T. Schiex. Possibilistic constraint satisfaction problems, or \how to handle soft constraints?". Proc. 8th Conf. of Uncertainty in AI, pages 269-275, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... where one usually looks for an optimal solution, that is, a solution which has the highest preference level (according to the ordering of the preference levels) 3 For example, a typical soft constraint scenario, that we will use extensively in this paper, is the fuzzy constraint framework [8, 9, 6, 3], where preferences are real numbers between 0 and 1, with a higher number denoting a higher preference, and are combined via the minimum operator. Thus the preference value of a solution is the minimum among the preference values selected by this solution on the constraints, and the best ....

....problem to model preference based con gurators. In particular, we associate a preference to each cell of the board, and also to each pair of positions within an attack constraint. We use preference levels between 0 and 1, and we aim at maximizing the minimum preference, as in the fuzzy framework [8]. As usual, we have one variable for each row, with the n cells of the row as possible values, and thus constraints hold between two positions on the same column and on the same diagonal. Notice that this implies that there is no constraint between two positions on the same row. Preference ....

T. Schiex. Possibilistic constraint satisfaction problems, or \how to handle soft constraints?". In Proc. 8th Conf. of Uncertainty in AI, pages 269-275, 1992.

....problem is a pair hC; coni where con V and C is a set of constraints: con is the set of variables of interest for the constraint set C, which may also concern variables not in con. Note that a classical CSP is a SCSP with the c semi ring: SCSP = hffalse; trueg; false; truei. Fuzzy CSPs [6, 5] can be modeled in the SCSP framework by choosing the c semi ring SFCSP = h[0; 1] max; min; 0; 1i. Many other soft CSPs (probabilistic, weighted, can be modeled by using suitable semi rings (S prob = h[0; 1] max; 0; 1i, Sweight = hR; min; 0; 1i, Finding an optimal ....

....it onto an arbitrary ordering, since the choice is still left to the nal user. We might conclude therefore that the min model is to be preferred to the SLO model, as far as the approximation of the original model is concerned. However maximizing the minimum reward, as in any fuzzy framework [6, 5], has proved its usefulness in problem representation. In the end, we suggest choosing the semi ring considering the tradeo between linearization of the order and suitability of the representation provided. 6 Conclusions and future work We have proposed two remedies to the fact that consistency ....

T. Schiex. Possibilistic constraint satisfaction problems, or \How to handle soft constraints?", Proc. 8th Conf. of Uncertainty in AI, pp. 269-275, 1992. 13

....that c 1 v S c 2 . De nition 3 (soft constraint problem) A soft constraint satisfaction problem (SCSP) is a pair hC; coni where con V and C is a set of constraints. Note that a classical CSP is a SCSP where the chosen c semiring is: S CSP = hffalse; trueg; false; truei: Fuzzy CSPs [11,28,29] can instead be modeled in the SCSP framework by choosing the c semiring: S FCSP = h[0; 1] max; min; 0; 1i: Example 4 Figure 1 shows a fuzzy CSP. Variables are inside circles, constraints are represented by undirected arcs, and semiring values are written to the right of the corresponding ....

T. Schiex. Possibilistic constraint satisfaction problems, or \how to handle soft constraints?". In Proc. 8th Conf. of Uncertainty in AI, pages 269-275, 1992.

....of interest for the constraint set C, which however may concern also variables not in con. Figure 2 pictures a soft CSP, with the semiring values written to the right of the corresponding tuples, obtained from the classical one represented in gure 1 by using the fuzzy c semiring [DFP93,Rut94,Sch92] SFCSP = h[0; 1] max; min; 0; 1i: a 0.9 b 0.1 b, b 0 b, a 0 a, b 0.2 a, a 0.8 a 0.9 b 0.5 X Y Fig. 2. A fuzzy CSP Combining and projecting soft constraints Given two constraints c 1 = hdef 1 ; con 1 i and c 2 = hdef 2 ; con 2 i, their ....

T. Schiex. Possibilistic Constraint Satisfaction Problems, or \How to Handle Soft Constraints?". In Proc. of 8th Conference on Uncertainty in AI, pages 269-275, 1992.

....have that c 1 vS c 2 . De nition 3 (soft constraint problem) A soft constraint satisfaction problem (SCSP) is a pair hC; coni where con V and C is a set of constraints. Note that a classical CSP is a SCSP where the chosen c semiring is: SCSP = hffalse; trueg; false; truei: Fuzzy CSPs [8, 19, 20] can instead be modeled in the SCSP framework by choosing the c semiring: SFCSP = h[0; 1] max; min; 0; 1i: Figure 1 shows a fuzzy CSP. Variables are inside circles, constraints are represented by undirected arcs, and semiring values are written to the right of the corresponding tuples. Here we ....

T. Schiex. Possibilistic constraint satisfaction problems, or \how to handle soft constraints?". In Proc. 8th Conf. of Uncertainty in AI, pages 269-275, 1992.

....of feasible solution in problems, where all constraints can not be satis Thetaed. We have de Thetaned mappings of variables annotations to different frameworks [Rud98b, Rud98c, Rud98a] for solving over constrained problems # to constraint hierarchies [BFBW92, WB93] and possibilistic CSPs [Sch92] Particular mappings give us several interpretations of annotations and they may be used as examples of possible semantics of annotations in overconstrained problems. This work applies the same preferences as above for solving constraint satisfaction problems with optimizations where the whole ....

....[FW92] seeks a solution that satis Thetaes as many constraints as possible. Weighted constraint satisfaction considers weights for each constraint and minimizes weighted sum of unsatis Thetaed constraints. Both of these systems are used for solving over constrained problems. Possibilistic CSP [Sch92] assigns to each constraint some preference degree, which express necessity of its satisfaction. Fuzzy CSP [DFP96] considers constraint as a relation assigning to each tuple of values its level of preferences. Preference degrees in both approaches are combined with help of fuzzy sets, possibility ....

Thomas Schiex. Possibilistic constraint satisfaction problems or #How to handle soft constraints ?#. In 8 th International Conference on Uncertainty in Arti\Thetacial Intelligence, pages 268#275, Stanford, CA, July 1992.

....constraints the flexibility lies in the ability to discard constraints involved in inconsistencies, provided that they are not too important. Generally, a weight is associated with each constraint and the request is to minimize the greatest priority levels of the violated constraints [10][11]. More generally, Brewka et al. 12] and Borning et al. 13] identify different forms of constraint relaxation, viewing each constraint as a strict partial order on value assignment and weighting the importance of constraints; in particular, Brewka et al. provide a formal semantics in relation to ....

....in a classical CSP. As a general model based on possibility theory, the FCSP approach generalizes the frameworks that model softness by means of fuzzy sets [4] 7] 8] as well as those dealing with constraint priorities by searching to minimize the priority of the violated constraints [10] 7] 13] [11]. More precisely, some of them use an inclusion based refinement of the min induced ordering [13] or a lexicographic refinement [10] 7] which is itself a refinement of the inclusion based ordering. See [33] 34] for a discussion on the selection of preferred solutions in FCSP by means of these ....

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T. Schiex, "Possibilistic constraint satisfaction problems or how to handle soft constraints," in Proc. of the 8th Conf. on Uncertainty in Artificial Intelligence, Stanford, CA, July 17-19, 1992, edited by D. Dubois, M.P. Wellman, B. D'Ambrosio and P. Smets, Morgan & Kaufmann: San Mateo, CA, pp. 268-275, 1992.

....that C 0 and C 00 are two subsets of C, then the satisfaction of C 0 is preferred to be satisfied rather than C 00 if and only if C 0 C 00 . Such a preference ordering generally depends on some attributes of the constraints, e.g. a weight [Freuder and Wallace, 1992] or a priority [Schiex, 1992, Dubois et al. 1994] As an example assume for each constraint c a weight (c) is given. Then a preference ordering defines an instance of the well known partial constraint satisfaction problems (PCSP) if C 0 C 00 iff X c 0 2C (c) X c 00 2C 00 (c 00 ) holds. Such ....

....Hierarchies. ECAI 96 Workshop 4 1. Expert specifications form the most important level comprising soft constraints, because an expert should be able to rule everything. Preferences may be given as a partial ordering among constraints c . 2. Prioritized or possibilistic constraints [Schiex, 1992, Dubois et al. 1994] may be appropriate to represent defaults. 3. Cost optimization can be specified by weighted constraints . The configuration system shall perform as follows: Interaction: After each refinement step the expert is informed about occurring conflicts like this: the last ....

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Thomas Schiex. Possibilistic constraint satisfaction problems or: How to handle soft constraints? In Proceedings of the Eighth Conference on Uncertainty in AI, pages 269--275, Stanford (CA), USA, 1992.

.... may be obtained by determining the mentioned properties in a similar way to Semiring based CSP [BMR97a, BMR97b] and Valued CSP [SFV95] both of these frameworks are introduced and compared in [BFM 96] In the following, we describe a mapping of annotations to the possibilistic CSP [DFP96, Sch92] and to the hierarchical CSP [BFBW92, WB93] These mappings may be used as examples of possible semantics of variables annotations. 3.1 Possibilistic System The interpretation of annotations through possibilistic CSP emphasizes global variables annotations whereas the importance of particular ....

Thomas Schiex. Possibilistic constraint satisfaction problems or #How to handle soft constraints ?#. In 8 th International Conference on Uncertainty in Arti\Thetacial Intelligence, pages 268#275, Stanford, CA, July 1992.

....For each value d in a variable x, this information on con icts with fuzzy constraints can be recorded by a fuzzy set h ; Ci in the set of all constraints C, where (c) is the estimate on the compliance of x d with fuzzy constraint c 2 C. The previously proposed frameworks of fuzzy constraints [16, 4, 5, 9] employ the idempotent max operator for combination. This restriction allows the application of AC3 like algorithms for maintaining consistent domains [19] but leads to the drowning e ect . In contrast, FHCSPs deploy the operator on R n which is not idempotent. Consequently, the propagation ....

Thomas Schiex. Possibilistic constraint satisfaction problems or: How to handle soft constraints? In Proceedings of the Eighth Conference on Uncertainty in AI, pages 269-275, Stanford (CA), USA, 1992.

....ConPlan project considered assumptions only in this limited way. This made the representation of optional requirements easier but not simple. Generally, several approaches have been proposed to specify the merit of labelings by soft constraints: ffl prioritized resp. possibilistic constraints [ Schiex, 1992 ] ffl weighted constraints (PCSP) Freuder, 1989, Freuder and Wallace, 1992 ] ffl constraint hierarchies [ Borning et al. 1989, Wilson, 1993 ] ffl fuzzy constraints [ Guesgen, 1994, Dubois et al. 1993 ] Each of these approaches has its own characteristics and, thus, can be ....

....can be found without any backtracking by assigning only labels of maximal compatibility. Otherwise, locally consistent compatibilities can be exploited to improve the assign best first heuristic. 4 The term bound has also been used to denote consistent compatibilities [Snow and Freuder, 1990, Schiex, 1992]. Locally consistent compatibilities can be considered as an optimistic estimation of the most important priority level, that can be satisfied completely. Thus, it is possible to compute them before starting the searching algorithm, and exploit them in conjunction with forward checking results. ....

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T. Schiex. Possibilistic constraint satisfaction problems or: How to handle soft constraints ? In Proceedings of the Eighth Conference on Uncertainty in AI, pages 269--275, Stanford (CA), USA, 1992.

....deal with soft constraints is to generalise the notion of crisp constraint. A crisp constraint is characterised by the set of tuples for which the constraint holds. Hence, it is a natural extension that a fuzzy set characterises a fuzzy constraint. This possibility has recently received attention [3, 4, 11, 12, 17]. In the case of a fuzzy constraint, different tuples satisfy the given constraint to a different degree. We find the fuzzy set theoretical approach to deal with soft constraints appealing for two reasons. From a technical point of view, the existing concepts and techniques of fuzzy set theory can ....

Schiex, T.: Possibilistic constraint satisfaction problems or how to handle soft constraints, Proc. of the 8th Con. on Uncertainity in AI, Stanford, 1992. pp 268-275.

....with the treatment of certainty qualified assertions in possibility theory (Dubois and Prade [15] and the principle of minimum specificity, interpreting priority levels as necessity degrees. When constraints are crisp, they are in full agreement with possibilistic logic (Lang [25] Schiex [31]; Dubois, Lang and Prade [13] This treatment of prioritized constraints relies on the assumption that it is not possible to fully violate a constraint that is not imperative. As a consequence priority weights can only decrease the importance of constraints, since, when no weight is attached, ....

Schiex T. Possibilistic constraint satisfaction problems or how to handle soft constraints. Proc. of the 8th Conf. on Uncertainty in Artificial Intelligence, Stanford, July 17-19 (1992) 268-275.

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T. Schiex. Possibilistic constraint satisfaction problems, or \how to handle soft constraints?". Proc. 8th Conf. of Uncertainty in AI, pages 269-275, 1992.

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T. Schiex. Possibilistic constraint satisfaction problems, or \how to handle soft constraints?". In Proc. 8th Conf. of Uncertainty in AI, pages 269-275, 1992.

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T. Schiex. Possibilistic constraint satisfaction problems, or \how to handle soft constraints?". In Proc. 8th Conf. of Uncertainty in AI, pages 269-275, 1992.

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T. Schiex. Possibilistic constraint satisfaction problems, or how to handle soft constraints. In Proceedings of the 14th Conference on Uncertainty in Artificial Intelligence, pages 268--275, 1992.

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T. Schiex. Possibilistic constraint satisfaction problems, or \how to handle soft constraints?". In Proc. 8th Conf. of Uncertainty in AI, pages 269-275, 1992.

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T. Schiex. Possibilistic constraint satisfaction problems, or how to handle soft constraints. Proceedings of the Eighth Conference on Uncertainty in Arti cial Intelligence, pages 268-275, 1992.

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T. Schiex. Possibilistic constraint satisfaction problems, or \how to handle soft constraints?". Proc. 8th Conf. of Uncertainty in AI, pages 269-275, 1992.

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