R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1):283--308, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is organized as follows. In section 2 we present the basic concepts used in the rest of the paper. In section 3 we introduce a family of encodings called the k AC encodings where k is a parameter. These encodings enable a large family of consistencies, the so called relational k arc consistency [DvB97] to be established by unit propagation on the SAT encoding. They work with any constraint arity. Section 4 focuses on binary networks, and shows that these encodings can also be used to establish any (i,j) consistency (another large family of consistencies [Fre85] We also show that unit ....

....premiss is falsi ed, or S = S[R] and then S[T ] is one of its support and belongs to C s conclusion. C is then satis ed since both premiss and conclusion are satis ed. 2 Unit propagation on the k AC Clauses corresponds exactly to enforcing relational k arc consistency. Relational arc consistency [DvB97] extends the concept of local consistency, which usually concerns variables, to constraints. A constraint is relationally arc consistent if any instantiation which is allowed on a subset of its variables extends to a consistent instantiation on the whole. Relational k arc consistency is the ....

R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1):283308, 1997.

....domains of the variables will be the available time intervals as stated in Table 1. Since two speeches cannot be held at the same time in the same conference room, the period for two di#erent speakers must be di#erent. The constraints for this scheduling problem thus become: 2, 5] [1, 6], alldifferent(x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) To find a solution to a model as in the previous example, a constraint solver essentially builds a search tree from all possible variable values. In general, finding a solution for such problems is NP complete, and this search tree can grow ....

....# n , alldifferent(x 1 , xn ) Our next goal is to find a consistency notion for the set of disequalities that is equivalent to the hyper arc consistency notion for the alldifferent constraint. Relational consistency can be used for this. Definition 6 (Relational (1, m) consistency, [6]) A set of constraints S = C 1 , Cm is relationally (1, m) consistent i# all domain values d D i of variables appearing in S, appear in a solution to the m constraints, evaluated simultaneously. A CSP P = X , is relationally (1, m) consistent i# every set of m constraints S ....

R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173:283--308, 1997.

....# n , alldifferent(x 1 , x n ) Our next goal is to find a consistency notion for the set of disequalities that is equivalent to the hyper arc consistency notion for the alldifferent constraint. Relational consistency can be used for this. DEFINITION 9 (Relational (1, m) consistency, [8]) A set of constraints S = C 1 , Cm is relationally (1, m) consistent i# all domain values d D i of variables appearing in S, appear in a solution to the m constraints, evaluated simultaneously. A CSP P = X , 1, m) consistent i# every set of m constraints S (1, ....

R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173:283--308, 1997.

....obtained, called kB consistency. kB consistency has been successfully applied to difficult nonlinear problems, such as a challenging transistor design application [Puget and van Hentenryck, 1998] The important advantage of kB over path consistency [Montanari, 1974] and its variants [Dechter and van Beek, 1996; Sam Haroud and Faltings, 1996] is that it does not require the use of the costly multi dimensional representations of the constraints needed in these methods. Let d be the number of elements in the widest initial interval, n be the number of variables, and m be the number of constraints; the ....

....encoded on a given number of bits. 2.1 Basic terminology Let be a finite set of variables. A point t is defined by a coordinate t x on each variable x, i.e. points are mappings from to D. A constraint c is a set of tuples. For readers more familiar with database theoretic concepts [Dechter and van Beek, 1996] , variables may also be thought of as field names, points as tuples, and constraints as relations or tables. Points may also be seen as logical valuations. A CSP is a set of constraints. The set of solutions to the set Sol(C) of the points which belong to all its constraints, i.e. Sol(C) ....

R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1), 1996.

....we have to consider each of the primitive QCPs: 0, 18] x y = sum) 0, 9] 9.k = k # ) 0, 9] sum k # = mod) Note that the term arc consistency is to be understood in the sense of consistent w.r.t. each constraint , a definition used for instance by [10], and hence somewhat di#erent from the more classical, binary definition. It is also possible to generalize strong consistencies, for instance the following relational consistency (other state of the art techniques could be considered) Definition 6. Quantified k relational consistency] Consider ....

R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1):283--308, 1997.

....their layout. The layout is given by the description of graphical objects and graphical constraints between the objects. In general, graphical constraints express relations between graphical objects concerning their positions and sizes. We use the notion of constraint satisfaction problem [DvB97] in order to de ne a set of constraint variables correlating with certain object properties (positions and sizes) and equations (graphical constraints) over the constraint variables. The problem is to nd a solution for all the constraints. For example, we can specify that a circle is to be ....

R. Dechter and P van Beek. Local and Global Relational Consistency. Theoretical Computer Science, 173:283-308, 1997.

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R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, pages 283--308, 1997.

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Rina Dechter and Peter van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1):283--308, 20 February 1997.

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R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, pages 283--308, 1997.

....domain of each other variable. Path consistency is achieved by introducing new constraints or nogoods which disallow certain pairs of values. Relational based consistency enforcing algorithms allow exible extensions of consistency algorithms that are constraint based rather than variable based [22]. Constraint propagation can be used as a CSP solution procedure. If a problem can be made k consistent, for all k from 1 to n 1, then solutions can easily be found in the transformed problem, without backtracking. However, enforcing k consistency requires in general exponential time and ....

R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, pages 283-308, 1997.

....do not violate any clause. Deciding if a theory is satisfiable is known to be NP complete [26] It can be shown that the join project operation used to process and eliminate a variable by adaptive consistency over relational constraints translates to pairwise resolution when applied to clauses [23]. This yields a bucket elimination 9 A D E w = 3 B C D E D C C D E Input B C A D Bucket Bucket Bucket E o Extension Directional E C A B B A B E C Figure 7: A schematic execution of directional resolution using ordering d = E; D;C;B;A algorithm for propositional ....

....is not bounded exponentially by the induced width, however. The reason is that the number of feasible linear inequalities that can be specified over a subset of i variables cannot be bounded exponentially by i. For a schematic execution of the algorithm see Figure 10, and for more details see [23]. 12 A B E A B C A B D D = 0 C = 0 0 0 1 C A 0 1 B 1 0 1 B 0 1 1 D C A 0 1 0 0 1 1 D 0 E on conditioning E 0 Search tree: Figure 11: The search tree of the graph coloring problem 2.4 Conditioning When a problem has a high induced width the bucket elimination ....

R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, pages 283--308, 1997.

....are quite promising. Our approach leaves plenty of room for future improvement, which are likely to make it more cost effective in practice. For instance, it can be modified to treat separately hard and soft constraints, since hard constraints can be more efficiently processed and propagated [10]. As a matter of fact, even if the original problem has no hard constraints, our approach can be used to infer them (i.e, detect infeasible tuples) A second line of improvement is to exploit lower bound redundancy. Namely, to use tree decompositions with several solutionvertices for each task. ....

R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1):283--308, 20 February 1997.

....are quite promising. Our approach leaves plenty of room for future improvement, which are likely to make it more cost effective in practice. For instance, it can be modified to treat separately hard and soft constraints, since hard constraints can be more efficiently processed and propagated [10]. As a matter of fact, even if the original problem has no hard constraints, our approach can be used to infer them (i.e, detect infeasible tuples) A second line of improvement is to exploit lower bound redundancy. Namely, to use tree decompositions with several solutionvertices for each task. ....

Rina Dechter and Peter van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1):283--308, 20 February 1997.

....are quite promising. Our approach leaves plenty of room for future improvement, which are likely to make it more cost effective in practice. For instance, it can be modified to treat separately hard and soft constraints, since hard constraints can be more efficiently processed and propagated [10]. As a matter of fact, even if the original problem has no hard constraints, our approach can be used to infer them (i.e, detect infeasible tuples) A second line of improvement is to exploit lower bound redundancy. Namely, to use tree decompositions with several solutionvertices for each task. ....

Rina Dechter and Peter van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1):283--308, 20 February 1997.

.... for probabilistic inference [7, 57, 55, 59] Our approach is inspired by adaptive consistency, a full bucket elimination algorithm whose approximation, directional i consistency and its relational variant directional relationalconsistency (i,m) DRC (i;m) enforce bounded levels of consistency [24]. For example, directional relational arc consistency, DRC 1 , is similar to mini bucket(m = 1) directional path consistency, DRC 2 , corresponds to mini bucket(m = 2) and so on. Note that mini bucket approximations can be used as heuristics for subsequent search, similar to pre processing by ....

R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, pages 283--308, 1997.

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R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1):283--308, 1997.

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R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173:283--308, 1997.

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R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1):283--308, 1997.

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R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1):283308, 1997.

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R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1):283--308, 1997.

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R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173:283--308, 1997.

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R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1):283308, 1997.

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R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1):283--308, 1997.

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R. Dechter & P. van Beek, Local and Global Relational Consistency,Theoretical Computer Science 173(1): 283--308, 1997.

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R. Dechter and P. van Beek. Local and Global Relational Consistency. Theoretical Computer Science, 173:283--308, 1997.

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