## Directional Resolution: The Davis-Putnam Procedure, Revisited (1994)

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Venue: | IN PROCEEDINGS OF KR-94 |

Citations: | 100 - 21 self |

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@INPROCEEDINGS{Dechter94directionalresolution:,

author = {Rina Dechter and Irina Rish},

title = {Directional Resolution: The Davis-Putnam Procedure, Revisited},

booktitle = {IN PROCEEDINGS OF KR-94},

year = {1994},

pages = {134--145},

publisher = {Morgan Kaufmann}

}

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### Abstract

The paper presents an algorithm called directional resolution, a variation on the original Davis-Putnam algorithm, and analyzes its worst-case behavior as a function of the topological structure of propositional theories. The concepts of induced width and diversity are shown to play a key role in bounding the complexity of the procedure. The importance of our analysis lies in highlighting structure-based tractable classes of satisfiability and in providing theoretical guarantees on the time and space complexity of the algorithm. Contrary to previous assessments, we show that for many theories directional resolution could be an effective procedure. Our empirical tests confirm theoretical prediction, showing that on problems with a special structure, namely k-tree embeddings (e.g. chains, (k,m)-trees), directional resolution greatly outperforms one of the most effective satisfiability algorithms known to date, the popular Davis-Putnam procedure. Furthermore, combining a bounded...

### Citations

1334 |
Local computations with probabilities on graphical structures and their applications to expert systems
- Lauritzen, Spiegelhalter
- 1988
(Show Context)
Citation Context ...tional resolution belongs to a family of elimination algorithms rst analyzed for optimization tasks in dynamic programming [2] and later used in constraint satisfaction [18, 7] and in belief networks =-=[20]-=-. The complexity of all those elimination algorithms is a function of the induced width w3 of the undirected graph characteristic of each problem instance. Although it is known that determining the w3... |

1107 | A computing procedure for quantification theory - Davis, Putnam - 1960 |

763 |
A machine program for theorem proving
- Davis, Logeman, et al.
- 1962
(Show Context)
Citation Context ..., because the algorithm was immediately overshadowed by a competitor with nearly the same name: The Davis-Putnam Procedure. This competing algorithm, proposed in 1962 by Davis, Logemann, and Loveland =-=[5]-=-, searches through the space of possible truth assignments while performing unit resolution until quiesience at each step. We will refer to the rst algorithm as DP-elimination and to the second as DP-... |

698 | A New Method for Solving Hard Satisfiability Problems - Selman, Levesque, et al. - 1996 |

397 |
Network-based heuristics for constraint-satisfaction problems
- Dechter, Pearl
- 1988
(Show Context)
Citation Context ... this parent set is the width of A relative tod. The width w(d) of an ordering d is the maximum width of nodes along the ordering, and the width w of a graph is the minimal width of all its orderings =-=[11, 7]-=-. Lemma 1: Given an interaction graph G(') and an ordering d, ifA isanode having k 01 parents, then there arenomore than 3 k size of the corresponding theory is O(n 1 3 w ). clauses in the bucket of A... |

336 |
Complexity of finding embedding in a k-tree
- Arnborg, Corneil, et al.
- 1987
(Show Context)
Citation Context ...last theorem, theories with the bounded induced width would constitute a tractable class for directional resolution. It is known that the induced width of a graph embedded in a k-tree is bounded by k =-=[1]-=-. Here is a recursive de nition of k-trees. De nition 3: (k-trees) 1. A clique of size k (complete graph with k vertices) is a k-tree. 2. Given a k-tree de ned on Q 1; :::; Q i01, ak-tree on Q 1; :::;... |

279 | C.: A Sufficient Condition for Backtrack-Free Search - Freuder - 1982 |

278 | Enhancement schemes for constraint processing backjumping, learning and cutset decomposition - Dechter - 1990 |

224 | and easy distributions of sat problems - Mitchell, Selman, et al. - 1992 |

202 | Experimental results on the crossover point in satisfiability problems
- Crawford, Auton
- 1993
(Show Context)
Citation Context ... with the chain structure that uses a loop construction instead of recursion in order to increase space e ciency. The algorithm has been also augmented with the 2-literal clause heuristic proposed in =-=[4]-=-. The heuristic suggests to instantiate next a variable that would cause the largest number of unit propagations. The number of possible unit propagations is approximated by the number of 2literal cla... |

183 |
Nonserial Dynamic Programming
- Bertele, Brioschi
- 1972
(Show Context)
Citation Context ...of '8 in example 3:'8 = f(A1;A2; :A3), (:A2;A4), (:A2;A3;:A4), (A3;A4;:A5), (:A4;A6), (:A4;A5;:A6), (A5;A6;:A7), (:A6;A8), (:A6;A7;:A8)g width possible. Several heuristic orderings are available (see =-=[2]-=-). Important special tractable classes are those having w3 = 1 (namely, the interaction graph is a tree) and those having w 3 = 2, called series parallel networks. These classes can be recognized in l... |

58 |
A new method for solving hard satis ability problems
- Selman, Levesque, et al.
- 1992
(Show Context)
Citation Context ...plosion), the authors of [5] may haveleft the impression that the two algorithms are basically identical. Indeed, from then on, most work on the Davis-Putnam procedure quotes the backtracking version =-=[15, 19]-=-, wrongly suggesting that this is the algorithm presented in [6]. In this paper, we wish to \revive" the DP-elimination algorithm by studying its virtues theoretically and by subjecting it to a more e... |

57 | Tree-Clustering Schemes for Constraint-Processing - Dechter, Pearl |

54 | A su cient condition for backtrack-free search - Freuder |

41 |
A Computing Procedure for Quanti cation Theory
- Davis, Putnam
- 1960
(Show Context)
Citation Context ...nti c Research grant AFOSR 900136, by Toshiba of America, and by a Xerox grant. 11 Introduction In 1960, Davis and Putnam presented a resolution algorithm for determining propositional satis ability =-=[6]-=-. They proved that a restricted amount of resolution performed systematically along some ordering of the variables in a propositional theory is su cient for deciding satis ability. This algorithm, in ... |

39 |
A new method for solving constraint satisfaction problems
- Seidel
- 1981
(Show Context)
Citation Context ...ated work and conclusions Directional resolution belongs to a family of elimination algorithms rst analyzed for optimization tasks in dynamic programming [2] and later used in constraint satisfaction =-=[18, 7]-=- and in belief networks [20]. The complexity of all those elimination algorithms is a function of the induced width w3 of the undirected graph characteristic of each problem instance. Although it is k... |

34 | Complexity of nding embedding in a k-tree - Arnborg, Corneil, et al. - 1987 |

31 | Directed constraint networks: A relational framework for causal modeling
- Dechter, Pearl
- 1991
(Show Context)
Citation Context ...ility of DP-elimination for many known tractable classes of satis ability and constraint satisfaction problems (e.g., 2-cnfs, Horn clauses, causal theories and theories having a bounded induced width =-=[7, 8]-=-). Third, we introduce a new parameter, called diversity, that gives rise to new tractable classes. 2On the empirical side, we qualify prior empirical tests in [5] by showing that for uniform random ... |

25 |
Default logic, propositional logic and constraints
- Ben-Eliyahu, Dechter
- 1991
(Show Context)
Citation Context ... ability is a special case of constraint satisfaction, the induced28width could be obtained by mapping a propositional formula into the relational framework of a constraint satisfaction problem (see =-=[3]-=-), and then applying adaptive consistency, the elimination algorithm tailored for constraint satisfaction problems [7, 18]. We haverecently shown, however, that this kind of pairwise elimination opera... |

22 |
Regular resolution versus unrestricted resolution
- Goerdt
- 1993
(Show Context)
Citation Context ...heory is su cient for deciding satis ability. This algorithm, in its original form, has received limited attention, and analyses of its performance have emphasized its worst-case exponential behavior =-=[12, 14]-=-, while neglecting the algorithm's virtues. This happened, in our view, because the algorithm was immediately overshadowed by a competitor with nearly the same name: The Davis-Putnam Procedure. This c... |

17 | On the Minimality and Decomposability of Row-convex Constraint Networks - Beek, Dechter - 1995 |

17 | Constraint tightness versus global consistency - Beek, Dechter - 1994 |

16 |
Enhancement Schemes for Constraint
- Dechter
- 1990
(Show Context)
Citation Context ...cking. Choosing the right ordering would help but this may be hard to recognize without some preprocessing. Other variants of backtracking that are capable of exploiting the structure like backjumping=-=[13, 9]-=- would avoid such useless re-instantiation of variables. Experiments with backjumping on the instances in Table 3 con rmed that it outperforms DP-backtracking by far ( Figure 11). When experimenting w... |

7 |
Performance Measurement and Analysis of Certain Search Algorithms
- Gashnig
- 1979
(Show Context)
Citation Context ...cking. Choosing the right ordering would help but this may be hard to recognize without some preprocessing. Other variants of backtracking that are capable of exploiting the structure like backjumping=-=[13, 9]-=- would avoid such useless re-instantiation of variables. Experiments with backjumping on the instances in Table 3 con rmed that it outperforms DP-backtracking by far ( Figure 11). When experimenting w... |

5 |
On the complexity of timetable and multi-commodity flow
- Even, Itai, et al.
- 1976
(Show Context)
Citation Context ...this case, unrestricted resolution is also tractable). Clearly, this algorithm is not the most e ective one for satis ability of 2-cnfs. Satis ability for these theories can be decided in linear time =-=[10]-=-. However, as noted earlier, DR achieves more than satis ability, it compiles a theory that allows model generation in linear time. We summarize: Theorem 4: If ' is a 2-cnf theory, then algorithm dire... |

4 |
Private communication
- McAllester
- 1990
(Show Context)
Citation Context ...l be determined polynomially (for Horn theories) if the algorithm terminates once no unit clauses are available. However, executing the algorithm to full completion may result in long output theories =-=[McAllester]-=-. We nowshow that definite Horn theories of zero diversity can be given a simple graph interpretation, yielding a more accurate estimate of the extension's size for de nite and Horn theories. One may ... |

4 | Average Time Analysis of Simpli ed Davis-Putnum - Goldberg, Purdom, et al. - 1982 |

3 |
On the Complexity of Regular Resolution
- Galil
- 1977
(Show Context)
Citation Context ...heory is su cient for deciding satis ability. This algorithm, in its original form, has received limited attention, and analyses of its performance have emphasized its worst-case exponential behavior =-=[12, 14]-=-, while neglecting the algorithm's virtues. This happened, in our view, because the algorithm was immediately overshadowed by a competitor with nearly the same name: The Davis-Putnam Procedure. This c... |

2 | On the complexity of timetable and multicommodity ow - Even, Itai, et al. - 1976 |

1 |
Average Time Analysis of Simpli ed
- Goldberg, Purdom, et al.
- 1982
(Show Context)
Citation Context ...plosion), the authors of [5] may haveleft the impression that the two algorithms are basically identical. Indeed, from then on, most work on the Davis-Putnam procedure quotes the backtracking version =-=[15, 19]-=-, wrongly suggesting that this is the algorithm presented in [6]. In this paper, we wish to \revive" the DP-elimination algorithm by studying its virtues theoretically and by subjecting it to a more e... |

1 |
On the minimality and decomposability ofrow-convex constraint networks
- Beek, Dechter
- 1993
(Show Context)
Citation Context ...verecently shown, however, that this kind of pairwise elimination operation as performed by directional resolution is more e ective. And, while it can be extended to any row-convex constraint problem =-=[21]-=- or to every 1-tight relations [22] it cannot decide consistency for arbitrary multi-valued networks of relations. The paper makes three main contributions. First, we revive the old Davis-Putnam algor... |

1 | On the complexity ofregular resolution and the Davis-Putnam procedure", Theoretical Computer Science 4 - Galil - 1977 |

1 | Constraint tightness vs global consistency - Beek, Dechter - 1993 |

1 | On the Complexity of Timetable and MultiCommodity - Even, Itai, et al. - 1976 |

1 | Average Time Analysis - Goldberg, Purdom, et al. - 1982 |