678 research outputs found
Polynomials that Sign Represent Parity and Descartes' Rule of Signs
A real polynomial sign represents if
for every , the sign of equals
. Such sign representations are well-studied in computer
science and have applications to computational complexity and computational
learning theory. In this work, we present a systematic study of tradeoffs
between degree and sparsity of sign representations through the lens of the
parity function. We attempt to prove bounds that hold for any choice of set
. We show that sign representing parity over with the
degree in each variable at most requires sparsity at least . We show
that a tradeoff exists between sparsity and degree, by exhibiting a sign
representation that has higher degree but lower sparsity. We show a lower bound
of on the sparsity of polynomials of any degree representing
parity over . We prove exact bounds on the sparsity of such
polynomials for any two element subset . The main tool used is Descartes'
Rule of Signs, a classical result in algebra, relating the sparsity of a
polynomial to its number of real roots. As an application, we use bounds on
sparsity to derive circuit lower bounds for depth-two AND-OR-NOT circuits with
a Threshold Gate at the top. We use this to give a simple proof that such
circuits need size to compute parity, which improves the previous bound
of due to Goldmann (1997). We show a tight lower bound of
for the inner product function over .Comment: To appear in Computational Complexit
Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups
We study in this paper some connections between the Fraisse theory of
amalgamation classes and ultrahomogeneous structures, Ramsey theory, and
topological dynamics of automorphism groups of countable structures.Comment: 73 pages, LaTeX 2e, to appear in Geom. Funct. Ana
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
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