5,343 research outputs found
Lattices with non-Shannon Inequalities
We study the existence or absence of non-Shannon inequalities for variables
that are related by functional dependencies. Although the power-set on four
variables is the smallest Boolean lattice with non-Shannon inequalities there
exist lattices with many more variables without non-Shannon inequalities. We
search for conditions that ensures that no non-Shannon inequalities exist. It
is demonstrated that 3-dimensional distributive lattices cannot have
non-Shannon inequalities and planar modular lattices cannot have non-Shannon
inequalities. The existence of non-Shannon inequalities is related to the
question of whether a lattice is isomorphic to a lattice of subgroups of a
group.Comment: Ten pages. Submitted to ISIT 2015. The appendix will not appear in
the proceeding
On active and passive testing
Given a property of Boolean functions, what is the minimum number of queries
required to determine with high probability if an input function satisfies this
property or is "far" from satisfying it? This is a fundamental question in
Property Testing, where traditionally the testing algorithm is allowed to pick
its queries among the entire set of inputs. Balcan, Blais, Blum and Yang have
recently suggested to restrict the tester to take its queries from a smaller
random subset of polynomial size of the inputs. This model is called active
testing, and in the extreme case when the size of the set we can query from is
exactly the number of queries performed it is known as passive testing.
We prove that passive or active testing of k-linear functions (that is, sums
of k variables among n over Z_2) requires Theta(k*log n) queries, assuming k is
not too large. This extends the case k=1, (that is, dictator functions),
analyzed by Balcan et. al.
We also consider other classes of functions including low degree polynomials,
juntas, and partially symmetric functions. Our methods combine algebraic,
combinatorial, and probabilistic techniques, including the Talagrand
concentration inequality and the Erdos--Rado theorem on Delta-systems.Comment: 16 page
Quantitative information flow under generic leakage functions and adaptive adversaries
We put forward a model of action-based randomization mechanisms to analyse
quantitative information flow (QIF) under generic leakage functions, and under
possibly adaptive adversaries. This model subsumes many of the QIF models
proposed so far. Our main contributions include the following: (1) we identify
mild general conditions on the leakage function under which it is possible to
derive general and significant results on adaptive QIF; (2) we contrast the
efficiency of adaptive and non-adaptive strategies, showing that the latter are
as efficient as the former in terms of length up to an expansion factor bounded
by the number of available actions; (3) we show that the maximum information
leakage over strategies, given a finite time horizon, can be expressed in terms
of a Bellman equation. This can be used to compute an optimal finite strategy
recursively, by resorting to standard methods like backward induction.Comment: Revised and extended version of conference paper with the same title
appeared in Proc. of FORTE 2014, LNC
Partially Symmetric Functions are Efficiently Isomorphism-Testable
Given a function f: {0,1}^n \to {0,1}, the f-isomorphism testing problem
requires a randomized algorithm to distinguish functions that are identical to
f up to relabeling of the input variables from functions that are far from
being so. An important open question in property testing is to determine for
which functions f we can test f-isomorphism with a constant number of queries.
Despite much recent attention to this question, essentially only two classes of
functions were known to be efficiently isomorphism testable: symmetric
functions and juntas.
We unify and extend these results by showing that all partially symmetric
functions---functions invariant to the reordering of all but a constant number
of their variables---are efficiently isomorphism-testable. This class of
functions, first introduced by Shannon, includes symmetric functions, juntas,
and many other functions as well. We conjecture that these functions are
essentially the only functions efficiently isomorphism-testable.
To prove our main result, we also show that partial symmetry is efficiently
testable. In turn, to prove this result we had to revisit the junta testing
problem. We provide a new proof of correctness of the nearly-optimal junta
tester. Our new proof replaces the Fourier machinery of the original proof with
a purely combinatorial argument that exploits the connection between sets of
variables with low influence and intersecting families.
Another important ingredient in our proofs is a new notion of symmetric
influence. We use this measure of influence to prove that partial symmetry is
efficiently testable and also to construct an efficient sample extractor for
partially symmetric functions. We then combine the sample extractor with the
testing-by-implicit-learning approach to complete the proof that partially
symmetric functions are efficiently isomorphism-testable.Comment: 22 page
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