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Journal of Statistical Mechanics: Theory and Experiment Volume 2008 April 2008 Create an alert RSS this journal

Andrea Montanari et al J. Stat. Mech. (2008) P04004 doi:10.1088/1742-5468/2008/04/P04004

Clusters of solutions and replica symmetry breaking in random k-satisfiability

Andrea Montanari1,2, Federico Ricci-Tersenghi3 and Guilhem Semerjian4

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Abstract References Cited By Figures Tables

Figure 1

Figure 1. An example of the factor graph representation of a satisfiability formula for k = 3. The values Jia are encoded by drawing a solid (resp. dashed) edge between clause a and variable i if σi = + 1 (resp. –1) satisfies clause a. The distances between some of the variable nodes are di,j = di,j' = di,j' = 1 and dj,j' = 2. The neighborhoods are, for instance, \partial i=\{a, b, c\} , \partial a=\{i, j, j'\} , \partial_+i=\{a\} , \partial_-i=\{b, c\} , \partial_+i(a)=\emptyset , \partial_-i(a)=\{b, c\} , \partial_+i(b)=\{c\} , \partial_-i(b)=\{a\} .



Figure 2

Figure 2. The point-to-set correlation function for k = 4; from left to right α = 9.30, 9.33, 9.35 and 9.40.



Figure 3

Figure 3. The complexity Σ and the internal entropy phiint for the values m = 0, 1, and m = ms in the \mathsf {1RSB} regime, for k = 4.



Figure 4

Figure 4. The complexity Σ(phi) for k = 4 and several values of α: from top to bottom α = 9.3, 9.45, 9.6, 9.7, 9.8 and 9.9.



Figure 5

Figure 5. The value of the Parisi parameter ms in the thermodynamically relevant pure states of the \mathsf {1RSB} regime in random 4-SAT, and the freezing transition mf.



Figure 6

Figure 6. Intra-and inter-state overlaps for k = 4.



Figure 7

Figure 7. The point-to-set correlation function for k = 3; from left to right α = 3.60, 3.84, 3.86, 3.88.



Figure 8

Figure 8. The complexity Σ for k = 3 and m from 0 (highest curve) to 1 (lowest curve). For 0 < m < 1 the domain of existence of Σ may be slightly larger than the one shown in the plot (we have simulated only α values in multiples of 0.05).



Figure 9

Figure 9. The 1RSB estimate for the entropy of random 3-SAT, compared to the replica symmetric (RS) estimate and to the internal entropy of the m = 0 solution, corresponding to the maximum of the Σ(phi) curve.



Figure 10

Figure 10. The complexity Σ(phi) in random 3-SAT, for several values of α.



Figure 11

Figure 11. Condensation threshold in reduced units, 2–kαc(k). Symbols: numerical determination by population dynamics algorithm, see table 1. Lines: analytical large-k expansion, truncated at the three first orders, see equation (80).



Figure C.1

Figure C.1. Intra- and inter-state overlap, q0 and q1, for k = 3 and some values of the Parisi parameter m. Data below (resp. above) the RS line are for q0 (resp. q1). Full (resp. open) symbols refer to data measured while increasing (resp. decreasing) α.



Figure C.2

Figure C.2. The internal entropy should be a non-decreasing function of m if the solution is consistent. Filled (resp. empty) symbols refer to solutions with \partial _m \phi \gt 0 (resp. \partial_m \phi \lt 0 ), for k = 3.



Figure C.3

Figure C.3. The entropic complexity Σ(phi) for k = 3 and α = 4.2. The two different branches correspond to the consistent (full line) and inconsistent solution (dashed line).




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