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

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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].

. Traditionally, local consistency is dened as a relaxation of consistency which can be checked in polynomial time. It is accompanied by a corresponding \ltering" or \enforcing" algorithm that computes in polynomial time, and from any given CSP, an equivalent unique CSP which satises the local consistency property. The question whether the notion of local consistency can be extended to soft constraint frameworks has been addressed by several papers, in several settings [4, 14, 12]. The main positive conclusion of these works is that the notion of local consistency can be extended to soft constraints frameworks which rely on an idempotent violation combination operator. However, the question whether this can be done for non idempotent operators as eg, in the Max-CSP problem, is not clear and has lead to several dierent notions of arc consistency [14, 16, 1, 11, 10]. Each of these proposals lacks several of the original properties of local consistency. In this paper, we...

This paper presents a justification of two qualitative counterparts of the expected utility criterion for decision under uncertainty, which only require bounded, linearly ordered, valuation sets for expressing uncertainty and preferences. This is carried out in the style of Savage, starting with a set of acts equipped with a complete preordering relation. Conditions on acts are given that imply a possibilistic representation of the decision-maker uncertainty. In this framework, pessimistic (i.e., uncertainty-averse) as well as optimistic attitudes can be explicitly captured. The approach thus proposes an operationally testable description of possibility theory. 1

. 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...

Non-Axiomatic Reasoning System is an adaptive system that works with insu cient knowledge and resources. At the beginning of the paper, three binary term logics are de ned. The rst is based only on an inheritance relation. The second and the third suggest a novel way to process extension and intension, and they also have interesting relations with Aristotle's syllogistic logic. Based on the three simple systems, a Non-Axiomatic Logic is de ned. It has a term-oriented language and an experience-grounded semantics. It can uniformly represents and processes randomness, fuzziness, and ignorance. It can also uniformly carries out deduction, abduction, induction, and revision.

This paper is meant to survey the literature pertaining to this debate, and to try to overcome misunderstandings and to supply access to many basic references that have addressed the "probability versus fuzzy set" challenge. This problem has not a single facet, as will be claimed here. Moreover it seems that a lot of controversies might have been avoided if protagonists had been patient enough to build a common language and to share their scientific backgrounds. The main points made here are as follows. i) Fuzzy set theory is a consistent body of mathematical tools. ii) Although fuzzy sets and probability measures are distinct, several bridges relating them have been proposed that should reconcile opposite points of view ; especially possibility theory stands at the cross-roads between fuzzy sets and probability theory. iii) Mathematical objects that behave like fuzzy sets exist in probability theory. It does not mean that fuzziness is reducible to randomness. Indeed iv) there are ways of approaching fuzzy sets and possibility theory that owe nothing to probability theory. Interpretations of probability theory are multiple especially frequentist versus subjectivist views (Fine [31]) ; several interpretations of fuzzy sets also exist. Some interpretations of fuzzy sets are in agreement with probability calculus and some are not. The paper is structured as follows : first we address some classical misunderstandings between fuzzy sets and probabilities. They must be solved before any discussion can take place. Then we consider probabilistic interpretations of membership functions, that may help in membership function assessment. We also point out nonprobabilistic interpretations of fuzzy sets. The next section examines the literature on possibility-probability transformati...

This paper explores the implications of approximating a concept based on the Bayesian decision procedure, which provides a plausible unification of the fuzzy set and rough set approaches for approximating a concept. We show that if a given concept is approximated by one set, the same result given by the α-cut in the fuzzy set theory is obtained. On the other hand, if a given concept is approximated by two sets, we can derive both the algebraic and probabilistic rough set approximations. Moreover, based on the well known principle of maximum (minimum) entropy, we give a useful interpretation of fuzzy intersection and union. Our results enhance the understanding and broaden the applications of both fuzzy and rough sets. 1.

Methods for incorporating imprecision in engineering design decision-making are briefly reviewed and compared. A tutorial is presented on the Method of Imprecision (MoI), a formal method, based on the mathematics of fuzzy sets, for representing and manipulating imprecision in engineering design. The results of a design cost estimation example, utilizing a new informal cost specification, are presented. The MoI can provide formal information upon which to base decisions during preliminary engineering design and can facilitate set-based concurrent design.