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## Resolution Tunnels for Improved SAT Solver Performance (2005)

Venue: | In Proc. of 8th International Conference on Theory and Applications of Satisfiability Testing |

Citations: | 7 - 0 self |

### Citations

3497 | Graph-based algorithms for boolean function manipulation
- Bryant
- 1986
(Show Context)
Citation Context ...e performance of standard SAT solvers on these formulas by orders of magnitude. We obtain a new and much greater lower bound for one of the Van der Waerden numbers, specifically a bound of 1132 for W =-=(2, 6)-=-. We believe this bound to actually be the number we seek. The structure of propositional formulas for solving Van der Waerden numbers is similar to that of formulas arising from Bounded Model Checkin... |

1093 |
A machine-oriented logic based on the resolution principle
- ROBINSON
- 1965
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Citation Context ...al procedure that may be used to determine whether a given CNF expression has a model and to supply a certificate of unsatisfiability if it doesn’t. The idea predates the often cited work reported in =-=[22]-=- and for decades resolution has been one of the primary engines for CNF SAT solvers. In the last 10 years tree resolution, in the form of variants of DPLL [10], has given way to DAG resolution through... |

824 |
A machine program for theorem proving
- Davis, Logemann, et al.
- 1962
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Citation Context ...redates the often cited work reported in [22] and for decades resolution has been one of the primary engines for CNF SAT solvers. In the last 10 years tree resolution, in the form of variants of DPLL =-=[10]-=-, has given way to DAG resolution through the introduction of clause learning and recording during search. This and other ideas have led to a spectacular improvement in the performance of resolution-b... |

501 | Efficient Implementation of a BDD package
- Brace, Rudell, et al.
- 1990
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Citation Context ...lution. Fig. 2. Typical solution curve for ψ 34 2,4. This is the largest satisfiable formula for W (2, 4). Fig. 3. Typical solution curve for ψ 177 2,5 . This is the largest satisfiable formula for W =-=(2, 5)-=-. For both figures, the solution is shown as a sequence of 0’s and 1’s representing an assignment of values to variables in increasing order of index, from left to right. The top of each curve rises o... |

480 |
Binary Decision Diagrams
- Akers
- 1978
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Citation Context ... they are of little practical use. An unpublished general upper bound is [31] W (k, l) ≤ e e(1/k)ee(l+110) , and a general lower bound, due to the Lovász local lemma, is [31] � � l k W (k, l) > (1 + o=-=(1)-=-). elk Work on specific Van der Waerden numbers has sharpened some of these bounds as the results of Table 2 (taken from [11]) show. k \ l 3 4 5 6 7 8 2 9 35 178 > 695 > 3702 > 7483 3 27 > 291 > 1209 ... |

398 | Noise strategies for improving local search
- Selman, Kautz, et al.
- 1994
(Show Context)
Citation Context ...rticular structure. In the Satisfiability literature, the term “tunnel” has been applied to Stochastic Local Search algorithms, a class which includes members of the WalkSAT, GSAT, and other families =-=[14, 16, 17, 20, 25]-=-. In that context, one may think of a location as an assignment of values to variables and the height at a location as the number of constraints falsified by the assignment at that location. Then, for... |

283 | The interactability of resolution, - Haken - 1985 |

218 | Many hard examples for resolution - Chvátal, Szemerédi - 1988 |

204 | Short proofs are narrow — resolution made simple
- Ben-Sasson, Wigderson
(Show Context)
Citation Context ... for which Pn,k(l) is True. There is no known closed form expression for W (k, l) and all but five of the first few numbers are unknown. Table 1 shows all the known Van der Waerden numbers. In 1979 W =-=(3, 4)-=- became the most recent addition to this table. k \ l 3 4 5 2 9 35 178 3 27 4 76 Table 1. Known Van der Waerden numbers. 3s4 Upper and lower bounds on some of the remaining numbers have been derived b... |

195 | Evidence for invariants in local search.
- McAllester, Selman, et al.
- 1997
(Show Context)
Citation Context ...rticular structure. In the Satisfiability literature, the term “tunnel” has been applied to Stochastic Local Search algorithms, a class which includes members of the WalkSAT, GSAT, and other families =-=[14, 16, 17, 20, 25]-=-. In that context, one may think of a location as an assignment of values to variables and the height at a location as the number of constraints falsified by the assignment at that location. Then, for... |

178 | Symbolic model checking with partitioned transition relations - Burch, Clarke, et al. - 1991 |

135 | Towards an Understanding of Hill-Climbing Procedures for SAT,
- Gent, Walsh
- 1993
(Show Context)
Citation Context ...rticular structure. In the Satisfiability literature, the term “tunnel” has been applied to Stochastic Local Search algorithms, a class which includes members of the WalkSAT, GSAT, and other families =-=[14, 16, 17, 20, 25]-=-. In that context, one may think of a location as an assignment of values to variables and the height at a location as the number of constraints falsified by the assignment at that location. Then, for... |

124 | Representation of switching circuits by binary-decision programs. - Lee - 1959 |

102 |
Efficient local search for very large-scale satisfiability problems
- Gu
- 1992
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Citation Context |

99 | Simplified and improved resolution lower bounds
- Beame, Pitassi
- 1996
(Show Context)
Citation Context ...e performance of standard SAT solvers on these formulas by orders of magnitude. We obtain a new and much greater lower bound for one of the Van der Waerden numbers, specifically a bound of 1132 for W =-=(2, 6)-=-. We believe this bound to actually be the number we seek. The structure of propositional formulas for solving Van der Waerden numbers is similar to that of formulas arising from Bounded Model Checkin... |

83 | examples for resolution - Urquhart - 1987 |

66 | CUDD: Colorado university decision diagram package. ftp://vlsi .colorado.edu/pub/. - Somenzi - 1996 |

48 | On the complexity of unsatisfiability proofs for random k-CNF formulas
- Beame, Karp, et al.
- 1998
(Show Context)
Citation Context ... for which Pn,k(l) is True. There is no known closed form expression for W (k, l) and all but five of the first few numbers are unknown. Table 1 shows all the known Van der Waerden numbers. In 1979 W =-=(3, 4)-=- became the most recent addition to this table. k \ l 3 4 5 2 9 35 178 3 27 4 76 Table 1. Known Van der Waerden numbers. 3s4 Upper and lower bounds on some of the remaining numbers have been derived b... |

23 | On resolution with clauses of bounded size - Galil - 1977 |

21 | Search vs symbolic techniques in satisfiability solving - Pan, Vardi - 2004 |

20 | Random 3-SAT and BDDs: The plot thickens further - Aguirre, Vardi - 2001 |

14 | Hiding propositional constants in BDDs - Groote - 1996 |

13 |
Solving satisfiability in less than 2 n steps. Discrete Applied Mathematics 10
- Monien, Speckenmeyer
- 1985
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Citation Context ...y of φ is unaffected. Those values may not be inferred at all, but they are nevertheless safe to assign and doing so reduces φ somewhat. A safe assignment is a generalization of the notion of autarky =-=[21]-=-, defined for CNF formulas, to formulas that are conjunctionssof Boolean functions. The test implied in the theorem can be conducted with reasonable efficiency (see [30] for details). Although safe as... |

8 |
Algorithms for the Satisfiability problem: a survey
- Gu, Purdom, et al.
- 1997
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Citation Context |

3 | Extending Existential Quantification in Conjunctions of BDDs
- Weaver, Franco, et al.
(Show Context)
Citation Context ...d. There are several ways to do this, some safe and some risky. A reasonably efficient method for finding a safe tunnel, which is actually more like a cut, through the mountain has been identified in =-=[30]-=- for a class of non-CNF formulas. Given Boolean functions b1, b2, ..., bm, let φ = b1 ∧ · · · ∧ bm and let V ′ = {v1, · · · , vk} be a subset of variables occurring in φ. Re-index the functions so tha... |

2 | E.: Using ordered binary decision diagrams to solve highly structured satisfiability problems. Unpublished technical report CMU-CS1996 - Dransfield, Bryant - 1996 |

1 | der Waerden numbers. Available from http://home.comcast.net/ rm cooke/vdw.html - Cooke |

1 |
Using Answer-Set Programming to study van der Waerden numbers
- Dransfield, Liu, et al.
(Show Context)
Citation Context ...ower bound, due to the Lovász local lemma, is [31] � � l k W (k, l) > (1 + o(1)). elk Work on specific Van der Waerden numbers has sharpened some of these bounds as the results of Table 2 (taken from =-=[11]-=-) show. k \ l 3 4 5 6 7 8 2 9 35 178 > 695 > 3702 > 7483 3 27 > 291 > 1209 > 8885 > 43854 > 161371 4 76 > 1047 > 10436 > 90306 > 262326 5 > 125 > 2253 > 24044 > 177955 6 > 206 > 3693 > 56692 Table 2. ... |

1 |
Progressions in Every Two-coloration of Zn
- Song, Golomb, et al.
- 1992
(Show Context)
Citation Context ... numbers obtained by SAT solvers ([11]). We are interested in W (2, 6). When we employ aggressive tunneling techniques, we can push the lower bound to 1132 from the previously known best value of 696 =-=[27]-=-. The reason we have a bound instead of the actual number is that aggressive tunneling forces our SAT solver to be incomplete. We emphasize that although an incomplete solver precludes finding a refut... |

1 |
B.L.: Beweis einer Baudetschen Veermutung. Nieuw Archief voor Wiskunde 15
- Waerden
- 1927
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Citation Context ...lysis serves as an example for attempting aggressive tunnel heuristics for other hard problems. 2 Van der Waerden numbers and Satisfiability Van der Waerden numbers arise from a set partition problem =-=[29]-=-. Partition the set Sn = {1, ...n} of the first n positive consecutive integers into k classes. Let Pn,k(l) be a proposition that is True if and only if all partitions of Sn into k classes contain at ... |

1 |
et al.: van der Waerden Number. From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/vanderWaerdenNumber.html
- Weisstein
(Show Context)
Citation Context ...r Waerden numbers. 3s4 Upper and lower bounds on some of the remaining numbers have been derived but they are so far apart that they are of little practical use. An unpublished general upper bound is =-=[31]-=- W (k, l) ≤ e e(1/k)ee(l+110) , and a general lower bound, due to the Lovász local lemma, is [31] � � l k W (k, l) > (1 + o(1)). elk Work on specific Van der Waerden numbers has sharpened some of thes... |