. A constraint satisfaction problem involves finding values for variables subject to constraints on which combinations of values are allowed. In some cases it may be impossible or impractical to solve these problems completely. We may seek to partially solve the problem, in particular by satisfying a maximal number of constraints. Standard backtracking and local consistency techniques for solving constraint satisfaction problems can be adapted to cope with, and take advantage of, the differences between partial and complete constraint satisfaction. Extensive experimentation on maximal satisfaction problems illuminates the relative and absolute effectiveness of these methods. A general model of partial constraint satisfaction is proposed. 1 Introduction Constraint satisfaction involves finding values for problem variables subject to constraints on acceptable combinations of values. Constraint satisfaction has wide application in artificial intelligence, in areas ranging from temporal r...

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

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].

Computing, in its usual sense, is centered on manipulation of numbers and symbols. In contrast, computing with words, or CW for short, is a methodology in which the objects of computation are words and propositions drawn from a natural language, e.g., small, large, far, heavy, not very likely, the price of gas is low and declining, Berkeley is near San Francisco, it is very unlikely that there will be a significant increase in the price of oil in the near future, etc. Computing with words is inspired by the remarkable human capability to perform a wide variety of physical and mental tasks without any measurements and any computations. Familiar examples of such tasks are parking a car, driving in heavy traffic, playing golf, riding a bicycle, understanding speech and summarizing a story. Underlying this remarkable capability is the brain’s crucial ability to manipulate perceptions – perceptions of distance, size, weight, color, speed, time, direction, force, number, truth, likelihood and other characteristics of physical and mental objects. Manipulation of perceptions plays a key role in human recognition, decision and execution processes. As a methodology, computing with words provides a foundation for a computational theory of perceptions – a theory which may have an important bearing on how humans make – and machines might make – perception-based rational decisions in an environment of imprecision, uncertainty and partial truth. A basic difference between perceptions and measurements is that, in general, measurements are crisp whereas perceptions are fuzzy. One of the fundamental aims of science has been and continues to be that of progressing from perceptions to measurements. Pursuit of this aim has led to brilliant successes. We have sent men to the moon; we can build computers

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 ...

. 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...

interpretation of four 89-relations between I and J) 89 d ! = fhR I \ThetaJ ; R 2 i : R I \ThetaJ ` R 2 R I \ThetaJ is a total function R I \ThetaJ is one-one.g 89! = fhR I \ThetaJ ; R 2 i : R I \ThetaJ ` R 2 R I \ThetaJ is a total function.g 89 d = fhR I \ThetaJ ; R 2 i : R I \ThetaJ ` R 2 R I \ThetaJ is a total relation [8i; i 0 : I; 8j : JR I \ThetaJ (i; j) R I \ThetaJ (i 0 ; j) ! i = i 0 ]g 89 = fhR I \ThetaJ ; R 2 i : R I \ThetaJ ` R 2 R I \ThetaJ is a total relationg 14 IJCAI Spatial and Temporal Workshop For recurrence relations based on simple correlation, R I \ThetaJ is instantiated to COR I \ThetaJ , and R 2 to a relation in CR. Thus, the quantifier construction creates a class of relations which places restrictions on the temporal order of the pairs of correlated intervals. What distinguishes the 89 class from other operators is the condition that R I \ThetaJ ` R 2 . To summarize: the construction of recurrence relations using quantification, and the rel...

We define fuzzy constraint networks and prove a theorem about their relationship to fuzzy logic. Then we introduce Khayyam, a fuzzy constraint-based programming language in which any sentence in the first-order fuzzy predicate calculus is a well-formed constraint statement. Finally, using Khayyam to address an equipment selection application, we illustrate the expressive power of fuzzy constraint-based languages.

: The formal framework for decision making in a fuzzy environment is based on a general maxmin, bottleneck-like optimization problem, proposed by Zadeh. It is also the basis for extending the constraint satisfaction paradigm of Artificial Intelligence to accommodating flexible or prioritized constraints. This paper surveys refinements of the ordering of solutions supplied by the max-min formulation, namely the discrimin partial ordering and the leximin complete preordering. A general algorithm is given which computes all maximal solutions in the sense of these relations. It also sheds light on the structure of the set of best solutions. Moreover, classes of problems for which there is a unique best discrimin and leximin solution are exhibited, namely, continuous problems with convex domains, and so called isotonic problems. Noticeable examples of such problems are fuzzy linear programming problems and fuzzy PERT-like scheduling problems. Introduction Flexible constraint satisfaction p...

One of the limitation of the constraint network formalism lies in its inability of explicitly expressing a criteria to optimize. The introduction of several ad-hoc optimization mechanisms in constraint (logic) programming languages shows how important this restriction is. Several formalisms of varied generality have been proposed to remove this restriction: fuzzy constraint networks, partial constraint satisfaction, semi-ring constraint networks, valued constraint networks...