity of . This separation is optimal
up to the term in the exponent.
* Improved secret sharing schemes with reconstruction procedures in AC.Comment: 40 pages, 1 figur
Near-Optimal Lower Bounds on the Threshold Degree and Sign-Rank of AC^0
The threshold degree of a Boolean function is
the minimum degree of a real polynomial that represents in sign:
A related notion is sign-rank, defined for a
Boolean matrix as the minimum rank of a real matrix with
. Determining the maximum threshold degree
and sign-rank achievable by constant-depth circuits () is a
well-known and extensively studied open problem, with complexity-theoretic and
algorithmic applications.
We give an essentially optimal solution to this problem. For any
we construct an circuit in variables that has
threshold degree and sign-rank
improving on the previous best lower bounds of
and , respectively. Our
results subsume all previous lower bounds on the threshold degree and sign-rank
of circuits of any given depth, with a strict improvement
starting at depth . As a corollary, we also obtain near-optimal bounds on
the discrepancy, threshold weight, and threshold density of ,
strictly subsuming previous work on these quantities. Our work gives some of
the strongest lower bounds to date on the communication complexity of
.Comment: 99 page