## Possibilistic Constraint Satisfaction Problems or "How to handle soft constraints ?" (1992)

Venue: | In Proc. 8th Conf. of Uncertainty in AI |

Citations: | 85 - 8 self |

### BibTeX

@INPROCEEDINGS{Schiex92possibilisticconstraint,

author = {Thomas Schiex},

title = {Possibilistic Constraint Satisfaction Problems or "How to handle soft constraints ?"},

booktitle = {In Proc. 8th Conf. of Uncertainty in AI},

year = {1992},

pages = {269--275}

}

### Years of Citing Articles

### OpenURL

### Abstract

Many AI synthesis problems such as planning or scheduling may be modelized as constraint satisfaction problems (CSP). A CSP is typically defined as the problem of finding any consistent labeling for a fixed set of variables satisfying all given constraints between these variables. However, for many real tasks such as job-shop scheduling, time-table scheduling, design..., all these constraints have not the same significance and have not to be necessarily satisfied. A first distinction can be made between hard constraints, which every solution should satisfy and soft constraints, whose satisfaction has not to be certain. In this paper, we formalize the notion of possibilistic constraint satisfaction problems that allows the modeling of uncertainly satisfied constraints. We use a possibility distribution over labelings to represent respective possibilities of each labeling. Necessity-valued constraints allow a simple expression of the respective certainty degrees of each constraint. The main advantage of our approach is its integration in the CSP technical framework. Most classical techniques, such as Backtracking (BT), arc-consistency enforcing (AC) or Forward Checking have been extended to handle possibilistics CSP and are effectively implemented. The utility of our approach is demonstrated on a simple design problem.

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Citation Context ... define "best" labelings. The next section rapidly presents algorithmic issues for possibilistic CSP solving and shows how specific satisfaction (Backtrack) and consistency enforcing (Arc-co=-=nsistency [18]-=-) techniques may be built, taking into account the induced possibility distribution. In section 3, we give an example of application of possibilistic CSP to a simple design problem. Both representatio... |

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Citation Context ...ut its efficiency is yet to be evaluated (it is well known that horizontal ordering has a strong influence on the efficiency of Min-Max problems solving, e.g. in Alpha-Beta algorithm applied to games =-=[24]-=-). 2.4.3 Consistency enforcing A further step is to extend the various local consistency notions (node, arc, path and k consistency [18,21]) and their corresponding filtering algorithms[20,6]. A class... |

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Citation Context ...onstraints leading to "efficient" solving techniques. Non standard logics are manifold that allows the expression of probabilities [22], or preferences [27]. In particular, Zadeh's possibili=-=ty theory [28] has -=-already been successfully used for modeling uncertainty and preferences in the frame of propositional and first-order logic by Dubois, Prade and Lang leading to the so-called "possibilistic logic... |

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Citation Context ...problems solving, e.g. in Alpha-Beta algorithm applied to games [24]). 2.4.3 Consistency enforcing A further step is to extend the various local consistency notions (node, arc, path and k consistency =-=[18,21]-=-) and their corresponding filtering algorithms[20,6]. A classical n-ary CSP is said to be arc-consistent iff for every domain-variable j, the domain d j is not empty, and for every labels2 d j , for e... |

443 | Partial constraint satisfaction
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Citation Context ...@cert.fr or schiex@irit.fr give a "general" theoretical framework for expressing soft constraints (such approaches may be found in [26] using first and second order logic to express preferen=-=ces or in [12], relying -=-on a problem space and a general measure on this space), but to give a specific (and hopefully useful) meaning to such constraints leading to "efficient" solving techniques. Non standard log... |

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Citation Context ...ace), but to give a specific (and hopefully useful) meaning to such constraints leading to "efficient" solving techniques. Non standard logics are manifold that allows the expression of prob=-=abilities [22]-=-, or preferences [27]. In particular, Zadeh's possibility theory [28] has already been successfully used for modeling uncertainty and preferences in the frame of propositional and first-order logic by... |

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Citation Context ...pecific (and hopefully useful) meaning to such constraints leading to "efficient" solving techniques. Non standard logics are manifold that allows the expression of probabilities [22], or pr=-=eferences [27]-=-. In particular, Zadeh's possibility theory [28] has already been successfully used for modeling uncertainty and preferences in the frame of propositional and first-order logic by Dubois, Prade and La... |

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Citation Context ...lassical CSP ; (k; 0) is totally useless as it expresses that the necessity measure of k should be at least 0, which is always true. 1 Vague relations seen as a fuzzy set of authaurized labeling. See =-=[19,9,25]-=-. 2 In fact, such a predicate may be decomposed in a set of crisp predicates (ff-cuts of the vague constraint). In our case, the domains being finite, the set of ff-cuts is finite and a given fuzzy co... |

72 |
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Citation Context ...; The integration of fuzzy constraints (defined as a fuzzy set of authorized labelings) is almost immediate and leads to an even greater expressive power. As is has been shown for possibilistic logic =-=[10], the pre--=-order induced by necessity-valued constraints is a numerical "epistemic entrenchment" relation [13]. The consistency degree of a possibilistic CSP may be considered as an indicator of the co... |

12 |
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Citation Context ...ral extension of possibilistic CSPs may be considered : Many CSP techniques (AC-n, path or k-consistency, backjumping, learning, tree clustering, cycle cutset) and useful properties (Freuder theorems =-=[11]-=-) should be adapted or extended to possibilistic CSP ; The integration of fuzzy constraints (defined as a fuzzy set of authorized labelings) is almost immediate and leads to an even greater expressive... |

12 |
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Citation Context ...atisfaction problems may be solved. Our aim is not to e-mail : schiex@cert.fr or schiex@irit.fr give a "general" theoretical framework for expressing soft constraints (such approaches may be=-= found in [26]-=- using first and second order logic to express preferences or in [12], relying on a problem space and a general measure on this space), but to give a specific (and hopefully useful) meaning to such co... |

10 |
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Citation Context ...lassical CSP ; (k; 0) is totally useless as it expresses that the necessity measure of k should be at least 0, which is always true. 1 Vague relations seen as a fuzzy set of authaurized labeling. See =-=[19,9,25]-=-. 2 In fact, such a predicate may be decomposed in a set of crisp predicates (ff-cuts of the vague constraint). In our case, the domains being finite, the set of ff-cuts is finite and a given fuzzy co... |

9 |
Possibilistic logic as a logical framework for min-max discrete optimization problems and prioritized constraints
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(Show Context)
Citation Context ... done in the AC-5 [6] algorithm in classical CSP. 4 Related works The obviously related work is "possibilistic logic" [17] which has been a fundamental basis for possibilistic CSP definition=-=. J. Lang [16]-=- has applied propositionalpossibilistic logic to constraint satisfaction problems. In our opinion, our approach offers greater expressive power (let us recall that the encoding of the SEND+MORE=MONEY ... |

8 |
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Citation Context ...as already been successfully used for modeling uncertainty and preferences in the frame of propositional and first-order logic by Dubois, Prade and Lang leading to the so-called "possibilistic lo=-=gic" [17]. One of t-=-he desirable feature of possibilistic semantics is the tolerance to "partial" consistency, which allows a sort of paraconsistent reasoning. Another interesting feature of possibilistic logic... |

7 |
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Citation Context ...auses and 88 propositional variables) and more varied and powerful techniques (the only resolution technique used in propositional possibilistic logic being essentially a "backjumpinglike " =-=algorithm [23,14]-=-). Other related works include Hierarchical Constraint Logic Programming [3] that allows the expression of prioritized constraints in the body of an Horn clause. Satoh [26] proposes a formalization of... |

4 |
et Prade (H.). -- A class of fuzzy measures based on triangular norms. a general framework for the combination of uncertain information
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(Show Context)
Citation Context ...patibility, or by complete redefinition of the measure used. However, algorithmic issues will have to be reconsidered. Possibility and necessity measures may be seen as specific decomposable measures =-=[8]-=-. We think that most of this work could be easily extended to such measures (including probabilistic measures). Algorithmic issues will again have to be reconsidered. Possibilistic logic programming a... |

2 |
et Prade (H.). -- Towards possibilistic logic programming
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(Show Context)
Citation Context ... this work could be easily extended to such measures (including probabilistic measures). Algorithmic issues will again have to be reconsidered. Possibilistic logic programming as been experimented in =-=[7]-=-. The integration of Possibilistic logic programming and possibilistic CSP is a first step toward Possibilistic Constraint Logic Programming. Acknowledgements The author wants to thank Helene Fargier,... |

1 |
Bensana (E.) et Dubois (D.). -- Opal : a multi-knowledge based system for job shop scheduling
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(Show Context)
Citation Context ...me, relies on a possibility distribution over labelings. Let LV be the finite set of all possible labeling of the domain-variables in V . A possibility distribution on L V is a function �� from LV=-= to [0; 1]. �� is said to be n-=-ormalizedsif and only if 9l 2 L V such that ��(l) = 1. We define the sub-normalization degree of �� as the quantity SN(��) = 1 \Gamma max(f��(l)=l 2 Lg). Obviously, SN(��) = 0 iff ... |

1 |
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Citation Context ...more oriented towards production rules is very close to ours as it compute a solution that is the best compromise under a set of antagonist constraints. It is also close to the OPAL scheduling system =-=[2,1]-=- which has been extended to take in account fuzzy antagonist temporal constraints. 5 Further researches We are currently working on the conversion of the possibilistic AC1 like algorithm to more sophi... |

1 |
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Citation Context ... that the satisfaction of k is preferred to the satisfaction of k 0 . Therefore, possibilistic CSP are closely related to Hierarchical CSP as described in the frame of Constraint Logic Programming in =-=[3]-=-. 2.3 Possibilistic CSP : definition and semantics The only difference between a classical and a possibilistic CSP is the introduction of necessity-valued constraint instead of simple constraint. A ne... |

1 |
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(Show Context)
Citation Context ... Alternatively, one may stop the algorithm execution upon any event (time exhausted,: : : ) and get the best labeling found up to the occurrence of the event (getting closer to an "anytime algori=-=thm" [4]-=-). Every usual vertical heuristic ordering (max. cardinality, max. degree,: : : ), may be applied to this tree search. An horizontal heuristic is given by the current bounds obtained for the various l... |

1 |
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(Show Context)
Citation Context ...zed constraints in the body of an Horn clause. Satoh [26] proposes a formalization of soft constraints based on an interpretations ordering but does not provide any algorithmic issue. The system GARI =-=[5]-=- which is more oriented towards production rules is very close to ours as it compute a solution that is the best compromise under a set of antagonist constraints. It is also close to the OPAL scheduli... |

1 |
et van Hentenryck (Pascal). -- An efficient arc consistency algorithm for a class of csp problems
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(Show Context)
Citation Context ...d to games [24]). 2.4.3 Consistency enforcing A further step is to extend the various local consistency notions (node, arc, path and k consistency [18,21]) and their corresponding filtering algorithms=-=[20,6]-=-. A classical n-ary CSP is said to be arc-consistent iff for every domain-variable j, the domain d j is not empty, and for every labels2 d j , for every constraint k i such that j 2 V i , there is a l... |

1 |
et Makinson (D.). -- Revision of knowledge systems using epistemic entrenchment
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(Show Context)
Citation Context ...ich allows a sort of paraconsistent reasoning. Another interesting feature of possibilistic logic is the close relationships between necessity measures and Gardenfors "epistemic entrechment"=-= relation [13]. The main-=- idea is to encapsulate preferences (or respective certainty degree) among labelings in a "possibility distribution " over labelings. Such a distribution naturally induces two (possibility a... |

1 |
et Hertzberg (Joachim). -- Some fundamental properties of local constraint propagation
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(Show Context)
Citation Context ...ossibilistic AC1 like algorithm to more sophisticated schemes as AC4 [20]. A matter of study is also the fix-point semantics of the possibilistic arc-consistency as is has been done for classical CSP =-=[15]-=-. Several extension of possibilistic CSPs may be considered : Many CSP techniques (AC-n, path or k-consistency, backjumping, learning, tree clustering, cycle cutset) and useful properties (Freuder the... |

1 |
Robert A.) et Zucker (Steven W.). -- Scene labeling by relaxation operations
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(Show Context)
Citation Context ...lassical CSP ; (k; 0) is totally useless as it expresses that the necessity measure of k should be at least 0, which is always true. 1 Vague relations seen as a fuzzy set of authaurized labeling. See =-=[19,9,25]-=-. 2 In fact, such a predicate may be decomposed in a set of crisp predicates (ff-cuts of the vague constraint). In our case, the domains being finite, the set of ff-cuts is finite and a given fuzzy co... |