On the replica symmetric solution of the K - sat model
نویسنده
چکیده
In this paper we translate Talagrand’s solution of the K-sat model at high temperature into the language of asymptotic Gibbs measures. Using exact cavity equations in the infinite volume limit allows us to remove many technicalities of the inductions on the system size, which clarifies the main ideas of the proof. This approach also yields a larger region of parameters where the system is in a pure state and, in particular, for small connectivity parameter one can prove the replica symmetric formula for the free energy at any temperature.
منابع مشابه
The K-sat problem in a simple limit
The statistical mechanics of the random K-satisfiability (K-SAT) problem has been the object of many studies in the last years [1][2][3]. The K-SAT was the first problem to be shown to be Nondeterministic Polynomial (NP) complete [4] and the importance of such a model is that it provides a prototype for all the NP complete problems in complexity theory of computer science as well as in statisti...
متن کاملStatistical Mechanics of the Random K - SAT Model
The Random K-Satisfiability Problem, consisting in verifying the existence of an assignment of N Boolean variables that satisfy a set of M = αN random logical clauses containingK variables each, is studied using the replica symmetric framework of disordered systems. The detailed structure of the analytical solution is discussed for the different cases of interest K = 2, K ≥ 3 and K ≫ 1. We pres...
متن کاملBiased random satisfiability problems: from easy to hard instances.
In this paper we study biased random K -satisfiability ( K -SAT) problems in which each logical variable is negated with probability p . This generalization provides us a crossover from easy to hard problems and would help us in a better understanding of the typical complexity of random K -SAT problems. The exact solution of 1-SAT case is given. The critical point of K -SAT problems and results...
متن کاملCommunities of solutions in single solution clusters of a random K-satisfiability formula.
The solution space of a K-satisfiability (K-SAT) formula is a collection of solution clusters, each of which contains all the solutions that are mutually reachable through a sequence of single-spin flips. Knowledge of the statistical property of solution clusters is valuable for a complete understanding of the solution space structure and the computational complexity of the random K-SAT problem...
متن کاملInstability of one-step replica-symmetry-broken phase in satisfiability problems
Abstract We reconsider the one-step replica-symmetry-breaking (1RSB) solutions of two random combinatorial problems: k-XORSAT and k-SAT. We present a general method for establishing the stability of these solutions with respect to further steps of replica-symmetry breaking. Our approach extends the ideas of Ref. [1] to more general combinatorial problems. It turns out that 1RSB is always unstab...
متن کامل