### Table 1. Comparison of the sizes of the different CSP encodings for k-SAT.

"... In PAGE 8: ...f the size of the resulting CSP w.r.t. the number of variables (#variables), the size of the variable domains, the number of constraints (#constraints), and the size of the constraints, using the encodings presented in the previous section is shown in Table1 . Notice that out of all the encodings, only the place encoding is linear in all of these parameters (number of variables, domain size, and constraint size).... In PAGE 9: ...shown in Table1 , variables associated with clauses have domains of exponential size in the hidden variable encoding, while these domains are of linear size in the place encoding. We begin by showing that enforcing FC consistency on the place encoding results in an arc consistent CSP.... ..."

### Table 1. The k sat algorithm schema.

"... In PAGE 4: ... The General k sat Schema. In Table1 , we present the general algorithm schema k sat, on which KSAT [7] is based, for deciding the satis ability of formulas in K. The sat procedure (called at line 5) determines the satis ability of as a layer-0 proposition by returning a propositional assignment ; if is empty, backtracking takes place.... ..."

### Table 1 The k sat algorithm schema.

"... In PAGE 5: ....0.2 The General k sat Schema. In Table1 , we present the general algorithm schema k sat, on which KSAT [6] is based, for deciding the satis ability of formulas in K. The sat procedure (called at line 5) determines the satis ability of as a layer-0 proposition by returning a propositional assignment ; if is empty, backtracking takes place.... ..."

### Table 1. The k sat algorithm schema.

"... In PAGE 5: ... The General k sat Schema. In Table1 , we present the general algorithm schema k sat, on which KSAT [8] is based, for deciding the satis ability of for- mulas in K. The sat procedure (called at line 5) determines the satis ability of as a layer-0 proposition by returning a propositional assignment ; if is empty, backtracking takes place.... ..."

### Table 5: Hard random k{SAT formulas, e ciency of 256{processor algorithm

1996

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### Table 4 kSAT, kSAT+USB, and kSAT+USH on TANCS 2000 modalized easy/medium problems we consider only the formulas on which the system did not fail, and write \{ quot; when the system fails on all the samples. As it can be seen, caching produces dramatic improvements. Many tests that are not solved within the timelimit by kSAT, are solved when using a caching mechanism.Indeed, the sets in Tables 1 and 2 are just a few. Consider also Tables 3 and 4, showing kSAT, kSAT+USB and kSAT+USH results on the easy/medium and modalized easy/medium formulas respectively. These Tables make clear the qualitative di erences between kSATon one side, and kSAT+USB and kSAT+USH on the other. kSAT solves only 51 (9%) out of the 544 samples on which it has been tested. kSAT+USB and kSAT+USH solve 414 (76%) and 390 (71%)

### Table 1 Threshold values for random K-SAT. Bold numbers are the results of the population dynamics algorithm. (0) d is the value predicted by the rst moment expansion of the cavity equations (sec. 6.3), (r) c is the result of a series expansion in quot; = 2 K of the cavity equations up to order r (secs. 6.2 and A). Note that all reported values c(K) fall between the best rigorously known upper and lower bounds.

2006

"... In PAGE 22: ...he cavity equations up to order r (secs. 6.2 and A). Note that all reported values c(K) fall between the best rigorously known upper and lower bounds. Table1 shows the results. Since c for K = 3 is the most \prominent quot; thresh- old we spent a bit more CPU power to increase its accuracy.... In PAGE 27: ... Fig. 8 and Table1 show that the seventh order expansion gets actually very close to the numerical values. For K = 3 the deviation of the seventh order asymptotic expansion from the numerical value is less than 1%, and for K 4 this deviation is even smaller.... In PAGE 28: ... (39). The results are the values for (0) d in Table1 . The values for (0) d agree perfectly with the exact values d (within the error bars of the latter), even for K = 3, although the non-trivial distributions A(x) and B(y) that appear right above d are not -like.... In PAGE 35: ... The quality of the expansion up to seventh order can be seen in Fig. 8 and Table1 . Note that there exist also nonanalytic terms in quot;, because we dropped some corrections of order which in turn behaves as quot;1=(2 quot;).... ..."

Cited by 8

### Table 7 kSAT+USB and kSAT+USH on TANCS 2000 easy/medium problems

### Table 1.1: Some of the crossover points for random K-SAT problems. 13

1995