42,243 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
A composition theorem for parity kill number
In this work, we study the parity complexity measures
and .
is the \emph{parity kill number} of , the
fewest number of parities on the input variables one has to fix in order to
"kill" , i.e. to make it constant. is the depth
of the shortest \emph{parity decision tree} which computes . These
complexity measures have in recent years become increasingly important in the
fields of communication complexity \cite{ZS09, MO09, ZS10, TWXZ13} and
pseudorandomness \cite{BK12, Sha11, CT13}.
Our main result is a composition theorem for .
The -th power of , denoted , is the function which results
from composing with itself times. We prove that if is not a parity
function, then In other words, the parity kill number of
is essentially supermultiplicative in the \emph{normal} kill number of
(also known as the minimum certificate complexity).
As an application of our composition theorem, we show lower bounds on the
parity complexity measures of and . Here is the sort function due to Ambainis \cite{Amb06},
and is Kushilevitz's hemi-icosahedron function \cite{NW95}. In
doing so, we disprove a conjecture of Montanaro and Osborne \cite{MO09} which
had applications to communication complexity and computational learning theory.
In addition, we give new lower bounds for conjectures of \cite{MO09,ZS10} and
\cite{TWXZ13}
The Vampire and the FOOL
This paper presents new features recently implemented in the theorem prover
Vampire, namely support for first-order logic with a first class boolean sort
(FOOL) and polymorphic arrays. In addition to having a first class boolean
sort, FOOL also contains if-then-else and let-in expressions. We argue that
presented extensions facilitate reasoning-based program analysis, both by
increasing the expressivity of first-order reasoners and by gains in
efficiency
Cumulants and convolutions via Abel polynomials
We provide an unifying polynomial expression giving moments in terms of
cumulants, and viceversa, holding in the classical, boolean and free setting.
This is done by using a symbolic treatment of Abel polynomials. As a
by-product, we show that in the free cumulant theory the volume polynomial of
Pitman and Stanley plays the role of the complete Bell exponential polynomial
in the classical theory. Moreover via generalized Abel polynomials we construct
a new class of cumulants, including the classical, boolean and free ones, and
the convolutions linearized by them. Finally, via an umbral Fourier transform,
we state a explicit connection between boolean and free convolution
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