615 research outputs found
A Continuous-Discontinuous Second-Order Transition in the Satisfiability of Random Horn-SAT Formulas
We compute the probability of satisfiability of a class of random Horn-SAT
formulae, motivated by a connection with the nonemptiness problem of finite
tree automata. In particular, when the maximum clause length is 3, this model
displays a curve in its parameter space along which the probability of
satisfiability is discontinuous, ending in a second-order phase transition
where it becomes continuous. This is the first case in which a phase transition
of this type has been rigorously established for a random constraint
satisfaction problem
Hiding solutions in random satisfiability problems: A statistical mechanics approach
A major problem in evaluating stochastic local search algorithms for
NP-complete problems is the need for a systematic generation of hard test
instances having previously known properties of the optimal solutions. On the
basis of statistical mechanics results, we propose random generators of hard
and satisfiable instances for the 3-satisfiability problem (3SAT). The design
of the hardest problem instances is based on the existence of a first order
ferromagnetic phase transition and the glassy nature of excited states. The
analytical predictions are corroborated by numerical results obtained from
complete as well as stochastic local algorithms.Comment: 5 pages, 4 figures, revised version to app. in PR
The Scaling Window of the 2-SAT Transition
We consider the random 2-satisfiability problem, in which each instance is a
formula that is the conjunction of m clauses of the form (x or y), chosen
uniformly at random from among all 2-clauses on n Boolean variables and their
negations. As m and n tend to infinity in the ratio m/n --> alpha, the problem
is known to have a phase transition at alpha_c = 1, below which the probability
that the formula is satisfiable tends to one and above which it tends to zero.
We determine the finite-size scaling about this transition, namely the scaling
of the maximal window W(n,delta) = (alpha_-(n,delta),alpha_+(n,delta)) such
that the probability of satisfiability is greater than 1-delta for alpha <
alpha_- and is less than delta for alpha > alpha_+. We show that
W(n,delta)=(1-Theta(n^{-1/3}),1+Theta(n^{-1/3})), where the constants implicit
in Theta depend on delta. We also determine the rates at which the probability
of satisfiability approaches one and zero at the boundaries of the window.
Namely, for m=(1+epsilon)n, where epsilon may depend on n as long as |epsilon|
is sufficiently small and |epsilon|*n^(1/3) is sufficiently large, we show that
the probability of satisfiability decays like exp(-Theta(n*epsilon^3)) above
the window, and goes to one like 1-Theta(1/(n*|epsilon|^3)) below the window.
We prove these results by defining an order parameter for the transition and
establishing its scaling behavior in n both inside and outside the window.
Using this order parameter, we prove that the 2-SAT phase transition is
continuous with an order parameter critical exponent of 1. We also determine
the values of two other critical exponents, showing that the exponents of 2-SAT
are identical to those of the random graph.Comment: 57 pages. This version updates some reference
On the Satisfiability Threshold and Clustering of Solutions of Random 3-SAT Formulas
We study the structure of satisfying assignments of a random 3-SAT formula.
In particular, we show that a random formula of density 4.453 or higher almost
surely has no non-trivial "core" assignments. Core assignments are certain
partial assignments that can be extended to satisfying assignments, and have
been studied recently in connection with the Survey Propagation heuristic for
random SAT. Their existence implies the presence of clusters of solutions, and
they have been shown to exist with high probability below the satisfiability
threshold for k-SAT with k>8, by Achlioptas and Ricci-Tersenghi, STOC 2006. Our
result implies that either this does not hold for 3-SAT or the threshold
density for satisfiability in 3-SAT lies below 4.453.
The main technical tool that we use is a novel simple application of the
first moment method
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