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We study the satisfiability of randomly generated formulas formed by M clauses of exactly K literals over N Boolean variables. For a given value of N the problem is known to be most difficult when α = M/N is close to the experimental threshold α c separating the region where almost all formulas are SAT from the region where all formulas are UNSAT. Recent(More)
We study the satisfiability of random Boolean expressions built from many clauses with K variables per clause (K-satisfiability). Expressions with a ratio alpha of clauses to variables less than a threshold alphac are almost always satisfiable, whereas those with a ratio above this threshold are almost always unsatisfiable. We show the existence of an(More)
We study the problem of satisfiability of randomly chosen clauses, each with K Boolean variables. Using the cavity method at zero temperature, we find the phase diagram for the K=3 case. We show the existence of an intermediate phase in the satisfiable region, where the proliferation of metastable states is at the origin of the slowdown of search(More)
Using the cavity we derive the various threshold values for the number of clauses per variable of the random K-satisfiability problem, generalizing the previous results to K ≥ 4. We also give an analytic solution of the equations, and some closed expressions for these thresholds, in an expansion around large K. The stability of the solution is also(More)
Non-deterministic polynomial time (commonly termed`NP-complete') problems are relevant to many computational tasks of practical interestÐsuch as thètravelling salesman problem'Ðbut are dif®cult to solve: the computing time grows exponentially with problem size in the worst case. It has recently been shown that these problems exhibit`phase boundaries',(More)
The similarity in the three-dimensional structures of homologous proteins imposes strong constraints on their sequence variability. It has long been suggested that the resulting correlations among amino acid compositions at different sequence positions can be exploited to infer spatial contacts within the tertiary protein structure. Crucial to this(More)
The evolutionary trajectory of a protein through sequence space is constrained by its function. Collections of sequence homologs record the outcomes of millions of evolutionary experiments in which the protein evolves according to these constraints. Deciphering the evolutionary record held in these sequences and exploiting it for predictive and engineering(More)
Focusing on the optimization version of the random K -satisfiability problem, the MAX- K -SAT problem, we study the performance of the finite energy version of the survey propagation algorithm. We show that a simple (linear time) backtrack decimation strategy is sufficient to reach configurations well below the lower bound for the dynamic threshold energy(More)
This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape at solutions found by gradient descent. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative(More)