S. Kirkpatrick, G. Gyorgyi, N. Tishby, and L. Troyansky. The statistical mechanics of k-satisfaction. In Advances in Neural Information Processing Systems 6, pages 439--446. Morgan Kaufmann, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....K CNF formula is a random structure of binary variables which are subjected to a set of constrains due to the appearance of the same variables in many clauses. A K CNF formula can therefore be regarded as a system of spins with quenched randomness induced by the random clauses of the formula[11]. The number of constrains increases with the value of ff until conflicts between assignments of variables in different clauses appear ( frustration ) and the formula can no longer be satisfied. For K = 2 there exist a complete theory of this phenomena, based on the analysis of the known ....

....assignments q ij X i Delta X j = 1 Gamma 2d ij : 12) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 5 10 15 20 25 M N K=5, N=20 K=4, N=20 K=3, N=24 K=3, N=16 K=2, N=28 K=2, N=20 K=2, N=12 Figure 1: Theoretical vs. numerical values of q as a function of ff for K = 2 5[11]. The analytic saddle point value of q = 1 Gamma 2 d is the ensemble average over all pairs of satisfying assignments, for the ensemble of all formulae. Due to self averaging this value is also the average overlap between two satisfying assignments of a typical large random formula. Fig. 1 ....

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Kirkpatrick, Scott, G`eza Gy orgyi, Naftali Tishby and Lidror Troyansky, "The Statistical Mechanics of KSatisfaction ", Advances in Neural Information Processing Systems 6, (1993) 439--446.

....K CNF formula is a random structure of binary variables which are subjected to a set of constrains due to the appearance of the same variables in many clauses. A K CNF formula can therefore be regarded as a system of spins with quenched randomness induced by the random clauses of the formula[11]. The number of constrains increases with the value of ff until conflicts between assignments of variables in different clauses appear ( frustration ) and the formula can no longer be satisfied. For K = 2 there exist a complete theory of this phenomena, based on the analysis of the known ....

....of the two assignments q ij : 1 N X i Delta X j = 1 Gamma 2d ij : 12) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 5 10 15 20 25 q M N K=5, N=20 K=4, N=20 K=3, N=24 K=3, N=16 K=2, N=28 K=2, N=20 K=2, N=12 Figure 1: Theoretical vs. numerical values of q as a function of ff for K = 2 5[11]. The analytic saddle point value of q = 1 Gamma 2 d is the ensemble average over all pairs of satisfying assignments, for the ensemble of all formulae. Due to self averaging this value is also the average overlap between two satisfying assignments of a typical large random formula. Fig. 1 ....

[Article contains additional citation context not shown here]

Kirkpatrick, Scott, G`eza Gy orgyi, Naftali Tishby and Lidror Troyansky, "The Statistical Mechanics of KSatisfaction ", Advances in Neural Information Processing Systems 6, (1993) 439--446.

.... Phase transitions have proven to be a good location for problem instances which are hard on the average for complete search algorithms to solve [ CKT91 ] There are interesting relationships between phase transitions in these problems and similar early results in mathematics and physical systems [ KGTT94 ] Much of this research has helped shed light on the phenomena in computer science. Most interesting of all is the potential of research on phase transitions to shed light on the structure of NP Hard problems. These insights may help us design better algorithms which take advantage of these ....

....provides evidence that the maximum average complexity is achieved at or near the phase transition. CKT91 ] presented empirical results showing this relationship for Hamiltonian Cycle, Travelling Salesman, 3 SAT, and Graph Coloring. The transition for K SAT has been extensively studied in [ KGTT94 ] and [ SK95 ] Recent work has shown that the ever tightening phase transition curves often have a single form when re plotted against a function of the number of variables [ KGTT94 ] This phenomenon is called scale invariance and the so called scaling function explains how the phase ....

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S. Kirkpatrick, G. Gyorgyi, N. Tishby, and L. Troyansky. The statistical mechanics of k-satisfaction. Proceedings of the 7th Conference on Neural Information Processing Systems, 1994.

.... systems, a heuristical technique called finite size scaling have been developed to model phase transition phenomena [1] Finite size scaling also appears to be useful for modelling the behaviour of the phase transition in a variety of combinatorial problems including propositional satisfiability [15, 16, 8, 9], and the traveling salesman problem [10] Around the phase transition, finite size scaling predicts that problems of all sizes are indistinguishable except for a change of scale. This would suggest that, P rob(solution) f( Gamma c c : N 1= 4) where f is some fundamental function, ....

....for any other value of percentage solubility, and to extrapolate to any problem size. More significant still is the likelihood that we will see similar kinds of finite scaling in other randomly generated CSP s. This is likely because once similar kinds of scaling were observed in SAT problems [15] they were observed in many different classes of SAT problems [16, 8] We expect that similar kinds of predictions made from examining only small problems should be available for large problems in many different classes of CSP s. 6 Finite Size Scaling of Search Cost The main feature of CSP ....

S. Kirkpatrick, G. Gyorgyi, N. Tishby, and L. Troyansky. The statistical mechanics of k-satisfaction. In Advances in Neural Information Processing Systems 6, pages 439--446. Morgan Kaufmann, 1994.

.... systems, a heuristical technique called finite size scaling have been developed to model phase transition phenomena [1] Finite size scaling also appears to be useful for modelling the behaviour of the phase transition in a variety of combinatorial problems including propositional satisfiability [15, 16, 8, 9], and the traveling salesman problem [10] Around the phase transition, finite size scaling predicts that problems of all sizes are indistinguishable except for a change of scale. This would suggest that, P rob(solution) f( Gamma c ) N 1= 4) where f is some fundamental function, ....

....for any other value of percentage solubility, and to extrapolate this to any problem size. More significant still is the likelihood that we will see similar kinds of finite scaling in other randomly generated CSP s. This is likely because once similar kinds of scaling were observed in SAT problems [15] they were observed in many different classes of SAT problems [16, 8] We expect that similar kinds of predictions made from examining only small problems should be available for large problems in many different classes of CSP s. 6 Finite Size Scaling of Search Cost We showed in x5 that finite ....

S. Kirkpatrick, G. Gyorgyi, N. Tishby, and L. Troyansky. The statistical mechanics of k-satisfaction. In Advances in Neural Information Processing Systems 6, pages 439--446. Morgan Kaufmann, 1994.

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S. Kirkpatrick, G. Gyorgyi, N. Tishby, and L. Troyansky. The statistical mechanics of k-satisfaction. In Advances in Neural Information Processing Systems 6, pages 439--446. Morgan Kaufmann, 1994.

No context found.

S. Kirkpatrick, G. Gyorgi, N. Tishby, and L. Troyansky. The statistical mechanics of k-satisfaction. In J. D. Cowan, G. Tesauro, and J. Alspector, editors, Advances in Neural Information Processing Systems, volume 6, pages 439-446. Morgan Kaufmann Publishers, 1993.

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