tschiexOtoulouse.inra.fr fargierOirit.fr verfailOcert.fr In order to deal with over-constrained Constraint Satisfaction Problems, various extensions of the CSP framework have been considered by taking into account costs, uncertainties, preferences, priorities...Each extension uses a specific mathematical operator (+, max...) to aggregate constraint violations. In this paper, we consider a simple algebraic framework, related to Partial Constraint Satisfaction, which subsumes most of these proposals and use it to characterize existing proposals in terms of rationality and computational complexity. We exhibit simple relationships between these proposals, try to

We introduce a general framework for constraint solving where classical CSPs, fuzzy CSPs, weighted CSPs, partial constraint satisfaction, and others can be easily cast. The framework is based on a semiring structure, where the set of the semiring specifies the values to be associated to each tuple of values of the variable domain, and the two semiring operations (+ and x) model constraint projection and combination respectively. Local consistency algorithms, as usually used for classical CSPs, can be exploited in this general framework as well, provided that some conditions on the semiring operations are satisfied. We then show how this framework can be used to model both old and new constraint solving schemes, thus allowing one both to formally justify many informally taken choices in existing schemes, and to prove that the local consistency techniques can be used also in newly defined schemes. 1

In this paper we describe and compare two frameworks for constraint solving where classical CSPs, fuzzy CSPs, weighted CSPs, partial constraint satisfaction, and others can be easily cast. One is based on a semiring, and the other one on a totally ordered commutative monoid. While comparing the two approaches, we show how to pass from one to the other one, and we discuss when this is possible. The two frameworks have been independently introduced in [2], [3] and [35].

In this paper the issue of soft constraint satisfaction is discussed from a fuzzy set theoretical point of view. A fuzzy constraint is considered as a fuzzy relation. Different possible definitions for the degree of joint satisfaction of a set of fuzzy constraints are given, covering specific other soft constraint satisfaction problem (CSP) types such a partial and hierarchical CSP. It is shown that the classical CSP solving heuristics based on variable and value evaluations can be generalised and used to guide the solution construction process for solving fuzzy CSPs, and that the heuristic search can be replaced by branch-and-bound search. The solution process is illustrated with an example from the CSP literature. Finally, research issues are discussed. 1. Introduction In the recent years there has been a growing interest in soft constraint satisfaction. In general, in a soft CSP not all the given constraints need to be satisfied --- either because all of them cannot be met, theoret...

In classical Constraint Satisfaction Problems (CSPs) knowledge is embedded in a set of hard constraints, each one restricting the possible values of a set of variables. However constraints in real world problems are seldom hard, and CSP's are often idealizations that do not account for the preference among feasible solutions. Moreover some constraints may have priority over others. Lastly, constraints may involve uncertain parameters. This paper advocates the use of fuzzy sets and possibility theory as a realistic approach for the representation of these three aspects. Fuzzy constraints encompass both preference relations among possible instanciations and priorities among constraints. In a Fuzzy Constraint Satisfaction Problem (FCSP), a constraint is satisfied to a degree (rather than satisfied or not satisfied) and the acceptability of a potential solution becomes a gradual notion. Even if the FCSP is partially inconsistent, best instanciations are provided owing to the relaxation of ...

. Soft constraints extend classical constraints to represent multiple consistency levels, and thus provide a way to express preferences, fuzziness, and uncertainty. While there are many soft constraint solving algorithms, even distributed ones, by now there seems to be no concurrent programming framework where soft constraints can be handled. In this paper we show how the classical concurrent constraint (cc) programming framework can work with soft constraints, and we also propose an extension of cc languages which can use soft constraints to prune and direct the search for a solution. We believe that this new programming paradigm, called soft cc (scc), can be very useful in many webrelated scenarios. In fact, the language level allows web agents to express their interaction and negotiation protocols, and also to post their requests in terms of preferences, and the underlying soft constraint solver can nd an agreement among the agents even if their requests are incompatible. 1

: This paper proposes an extension of the constraint-based approach to job-shop scheduling, that accounts for the flexibility of temporal constraints and the uncertainty of operation durations. The set of solutions to a problem is viewed as a fuzzy set whose membership function reflects preference. This membership function is obtained by an egalitarist aggregation of local constraint-satisfaction levels. Uncertainty is qualitatively described is terms of possibility distributions. The paper formulates a simple mathematical model of jobshop scheduling under preference and uncertainty, relating it to the formal framework of constraint-satisfaction problems in Artificial Intelligence. A combinatorial search method that solves the problem is outlined, including fuzzy extensions of well-known look-ahead schemes. 1. Introduction There are traditionally three kinds of approaches to jobshop scheduling problems: priority rules, combinatorial optimization and constraint analysis. The first kind ...

Many AI tasks, such as product configuration, decision support, and the construction of autonomous agents, involve a process of constrained optimization, that is, optimization of behavior or choices subject to given constraints. In this paper we present an approach for constrained optimization based on a set of hard constraints and a preference ordering represented using a CP-network - a graphical model for representing qualitative preference information. This approach offers both pragmatic and computational advantages. First, it provides a convenient and intuitive tool for specifying the problem, and in particular, the decision maker's preferences. Second, it provides an algorithm for finding the most preferred feasible outcomes that has the following anytime property: the set of preferred feasible outcomes are enumerated without backtracking. In particular, the first feasible solution generated by this algorithm is optimal.

. We introduce two frameworks for constraint solving where classical CSPs, fuzzy CSPs, weighted CSPs, partial constraint satisfaction, and others can be easily cast. One is based on a semiring, and the other one on a totally ordered commutative monoid. We then compare the two approaches and we discuss the relationship between them. 1 Introduction Classical constraint satisfaction problems (CSPs) [19, 17] are a very expressive and natural formalism to specify many kinds of real-life problems. In fact, problems ranging from map coloring, vision, robotics, job-shop scheduling, VLSI design, etc., can easily be cast as CSPs and solved using one of the many techniques that have been developed for such problems or subclasses of them [8, 9, 18, 16, 19]. However, they also have evident limitations, mainly due to the fact that they are not very flexible when trying to represent real-life scenarios where the knowledge is not completely available nor crisp. In fact, in such situations, the abilit...

Many real life optimization problems contain both hard and soft constraints, as well as qualitative conditional preferences. However, there is no single formalism to specify all three kinds of information. We therefore propose a framework, based on both CP-nets and soft constraints, that handles both hard and soft constraints as well as conditional preferences efficiently and uniformly. We study the complexity of testing the consistency of preference statements, and show how soft constraints can faithfully approximate the semantics of conditional preference statements whilst improving the computational complexity. 1