Showing papers by "Henri Prade published in 1992"

Putting Rough Sets and Fuzzy Sets Together

Abstract: In this paper we argue that fuzzy sets and rough sets aim to different purposes and that it is more natural to try to combine the two models of uncertainty (vagueness for fuzzy sets and coarseness for rough sets) in order to get a more accurate account of imperfect information. First, the upper and lower approximations of a fuzzy set are defined, when the universe of discourse of a fuzzy sets is coarsened by means of an equivalence relation. We then come close to Caianiello’s C-calculus. Shafer’s concept of coarsened belief functions also belongs to the same line of thought and is reviewed here. Another idea is to turn the equivalence relation relation into a fuzzy similarity relation, for a more expressive modeling of coarseness. New results on the representation of similarity relations by means of a fuzzy partition of fuzzy clusters of more or less indiscernible points are surveyed. The properties of upper and lower approximations of fuzzy sets by similarity relations are thoroughly studied. Lastly the potential usefulness of the fuzzy rough set notions for logical inference in the presence of both fuzzy predicates and graded indiscernibility is indicated. Especially fuzzy rough sets may provide a nice semantic background for modal logic involving fuzzy modalities and/or fuzzy sentences.

Representing Default Rules in Possibilistic Logic.

Abstract: A key issue when reasoning with default rules is how to order them so as to derive plausible conclusions according to the more specific rules applicable to the situation under concern, to make sure that default rules are not systematically inhibited by more general rules, and to cope with the problem of irrelevance of facts with respect to exceptions. Pearl's system Z enables us to rank-order default rules. In this paper we show how to encode such a rank-ordered set of defaults in possibilistic logic. We can thus take advantage of the deductive machinery available in possibilistic logic. We point out that the notion of inconsistency tolerant inference in possibilistic logic corresponds to the bold inference ;1 in system Z. We also show how to express defaults by means of qualitative possibility relations. Improvements to the ordering provided by system Z are also proposed.

On the Combination of Evidence in Various Mathematical Frameworks

Abstract: The problem of combining pieces of evidence issued from several sources of information turns out to be a very important issue in artificial intelligence. It is encountered in expert systems when several production rules conclude on the value of the same variable, but also in robotics when information coming from different sensors is to be aggregated. Solutions proposed in the literature so far have often been unsatisfactory because relying on a single theory of uncertainty, a unique mode of combination, or the absence of analysis of the reasons for uncertainty. Besides dependencies and redundancies between sources must be dealt with especially in knowledge bases, where sources correspond to production rules.

Possibility theory as a basis for preference propagation in automated reasoning

Abstract: Possibility theory is proposed as a tool for encoding and propagating preference relations among possible interpretations or worlds, as well as certainty or priority degrees attached to logic sentences. The following points are particularly considered: (i) the representation of certainty- or possibility-qualified statements and its application to a typology of fuzzy rules; (ii) the principle of minimum specificity as the possibilistic counterpart of the maximal entropy principle; (iii) hypergraph methods for implementing the combination/projection paradigm of approximate reasoning; and (iv) the expression of the semantics of a set of certainty-weight logical formulas in possibilistic logic in terms of a possibility distribution on a set of interpretations. Simple examples of uncertain reasoning, analogical reasoning, interpolative reasoning, qualitative or temporal reasoning are provided in this framework. >

Dealing with multi-source information in possibilistic logic

Abstract: In this paper, both the uncertainty and the origin of pieces of information is handled in an extended possibilistic logic framework. Each formula is associated with a set (a fuzzy set more generally) which gathers labels of sources according to which the formula is (more or less) certainly true. In case of a single source of information, possibilistic logic is recovered. Soundness and completeness results of possibilistic logic are extended. Besides, the combination of the information provided by different sources (fusion), taking possibly into account the relative reliability of the sources, is discussed.

Fuzzy rules in knowledge-based systems

Abstract: The paper starts with ideas of possibility qualification and certainty qualification for specifying the possible range of a variable whose value is ill-known. The notion of possibility which is used for that purpose is not the standard one in possibility theory, although the two notions of possibility can be related. Based on these considerations four distinct types of rules with different semantics involving gradedness and uncertainty are then introduced. The combination operations which appear for taking advantage of the available knowledge are all derived from the intended semantics of the rules. The processing of these four types of rules is studied in detail. Fuzzy rules modelling preference in decision processes are also discussed.

A symbolic approach to reasoning with linguistic quantifiers

Abstract: This paper investigates the possibility of performing automated reasoning in probabilistic logic when probabilities are expressed by means of linguistic quantifiers. Each linguistic term is expressed as a prescribed interval of proportions. Then instead of propagating numbers, qualitative terms are propagated in accordance with the numerical interpretation of these terms. The quantified syllogism, modelling the chaining of probabilistic rules, is studied in this context. It is shown that a qualitative counterpart of this syllogism makes sense, and is relatively independent of the threshold defining the linguistically meaningful intervals, provided that these threshold values remain in accordance with the intuition. The inference power is less than that of a full-fledged probabilistic constraint propagation device but better corresponds to what could be thought of as commonsense probabilistic reasoning.

Positive and negative explanations of uncertain reasoning in the framework of possibility theory

Abstract: This paper presents an approach for developing the explanation capabilities of rule-based expert systems managing imprecise and uncertain knowledge. The treatment of uncertainty takes place in the framework of possibility theory where the available information concerning the value of a logical or numerical variable is represented by a possibility distribution which restricts its more or less possible values. We first discuss different kinds of queries asking for explanations before focusing on the two following types : i) how, a particular possibility distribution is obtained (emphasizing the main reasons only) ; ii) why in a computed possibility distribution, a particular value has received a possibility degree which is so high, so low or so contrary to the expectation. The approach is based on the exploitation of equations in max-min algebra. This formalism includes the limit case of certain and precise information.

Fuzzy boom in Japan

Abstract: This article is an abridged and translated version of a mission report initially written for the Scientific Service of the French Embassy in Japan. the mission took place in Japan from the 14th to the 21st of October, 1989. an introduction gives the necessary background concerning the applications of fuzzy sets to process control and expert systems; specialized hardwares for these applications are also introduced. Section II offers an account of the visits done during the mission. Section III synthesizes the lesson of the mission. the title makes use of the expression “fuzzy boom” which is often employed in Japan for describing the present blossoming of a great number of practical applications of fuzzy sets and their important repercussion in the media of this country.

The Semantics of Fuzzy “If … Then …” Rules

Abstract: This paper reports on an investigation into the semantics fuzzy “if... then ...” rules. Three kinds of interpretations olluzzy rules are pointed out. Each type of rule is claimed to be adapted to a specijic task: reasoning under uncertainty, reasoning by interpolation, and reasoning by analogy, respectively. The representation of each type of rule is carried out in the setting of possibility theory using inlormational principles. The luzzy controller synthesis technique is examined in the light of the obtained results.

Estimations of expectedness and potential surprise in possibility theory

Abstract: This note investigates how various ideas of 'expectedness' can be captured in the framework of possibility theory. Particularly, we are interested in trying to introduce estimates of the kind of lack of surprise expressed by people when saying 'I would not be surprised that...' before an event takes place, or by saying 'I knew it' after its realization. In possibility theory, a possibility distribution is supposed to model the relative levels of mutually exclusive alternatives in a set, or equivalently, the alternatives are assumed to be rank-ordered according to their level of possibility to take place. Four basic set-functions associated with a possibility distribution, including standard possibility and necessity measures, are discussed from the point of view of what they estimate when applied to potential events. Extensions of these estimates based on the notions of Q-projection or OWA operators are proposed when only significant parts of the possibility distribution are retained in the evaluation. The case of partially-known possibility distributions is also considered. Some potential applications are outlined.