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Tighter Relations Between Sensitivity and Other Complexity Measures
Sensitivity conjecture is a longstanding and fundamental open problem in the
area of complexity measures of Boolean functions and decision tree complexity.
The conjecture postulates that the maximum sensitivity of a Boolean function is
polynomially related to other major complexity measures. Despite much attention
to the problem and major advances in analysis of Boolean functions in the past
decade, the problem remains wide open with no positive result toward the
conjecture since the work of Kenyon and Kutin from 2004.
In this work, we present new upper bounds for various complexity measures in
terms of sensitivity improving the bounds provided by Kenyon and Kutin.
Specifically, we show that deg(f)^{1-o(1)}=O(2^{s(f)}) and C(f) < 2^{s(f)-1}
s(f); these in turn imply various corollaries regarding the relation between
sensitivity and other complexity measures, such as block sensitivity, via known
results. The gap between sensitivity and other complexity measures remains
exponential but these results are the first improvement for this difficult
problem that has been achieved in a decade.Comment: This is the merged form of arXiv submission 1306.4466 with another
work. Appeared in ICALP 2014, 14 page
Two Notions of Naturalness
My aim in this paper is twofold: (i) to distinguish two notions of
naturalness employed in BSM physics and (ii) to argue that recognizing this
distinction has methodological consequences. One notion of naturalness is an
"autonomy of scales" requirement: it prohibits sensitive dependence of an
effective field theory's low-energy observables on precise specification of the
theory's description of cutoff-scale physics. I will argue that considerations
from the general structure of effective field theory provide justification for
the role this notion of naturalness has played in BSM model construction. A
second, distinct notion construes naturalness as a statistical principle
requiring that the values of the parameters in an effective field theory be
"likely" given some appropriately chosen measure on some appropriately
circumscribed space of models. I argue that these two notions are historically
and conceptually related but are motivated by distinct theoretical
considerations and admit of distinct kinds of solution.Comment: 34 pages, 1 figur
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