11,176 research outputs found
On the Parity Problem in One-Dimensional Cellular Automata
We consider the parity problem in one-dimensional, binary, circular cellular
automata: if the initial configuration contains an odd number of 1s, the
lattice should converge to all 1s; otherwise, it should converge to all 0s. It
is easy to see that the problem is ill-defined for even-sized lattices (which,
by definition, would never be able to converge to 1). We then consider only odd
lattices.
We are interested in determining the minimal neighbourhood that allows the
problem to be solvable for any initial configuration. On the one hand, we show
that radius 2 is not sufficient, proving that there exists no radius 2 rule
that can possibly solve the parity problem from arbitrary initial
configurations. On the other hand, we design a radius 4 rule that converges
correctly for any initial configuration and we formally prove its correctness.
Whether or not there exists a radius 3 rule that solves the parity problem
remains an open problem.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
General Impossibility of Group Homomorphic Encryption in the Quantum World
Group homomorphic encryption represents one of the most important building
blocks in modern cryptography. It forms the basis of widely-used, more
sophisticated primitives, such as CCA2-secure encryption or secure multiparty
computation. Unfortunately, recent advances in quantum computation show that
many of the existing schemes completely break down once quantum computers reach
maturity (mainly due to Shor's algorithm). This leads to the challenge of
constructing quantum-resistant group homomorphic cryptosystems.
In this work, we prove the general impossibility of (abelian) group
homomorphic encryption in the presence of quantum adversaries, when assuming
the IND-CPA security notion as the minimal security requirement. To this end,
we prove a new result on the probability of sampling generating sets of finite
(sub-)groups if sampling is done with respect to an arbitrary, unknown
distribution. Finally, we provide a sufficient condition on homomorphic
encryption schemes for our quantum attack to work and discuss its
satisfiability in non-group homomorphic cases. The impact of our results on
recent fully homomorphic encryption schemes poses itself as an open question.Comment: 20 pages, 2 figures, conferenc
Lifeworld Analysis
We argue that the analysis of agent/environment interactions should be
extended to include the conventions and invariants maintained by agents
throughout their activity. We refer to this thicker notion of environment as a
lifeworld and present a partial set of formal tools for describing structures
of lifeworlds and the ways in which they computationally simplify activity. As
one specific example, we apply the tools to the analysis of the Toast system
and show how versions of the system with very different control structures in
fact implement a common control structure together with different conventions
for encoding task state in the positions or states of objects in the
environment.Comment: See http://www.jair.org/ for any accompanying file
Non-linear Log-Sobolev inequalities for the Potts semigroup and applications to reconstruction problems
Consider a Markov process with state space , which jumps continuously to
a new state chosen uniformly at random and regardless of the previous state.
The collection of transition kernels (indexed by time ) is the Potts
semigroup. Diaconis and Saloff-Coste computed the maximum of the ratio of the
relative entropy and the Dirichlet form obtaining the constant in
the -log-Sobolev inequality (-LSI). In this paper, we obtain the best
possible non-linear inequality relating entropy and the Dirichlet form (i.e.,
-NLSI, ). As an example, we show . The more precise NLSIs have been shown by Polyanskiy and Samorodnitsky to
imply various geometric and Fourier-analytic results.
Beyond the Potts semigroup, we also analyze Potts channels -- Markov
transition matrices constant on and off diagonal. (Potts
semigroup corresponds to a (ferromagnetic) subset of matrices with positive
second eigenvalue). By integrating the -NLSI we obtain the new strong data
processing inequality (SDPI), which in turn allows us to improve results on
reconstruction thresholds for Potts models on trees. A special case is the
problem of reconstructing color of the root of a -colored tree given
knowledge of colors of all the leaves. We show that to have a non-trivial
reconstruction probability the branching number of the tree should be at least
This extends previous
results (of Sly and Bhatnagar et al.) to general trees, and avoids the need for
any specialized arguments. Similarly, we improve the state-of-the-art on
reconstruction threshold for the stochastic block model with balanced
groups, for all . These improvements advocate information-theoretic
methods as a useful complement to the conventional techniques originating from
the statistical physics
How to Write a Proof: Patterns of Justification in Strategic Documents for Educational Reform
Writing strategic documents is a major practice of many actors striving to see their educational ideas realised in the curriculum. In these documents, arguments are systematically developed to create the legitimacy of a new educational goal and competence to make claims about it. Through a qualitative analysis of the writing strategies used in these texts, I show how two of the main actors in the Czech educational discourse have developed a proof that a new educational goal is needed. I draw on the connection of the relational approach in the sociology of education with Lyotard’s analytical semantics of instances in the event. The comparison of the writing strategies in the two documents reveals differences in the formation of a particular pattern of justification. In one case the texts function as a herald of pure reality, and in the other case as a messenger of other witnesses. This reveals different regimens of proof, although both of them were written as prescriptive directives – normative models of the educational world
Approximately Optimal Mechanism Design: Motivation, Examples, and Lessons Learned
Optimal mechanism design enjoys a beautiful and well-developed theory, and
also a number of killer applications. Rules of thumb produced by the field
influence everything from how governments sell wireless spectrum licenses to
how the major search engines auction off online advertising. There are,
however, some basic problems for which the traditional optimal mechanism design
approach is ill-suited --- either because it makes overly strong assumptions,
or because it advocates overly complex designs. The thesis of this paper is
that approximately optimal mechanisms allow us to reason about fundamental
questions that seem out of reach of the traditional theory.
This survey has three main parts. The first part describes the approximately
optimal mechanism design paradigm --- how it works, and what we aim to learn by
applying it. The second and third parts of the survey cover two case studies,
where we instantiate the general design paradigm to investigate two basic
questions. In the first example, we consider revenue maximization in a
single-item auction with heterogeneous bidders. Our goal is to understand if
complexity --- in the sense of detailed distributional knowledge --- is an
essential feature of good auctions for this problem, or alternatively if there
are simpler auctions that are near-optimal. The second example considers
welfare maximization with multiple items. Our goal here is similar in spirit:
when is complexity --- in the form of high-dimensional bid spaces --- an
essential feature of every auction that guarantees reasonable welfare? Are
there interesting cases where low-dimensional bid spaces suffice?Comment: Based on a talk given by the author at the 15th ACM Conference on
Economics and Computation (EC), June 201
Recommended from our members
Disciplines and Styles in Pure Mathematics, 1800-2000
This workshop addressed issues of discipline and style in number theory, algebra, geometry, topology, analysis, and mathematical physics. Most speakers presented case studies, but some offered global surveys of how stylistic shifts informed the transition and transformation of special research fields. Older traditions in established research communities were considered alongside newer trends, including changing views regarding the role of proof
- …