34,595 research outputs found
Linear recurrences and asymptotic behavior of exponential sums of symmetric boolean functions
In this paper we give an improvement of the degree of the homogeneous linear
recurrence with integer coefficients that exponential sums of symmetric Boolean
functions satisfy. This improvement is tight. We also compute the asymptotic
behavior of symmetric Boolean functions and provide a formula that allows us to
determine if a symmetric boolean function is asymptotically not balanced. In
particular, when the degree of the symmetric function is a power of two, then
the exponential sum is much smaller than .Comment: 18 pages, 3 figure
Spectral Norm of Symmetric Functions
The spectral norm of a Boolean function is the sum
of the absolute values of its Fourier coefficients. This quantity provides
useful upper and lower bounds on the complexity of a function in areas such as
learning theory, circuit complexity, and communication complexity. In this
paper, we give a combinatorial characterization for the spectral norm of
symmetric functions. We show that the logarithm of the spectral norm is of the
same order of magnitude as where ,
and and are the smallest integers less than such that
or is constant for all with . We mention some applications to the decision tree and communication
complexity of symmetric functions
Quadratization of Symmetric Pseudo-Boolean Functions
A pseudo-Boolean function is a real-valued function
of binary variables; that is, a mapping from
to . For a pseudo-Boolean function on
, we say that is a quadratization of if is a
quadratic polynomial depending on and on auxiliary binary variables
such that for
all . By means of quadratizations, minimization of is
reduced to minimization (over its extended set of variables) of the quadratic
function . This is of some practical interest because minimization of
quadratic functions has been thoroughly studied for the last few decades, and
much progress has been made in solving such problems exactly or heuristically.
A related paper \cite{ABCG} initiated a systematic study of the minimum number
of auxiliary -variables required in a quadratization of an arbitrary
function (a natural question, since the complexity of minimizing the
quadratic function depends, among other factors, on the number of
binary variables). In this paper, we determine more precisely the number of
auxiliary variables required by quadratizations of symmetric pseudo-Boolean
functions , those functions whose value depends only on the Hamming
weight of the input (the number of variables equal to ).Comment: 17 page
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