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Toward Better Depth Lower Bounds: A KRW-like theorem for Strong Composition
One of the major open problems in complexity theory is proving
super-logarithmic lower bounds on the depth of circuits (i.e.,
). Karchmer, Raz, and Wigderson
(Computational Complexity 5(3/4), 1995) suggested to approach this problem by
proving that depth complexity of a composition of functions is
roughly the sum of the depth complexities of and . They showed that the
validity of this conjecture would imply that
.
The intuition that underlies the KRW conjecture is that the composition
should behave like a "direct-sum problem", in a certain sense,
and therefore the depth complexity of should be the sum of the
individual depth complexities. Nevertheless, there are two obstacles toward
turning this intuition into a proof: first, we do not know how to prove that
must behave like a direct-sum problem; second, we do not know how
to prove that the complexity of the latter direct-sum problem is indeed the sum
of the individual complexities.
In this work, we focus on the second obstacle. To this end, we study a notion
called "strong composition", which is the same as except that it
is forced to behave like a direct-sum problem. We prove a variant of the KRW
conjecture for strong composition, thus overcoming the above second obstacle.
This result demonstrates that the first obstacle above is the crucial barrier
toward resolving the KRW conjecture. Along the way, we develop some general
techniques that might be of independent interest
Vanishing Abelian integrals on zero-dimensional cycles
In this paper we study conditions for the vanishing of Abelian integrals on
families of zero-dimensional cycles. That is, for any rational function ,
characterize all rational functions and zero-sum integers such
that the function vanishes identically. Here
are continuously depending roots of . We introduce a notion of
(un)balanced cycles. Our main result is an inductive solution of the problem of
vanishing of Abelian integrals when are polynomials on a family of
zero-dimensional cycles under the assumption that the family of cycles we
consider is unbalanced as well as all the cycles encountered in the inductive
process. We also solve the problem on some balanced cycles.
The main motivation for our study is the problem of vanishing of Abelian
integrals on single families of one-dimensional cycles. We show that our
problem and our main result are sufficiently rich to include some related
problems, as hyper-elliptic integrals on one-cycles, some applications to
slow-fast planar systems, and the polynomial (and trigonometric) moment problem
for Abel equation. This last problem was recently solved by Pakovich and
Muzychuk (\cite{PM} and \cite{P}). Our approach is largely inspired by their
work, thought we provide examples of vanishing Abelian integrals on zero-cycles
which are not given as a sum of composition terms contrary to the situation in
the solution of the polynomial moment problem.Comment: 35 pages, 1 figure; one reference added; abstract, introduction and
structure change
New solutions to the Hurwitz problem on square identities
The Hurwitz problem of composition of quadratic forms, or of "sum of squares
identity" is tackled with the help of a particular class of
-graded non-associative algebras generalizing the octonions.
This method provides an explicit formula for the classical Hurwitz-Radon
identity and leads to new solutions in a neighborhood of the Hurwitz-Radon
identity.Comment: 13 pages, 2 figures, final version to appear in J. Pure Appl. Al
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