A counterpart to von Neumann and Morgenstern' expected utility theory is proposed in the framework of possibility theory. The existence of a utility function, representing a preference ordering among possibility distributions (on the consequences of decision-maker's actions) that satisfies a series of axioms pertaining to decision-maker's behavior, is established. The obtained utility is a generalization of Wald's criterion, which is recovered in case of total ignorance; when ignorance is only partial, the utility takes into account the fact that some situations are more plausible than others. Mathematically, the qualitative utility is nothing but the necessity measure of a fuzzy event in the sense of possibility theory (a so-called Sugeno integral). The possibilistic representation of uncertainty, which only requires a linearly ordered scale, is qualitative in nature. Only max, min and order-reversing operations are used on the scale. The axioms express a risk-averse behavior of the d...

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

The paper focuses mainly on extraction of important topographic objects, like buildings and roads, that have received much attention the last decade. As main input data, aerial imagery is considered, although other data, like from laser scanner, SAR and high-resolution satellite imagery, can be also used. After a short review of recent image analysis trends, and strategy and overall system aspects of knowledge-based image analysis, the paper focuses on aspects of knowledge that can be used for object extraction: types of knowledge, problems in using existing knowledge, knowledge representation and management, current and possible use of knowledge, upgrading and augmenting of knowledge. Finally, an overview on commercial systems regarding automated object extraction and use of a priori knowledge is given. In spite of many remaining unsolved problems and need for further research and development, use of knowledge and semi-automation are the only viable alternatives towards development of useful object extraction systems, as some commercial systems on building extraction and 3D city modelling as well as advanced, practically oriented research have shown.

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...

We introduce a method for inducing the structure of (causal) possibilistic networks from databases of sample cases. In comparison to the construction of Bayesian belief networks, the proposed framework has some advantages, namely the explicit consideration of imprecise (setvalued) data, and the realization of a controlled form of information compression in order to increase the efficiency of the learning strategy as well as approximate reasoning using local propagation techniques. Our learning method has been applied to reconstruct a non-singly connected network of 22 nodes and 24 arcs without the need of any a priori supplied node ordering. 14.1 Introduction Bayesian networks provide a well-founded normative framework for knowledge representation and reasoning with uncertain, but precise data. Extending pure probabilistic settings to the treatment of imprecise (set-valued) information usually restricts the computational tractability of the corresponding inference mechanisms. It is t...

This paper presents an approximate algorithm to obtain a posteriori intervals of probability, when available information is also given with intervals. The algorithm uses probability trees as a means of representing and computing with the convex sets of

In this paper we propose a propositional temporal language based on fuzzy temporal constraints which turns out to be expressive enough for domains like many coming from medicine where knowledge is of propositional nature and an explicit handling of time, imprecision and uncertainty are required. The language is provided with a natural possibilistic semantics to account for the uncertainty issued by the fuzziness of temporal constraints. We also present an inference system based on specific rules dealing with the temporal constraints and a general fuzzy modus ponens rule whereby behaviour is shown to be sound. The analysis of the different choices as fuzzy operators leads us to identify the well-known Lukasiewicz implication as very appropriate to define the notion of possibilistic entailment, an essential element of our inference system.

The paper improves a previously proposed axiomatic setting for qualitative decision under uncertainty in the von Neumann and Morgenstern' style, where only ordinal scales are required for assessing uncertainty and utility. A deeper analysis shows that only four axioms are needed, postulating i) that the choice is governed by a complete preorder, ii) that the decision maker is uncertainty-averse, iii) independence and iv) a (new) continuity assumption. Because uncertaintyaversion is assumed, the qualitative utility function enforced by these four postulates is pessimistic on a subset of highly plausible states. Substituting uncertainty-aversion by an uncertainty-prone postulate and suitably modifying the continuity postulate leads to a dual optimistic qualitative utility function (a decision is good as soon as it may have good consequences plausibly). This second function can help breaking ties between decisions having the same pessimistic evaluation. Besides, a case-based counterpart t...

This paper presents an axiomatic framework for qualitative decision under uncertainty in a finite setting. The corresponding utility is expressed by a sup-min expression, called Sugeno (or fuzzy) integral. Technically speaking, Sugeno integral is a median, which is indeed a qualitative counterpart to the averaging operation underlying expected utility. The axiomatic justification of Sugeno integral-based utility is expressed in terms of preference between acts as in Savage decision theory. Pessimistic and optimistic qualitative utilities, based on necessity and possibility measures, previously introduced by two of the authors, can be retrieved in this setting by adding appropriate axioms. 1

In this paper we propose a natural approach to handle imprecise numbers as they arise for example from measurements. Fuzzy sets turn out to be a canonical representation for such imprecise numbers that are induced by taking different tolerance or error bounds into account. Fuzzy sets are induced by scaling factors that describe the magnitude of the imprecision. On the other, the scaling factors can be derived from given fuzzy sets so that we have a correspondence between scaling factors and fuzzy sets. When these concepts are applied to control problems, the max--min rule is rediscovered as an interpolations technique. Viewing fuzzy control as an interpolation technique in vague environments enables us to validate various concepts for the design and tuning of fuzzy controllers and suggests new also new methods based on clear semantics. Keywords: Vague environment; interpolation; fuzzy control. 1 Introduction The formal definition of a fuzzy set as a mapping from an underlying univers...