## Local and global relational consistency (1997)

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Venue: | THEORETICAL COMPUTER SCIENCE |

Citations: | 64 - 13 self |

### BibTeX

@ARTICLE{Dechter97localand,

author = {Rina Dechter and Peter van Beek},

title = {Local and global relational consistency},

journal = {THEORETICAL COMPUTER SCIENCE},

year = {1997},

volume = {173},

pages = {283--308}

}

### Years of Citing Articles

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### Abstract

Local consistency has proven to be an important concept in the theory and practice of constraint networks. In this paper, we present a new definition of local consistency, called relational consistency. The new definition is relation-based, in contrast with the previous definition of local consistency, which we characterize as variable-based. We show the conceptual power of the new definition by showing how it unifies known elimination operators such as resolution in theorem proving, joins in relational databases, and variable elimination for solving linear inequalities. Algorithms for enforcing various levels of relational consistency are introduced and analyzed. We also show the usefulness of the new definition in characterizing relationships between properties of constraint networks and the level of local consistency needed to ensure global consistency.

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Citation Context ... search to nd a solution, such an inconsistency can lead to a dead end in the search. This insight has led to the de nition of conditions that characterize the level of local consistency of a network =-=[17, 20]-=- and to the development of algorithms for enforcing local consistency conditions by removing local inconsistencies (e.g., [3, 7, 11, 17, 20]). In this paper, we present a new de nition of local consis... |

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Citation Context ...nequalities over nite set of integers can be globally solved byDRC3. ACNF formula can be globally solved byDRC3. Two special cases are a restricted and discrete version of Dechter, Meiri, and Pearl's =-=[6]-=- continuous, bounded di erence framework for temporal reasoning and a restricted and discrete version of Vilain and Kautz's [27] qualitative framework for temporal reasoning. Another known class that ... |

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Citation Context ...nd global consistency Relational consistency and the DRCm algorithms can also easily capture the tractable classes of acyclic and causal networks. It is well known that acyclic networks are tractable =-=[19, 8]-=-. Lemma 35. If a network is acyclic then there exists an ordering of the variables for which each bucket has a single relation. Single-bucket networks contain the class of acyclic networks and causal ... |

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Citation Context ... search to nd a solution, such an inconsistency can lead to a dead end in the search. This insight has led to the de nition of conditions that characterize the level of local consistency of a network =-=[17, 20]-=- and to the development of algorithms for enforcing local consistency conditions by removing local inconsistencies (e.g., [3, 7, 11, 17, 20]). In this paper, we present a new de nition of local consis... |

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Citation Context ...ditions that characterize the level of local consistency of a network [17, 20] and to the development of algorithms for enforcing local consistency conditions by removing local inconsistencies (e.g., =-=[3, 7, 11, 17, 20]-=-). In this paper, we present a new de nition of local consistency called relational consistency, introduced recently [24]. We show the usefulness and conceptualspower of the new de nition in generaliz... |

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Citation Context ...ses are a restricted and discrete version of Dechter, Meiri, and Pearl's [6] continuous, bounded di erence framework for temporal reasoning and a restricted and discrete version of Vilain and Kautz's =-=[27]-=- qualitative framework for temporal reasoning. Another known class that can be shown to be row-convex is implicational constraints [15]. For more details see [25]. 5.3 Constraint tightness and global ... |

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Citation Context ...elational calculus and algebra can be developed without the niteness assumptions for relations. We will use the term unrestricted relation, for nite or in nite sets of points in a k-dimensional space =-=[24]-=-. In particular, it was shown that relational calculus is identical to relational algebra for countable domains and that relational algebra for in nite relations is exactly the same as for nite relati... |

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Citation Context ...lent to applying extended 2-composition between each relation and the singleton relation. This special case-handling for instantiation exploits the computational e ect of conditioning as described in =-=[8, 15]-=-. In step 7 of the algorithm, if the size of S j t=1 St,fxpg is smaller than i, we apply the operation to A = S j t=1 St,fxpg. In step 8, if more than one relation is recorded on the same subset of va... |

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Citation Context ...closure ofRwith respect to extended m-composition. Algorithm RCm is clearly computationally expensive though it can be improved in a manner parallel to the improvements of path-consistency algorithms =-=[18]-=-. Such improvements are not of much interest since enforcing relational consistency is likely to remain exponential for m 3, unless the constraints aresbinary. We will see that even for m =3,RC3 solve... |

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Citation Context ...ll its treeembeddings.sIt is known that nding the minimal width embedding is NP-complete [1], nevertheless every ordering of the variables o, yields a simple to compute upper bound denoted w (o) (see =-=[8]-=-). The complexity ofDRCm along o can be bounded as a function of w (o) of its constraint graph. Speci cally [8], Theorem 16. The time complexity and size of the network generated byDRCm along ordering... |

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Citation Context ...ditions that characterize the level of local consistency of a network [17, 20] and to the development of algorithms for enforcing local consistency conditions by removing local inconsistencies (e.g., =-=[3, 7, 11, 17, 20]-=-). In this paper, we present a new de nition of local consistency called relational consistency, introduced recently [24]. We show the usefulness and conceptualspower of the new de nition in generaliz... |

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Citation Context ...s ed. A consistent instantiation is also called a solution. The order of the variables constrained by a relation is not important; that is, we follow the set-of-mappings formulation of relations (see =-=[34]-=-). The notion of a consistent instantiation of a subset of the variables can be de ned in several 4 Note that all the de nitions and algorithms are applicable to relations without the niteness assumpt... |

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Citation Context ...plexity ofDirectional-Relational-Consistency can be bounded as a function of the topological structure of the problem via parameters like the induced width of the graph [11], also known as tree-width =-=[1, 2]-=-. De nition17 (width, tree-width). A constraint networkRcan be associated with a constraint graph, where each nodeisavariable and two variables that appear in one constraint are connected. A general g... |

203 | A Generic Arc-Consistency Algorithm and its Specializations
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Citation Context ... is globally consistent. These results were extended recently by van Beek and Dechter to the property of tightness and looseness of the constraint networks [24, 23]. VansHentenryck, Deville, and Teng =-=[26]-=- show that arc consistency is su cient to test whether a network is satis able if the relations are from a restricted class of functional and monotone constraints. These properties were generalized re... |

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Citation Context ...e the level of local consistency is adjusted on a node-by-node basis. Dechter and Pearl [8] generalize the results on trees to hyper-trees which are called acyclic databases in the database community =-=[2]-=-. For work on identifying properties of the constraints, Montanari [20] shows that path consistency is su cient to guarantee that a binary network is globally consistent if the relations are monotone.... |

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Citation Context ...thms we present in this paper belong to the class of variable elimination algorithms, formulated recently within the bucket elimination framework [15], which generalize non-serial dynamic programming =-=[5]-=-. We have recently shown how a collection of probabilistic and combinatorial optimization tasks can be formulated within this framework [15]. Such algorithms were also presented for various graph-base... |

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Citation Context ...ecently shown how a collection of probabilistic and combinatorial optimization tasks can be formulated within this framework [15]. Such algorithms were also presented for various graph-based tasks by =-=[1, 3]-=-. All the algorithms possess similar properties of compiling a theory into a backtrack-free one (or greedy) and their complexityis dependent on the same graph properties. Speci cally they all have a c... |

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Citation Context ... 24. The level of required consistency is equal only when m = d,1 and is less when m<d, 1. As well, the theorem can sometimes be usefully applied if d n,1, whereas Theorem 24 cannot. Example 6. Nadel =-=[21]-=- introduces a variant ofthen-queens problem called confused n-queens and uses it to empirically compare backtracking algorithms. In [24], we use Theorem 31 to show that these problems are quite easy (... |

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Citation Context ...lobally consistent networks have the property that any consistent instantiation of a subset of the variables can be extended to a consistent instantiation of all of the variables without backtracking =-=[5]-=-. It is frequently enough to have a globally consistent network along a single ordering of the variables as long as this ordering is known in advance. De nition10 globally solved. Wesay that a problem... |

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Citation Context ...ying: models(_) =ECQ(models( ); models( )). Incorporating resolution into DRC3 results in algorithm Directional Resolution which is the core of the well known Davis Putnam algorithm for satis ability =-=[4, 10]-=-. As is well known, and as will follow from our theory, the algorithm globally solves any CNF theory. Directional-Resolution ('; o) 1. Initialize: generate an ordered partition of clauses to buckets. ... |

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Citation Context ...ditions that characterize the level of local consistency of a network [17, 20] and to the development of algorithms for enforcing local consistency conditions by removing local inconsistencies (e.g., =-=[3, 7, 11, 17, 20]-=-). In this paper, we present a new de nition of local consistency called relational consistency, introduced recently [24]. We show the usefulness and conceptualspower of the new de nition in generaliz... |

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On Reading Sketch Maps
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Citation Context ...onsistent i it is relational (j; m)-consistent for every j i. Strong relational (n; m)-consistency is identical to relational m-consistency. The relational based de nition of arc-consistency given in =-=[30]-=- is identical to relational (1,1)-consistency. De nition12 (directional relational consistency). Given an ordering of the variables, o = x1; ...;xn, a network is m-directionally relationally consisten... |

63 |
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Citation Context ...s shown to have a polynomial implementation over the Rationals that uses a special data structure that bounds the number of inequalities over any pair of variables and leads to a polynomial algorithm =-=[18]-=-. Over the integers the binary linear problem is NP-complete [27]. A more restricted case of binary monotone inequalities of the form axi, bxj c, where a; b; c positive integers, was shown to be weakl... |

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Citation Context ...at uses a special data structure that bounds the number of inequalities over any pair of variables and leads to a polynomial algorithm [18]. Over the integers the binary linear problem is NP-complete =-=[27]-=-. A more restricted case of binary monotone inequalities of the form axi, bxj c, where a; b; c positive integers, was shown to be weakly NP-complete since there exists a pseudo-polynomial algorithm [1... |

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Citation Context ...de nes three properties of networks that characterize local consistency of networks: node, arc, and path consistency. Freuder [11] generalizes this to k-consistency. De nition9 k-consistency; Freuder =-=[11, 12]-=-. A network is k-consistent if and only if given any instantiation of any k,1 distinct variables satisfying all of the direct relations among those variables, there exists an instantiation of any kth ... |

50 | On the Minimality and Global Consistency of Row-Convex Constraint Networks
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Citation Context ...paper are domain size and row convexity. De nition5 (k-valued domains). Anetwork of constraints is k-valued if the domain sizes of all variables are bounded by k. De nition 6 (row convex constraints (=-=[38]-=-)). A binary constraint R on a set fx1�x2g of variables with associated domains D1 and D2, isrow convex if there exists an ordering of D2 such that, for every a1 2 D1, the set fx2 j (a1�x2) 2 Rg can b... |

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Citation Context ...f input relations, and t bounds the number of tuples in each relation. Example 3. Crossword puzzles have been used experimentally in evaluating backtracking algorithms for solving constraint networks =-=[13]-=-. We use an example puzzle to illustrate algorithm DRC3 (see Figure 1). One possible constraint network formulation of the problem is as follows: there is a variable for each square that can hold a ch... |

41 |
A Computing Procedure for Quanti cation Theory
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Citation Context ...ying: models(_) =ECQ(models( ); models( )). Incorporating resolution into DRC3 results in algorithm Directional Resolution which is the core of the well known Davis Putnam algorithm for satis ability =-=[4, 10]-=-. As is well known, and as will follow from our theory, the algorithm globally solves any CNF theory. Directional-Resolution ('; o) 1. Initialize: generate an ordered partition of clauses to buckets. ... |

37 |
Fast Parallel Constraint Satisfaction
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Citation Context ...ning and a restricted and discrete version of Vilain and Kautz's [27] qualitative framework for temporal reasoning. Another known class that can be shown to be row-convex is implicational constraints =-=[15]-=-. For more details see [25]. 5.3 Constraint tightness and global consistency For some networks, Theorem 24 is tight in that the level of local consistency speci ed by the theorem is really required (g... |

34 |
Complexity of nding embedding in a k-tree
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(Show Context)
Citation Context ...mplexity ofDirectional-Relational-Consistency can be bounded as a function of the topological structure of the problem via parameters like the induced width of the graph [7], also known as tree-width =-=[1]-=-. De nition15 width, tree-width. A constraint networkRcan be associated with a constraint graph, where each nodeisavariable and two variables that appear in one constraint are connected. A general gra... |

31 | Directed constraint networks: A relational framework for causal modeling
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Citation Context ...ricted class of functional and monotone constraints. These properties were generalized recently to the property ofrow-convexity [25]. Finally, for work that falls into both classes, Dechter and Pearl =-=[9]-=- present e ective procedures for determining whether a constraint network can be formulated as a causal theory and thus a solution can be found without backtracking. Whether a constraint network can b... |

21 | Constraint Query Algebras
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- 1997
(Show Context)
Citation Context ...; c positive integers, was shown to be weakly NP-complete since there exists a pseudo-polynomial algorithm [18]. A polynomial algorithm that globally solves the problem over the rationals is given in =-=[17]-=-. For bounded integer domains the general binary linear problem can be expressed in a relational form and since DRC2 is polynomial over binary constraints, the class can be solved in polynomial time r... |

20 | On the Minimality and Decomposability of Constraint Networks
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(Show Context)
Citation Context ...s we have seen, in this case, the operator of extended 3-composition takes the form of pair-wise resolution yielding algorithm directional resolution [10].s5.2 Row-convexity and global consistency In =-=[22]-=-, we have shown that: Theorem 27. [22] LetRbe apath-consistent binary constraint network. If there exists an ordering of the domains D1; ...;Dn ofRsuch that the relations are row convex, the network i... |

17 | Constraint tightness versus global consistency
- Beek, Dechter
- 1994
(Show Context)
Citation Context ... consistency conditions by removing local inconsistencies (e.g., [3, 7, 11, 17, 20]). In this paper, we present a new de nition of local consistency called relational consistency, introduced recently =-=[24]-=-. We show the usefulness and conceptualspower of the new de nition in generalizing results for binary networks to nonbinary networks on the relationships between four properties of constraints| row-co... |

16 |
On the Consistency of General Constraint-Satisfaction Problems
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(Show Context)
Citation Context ...m-consistency is identical (up to minor preprocessing) to variable-based m-consistency; otherwise the conditions are di erent. The virtue in our de nition (relative to the one based on the dual graph =-=[14]-=-), is that it can be incorporated naturally into algorithms for enforcing desired levels of relational m-consistency, it allows a simple generalization of local consistency relationships, and it uni e... |

16 |
Elements of relational database theory Handbook of theoretical computer science (vol. B): formal models and semantics:1073–1156
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- 1990
(Show Context)
Citation Context ... particular, it was shown that relational calculus is identical to relational algebra for countable domains and that relational algebra for in nite relations is exactly the same as for nite relations =-=[25]-=- 6 . Therefore, the relational framework we havepresented applies as is to in nite relations. In this section we will demonstrate the applicability of our results to the special case of linear inequal... |

10 |
On fourier's algorithm for linear constraints
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(Show Context)
Citation Context ...for partial containment. 2. Incorporating linear elimination into DRC3 results in algorithm Directional Linear Elimination (abbreviated DLE) which is the well known Fourier elimination algorithm (see =-=[16]-=-). It was shown that the algorithm decides the solvability of any set of linear inequalities over the Reals. Directional-Linear-Elimination ('; o) 1. Initialize: generate an ordered partition of the i... |

8 | On the inherent level of local consistency in constraint networks
- Beek
- 1994
(Show Context)
Citation Context ...l consistency su cient to ensure the network is globally consistent. These results were extended recently by van Beek and Dechter to the property of tightness and looseness of the constraint networks =-=[24, 23]-=-. VansHentenryck, Deville, and Teng [26] show that arc consistency is su cient to test whether a network is satis able if the relations are from a restricted class of functional and monotone constrain... |

5 |
The TheoryofRelational Databases
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(Show Context)
Citation Context ...ality and global consistency Relational consistency and the DRCm algorithms can also capture the tractable classes of acyclic and causal networks. It is well known that acyclic networks are tractable =-=[32, 12]-=-. De nition33 (acyclic networks). Anetwork of constraints is acyclic if it has achordal constraint graph and if each maximal clique is associated with a single constraint. It is easy to see that: Lemm... |

2 |
Enhancement schemes for constraint processing incorporating backjumping, learning, and cutset decomposition
- Dechter
- 1990
(Show Context)
Citation Context ...lent to applying extended 2-composition between each relation and the singleton relation. This special case-handling for instantiation exploits the computational e ect of conditioning as described in =-=[8, 15]-=-. In step 7 of the algorithm, if the size of S j t=1 St,fxpg is smaller than i, we apply the operation to A = S j t=1 St,fxpg. In step 8, if more than one relation is recorded on the same subset of va... |

1 |
Topological properties of time-space tradeo
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- 1996
(Show Context)
Citation Context ...an be incorporated into the bucketelimination scheme to allow trading space for time. The special case-handling of singleton values that we had introduced (i.e., instantiation) permits this extension =-=[16]-=- and will lead to similar time-space tradeo s. Since the algorithms may be quite time demanding, unless the problem is very sparse, practical considerations call for the use of approximations. Polynom... |

1 |
Characterizing tractable constraints Arti cial Intelligence 65
- Cooper, D, et al.
- 1994
(Show Context)
Citation Context ...nt to test whether a network is satis able if the relations are from a restricted class of functional and monotone constraints. These properties were generalized recently to implicational constraints =-=[26, 23]-=- and to row-convexity [38]. Finally, for work that falls into both classes, Dechter and Pearl [13] present e ective procedures for determining whether a constraint network can be formulated as a causa... |

1 | Topological properties of time-space tradeoff - Dechter - 1996 |