24 research outputs found
Going after the k-SAT Threshold
Random -SAT is the single most intensely studied example of a random
constraint satisfaction problem. But despite substantial progress over the past
decade, the threshold for the existence of satisfying assignments is not known
precisely for any . The best current results, based on the second
moment method, yield upper and lower bounds that differ by an additive , a term that is unbounded in (Achlioptas, Peres: STOC 2003).
The basic reason for this gap is the inherent asymmetry of the Boolean value
`true' and `false' in contrast to the perfect symmetry, e.g., among the various
colors in a graph coloring problem. Here we develop a new asymmetric second
moment method that allows us to tackle this issue head on for the first time in
the theory of random CSPs. This technique enables us to compute the -SAT
threshold up to an additive . Independently of
the rigorous work, physicists have developed a sophisticated but non-rigorous
technique called the "cavity method" for the study of random CSPs (M\'ezard,
Parisi, Zecchina: Science 2002). Our result matches the best bound that can be
obtained from the so-called "replica symmetric" version of the cavity method,
and indeed our proof directly harnesses parts of the physics calculations
Optimal Testing for Planted Satisfiability Problems
We study the problem of detecting planted solutions in a random
satisfiability formula. Adopting the formalism of hypothesis testing in
statistical analysis, we describe the minimax optimal rates of detection. Our
analysis relies on the study of the number of satisfying assignments, for which
we prove new results. We also address algorithmic issues, and give a
computationally efficient test with optimal statistical performance. This
result is compared to an average-case hypothesis on the hardness of refuting
satisfiability of random formulas
Generalised and Quotient Models for Random And/Or Trees and Application to Satisfiability
This article is motivated by the following satisfiability question: pick
uniformly at random an and/or Boolean expression of length n, built on a set of
k_n Boolean variables. What is the probability that this expression is
satisfiable? asymptotically when n tends to infinity?
The model of random Boolean expressions developed in the present paper is the
model of Boolean Catalan trees, already extensively studied in the literature
for a constant sequence (k_n)_{n\geq 1}. The fundamental breakthrough of this
paper is to generalise the previous results to any (reasonable) sequence of
integers (k_n)_{n\geq 1}, which enables us, in particular, to solve the above
satisfiability question.
We also analyse the effect of introducing a natural equivalence relation on
the set of Boolean expressions. This new "quotient" model happens to exhibit a
very interesting threshold (or saturation) phenomenon at k_n = n/ln n.Comment: Long version of arXiv:1304.561
Discovering, quantifying, and displaying attacks
In the design of software and cyber-physical systems, security is often
perceived as a qualitative need, but can only be attained quantitatively.
Especially when distributed components are involved, it is hard to predict and
confront all possible attacks. A main challenge in the development of complex
systems is therefore to discover attacks, quantify them to comprehend their
likelihood, and communicate them to non-experts for facilitating the decision
process. To address this three-sided challenge we propose a protection analysis
over the Quality Calculus that (i) computes all the sets of data required by an
attacker to reach a given location in a system, (ii) determines the cheapest
set of such attacks for a given notion of cost, and (iii) derives an attack
tree that displays the attacks graphically. The protection analysis is first
developed in a qualitative setting, and then extended to quantitative settings
following an approach applicable to a great many contexts. The quantitative
formulation is implemented as an optimisation problem encoded into
Satisfiability Modulo Theories, allowing us to deal with complex cost
structures. The usefulness of the framework is demonstrated on a national-scale
authentication system, studied through a Java implementation of the framework.Comment: LMCS SPECIAL ISSUE FORTE 201
Phase Transition Behavior of Cardinality and XOR Constraints
The runtime performance of modern SAT solvers is deeply connected to the
phase transition behavior of CNF formulas. While CNF solving has witnessed
significant runtime improvement over the past two decades, the same does not
hold for several other classes such as the conjunction of cardinality and XOR
constraints, denoted as CARD-XOR formulas. The problem of determining the
satisfiability of CARD-XOR formulas is a fundamental problem with a wide
variety of applications ranging from discrete integration in the field of
artificial intelligence to maximum likelihood decoding in coding theory. The
runtime behavior of random CARD-XOR formulas is unexplored in prior work. In
this paper, we present the first rigorous empirical study to characterize the
runtime behavior of 1-CARD-XOR formulas. We show empirical evidence of a
surprising phase-transition that follows a non-linear tradeoff between CARD and
XOR constraints