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1

Structures of Opposition in Fuzzy Rough Sets

Ciucci D., Dubois D., Prade H.

Fundamenta Informaticae

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2015

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Vol. 142, nr 1-4

1--19

EN

The square of opposition is as old as logic. There has been a recent renewal of interest on this topic, due to the emergence of new structures (hexagonal and cubic) extending the square. They apply to a large variety of representation frameworks, all based on the notions of sets and relations. After a reminder about the structures of opposition, and an introduction to their gradual extensions (exemplified on fuzzy sets), the paper more particularly studies fuzzy rough sets and rough fuzzy sets in the setting of gradual structures of opposition.

2

Rough Sets, Coverings and Incomplete Information

Couso I., Dubois D.

Fundamenta Informaticae

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2011

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Vol. 108, nr 3/4

223-247

EN

Rough sets are often induced by descriptions of objects based on the precise observations of an insufficient number of attributes. In this paper, we study generalizations of rough sets to incomplete information systems, involving imprecise observations of attributes. The precise role of covering-based approximations of sets that extend the standard rough sets in the presence of incomplete information about attribute values is described. In this setting, a covering encodes a set of possible partitions of the set of objects. A natural semantics of two possible generalisations of rough sets to the case of a covering (or a non transitive tolerance relation) is laid bare. It is shown that uncertainty due to granularity of the description of sets by attributes and uncertainty due to incomplete information are superposed, whereby upper and lower approximations themselves (in Pawlak’s sense) become ill-known, each being bracketed by two nested sets. The notion of measure of accuracy is extended to the incomplete information setting, and the generalization of this construct to fuzzy attribute mappings is outlined.

3

A Framework for Iterated Belief Revision Using Possibilistic Counterparts to Jeffrey's Rule

Benferhat S., Dubois D., Prade H., Williams M-A.

Fundamenta Informaticae

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2010

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Vol. 99, nr 2

147-168

EN

Intelligent agents require methods to revise their epistemic state as they acquire new information. Jeffrey’s rule, which extends conditioning to probabilistic inputs, is appropriate for revising probabilistic epistemic states when new information comes in the form of a partition of events with new probabilities and has priority over prior beliefs. This paper analyses the expressive power of two possibilistic counterparts to Jeffrey's rule for modeling belief revision in intelligent agents. We show that this rule can be used to recover several existing approaches proposed in knowledge base revision, such as adjustment, natural belief revision, drastic belief revision, and the revision of an epistemic state by another epistemic state. In addition, we also show that some recent forms of revision, called improvement operators, can also be recovered in our framework.

4

A Possibility-Theoretic View of Formal Concept Analysis

Dubois D., Dupin de Saint-Cyr F.

Fundamenta Informaticae

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2007

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Vol. 75, nr 1-4

195-213

EN

The paper starts from the standard relational view linking objects and properties in formal concept analysis, here augmented with four modal-style operators (known as sufficiency, dual sufficiency, necessity and possibility operators). Formal concept analysis is mainly based on the first operator, while the others come from qualitative data analysis and can be also related to rough set theory. A possibility-theoretic reading of formal concept analysis with these four operators is proposed. First, it is shown that four and only four operators are indeed needed in order to describe the nine situations that can occur when comparing a statement (or its negation) with a state of information. The parallel between possibility theory and formal concept analysis suggests the introduction of new notions such as normalization and conditioning in the latter framework, also leading to point out some meaningful properties. Moreover, the graded setting of possibility theory allows us to suggest the extension of formal concept analysis to situations with incomplete or uncertain information.

5

Quasi-Possibilistic Logic and its Measures of Information and Conflict

Dubois D., Konieczny S., Prade H.

Fundamenta Informaticae

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2003

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Vol. 57, nr 2-4

101--125

EN

Possibilistic logic and quasi-classical logic are two logics that were developed in artificial intelligence for coping with inconsistency in different ways, yet preserving the main features of classical logic. This paper presents a new logic, called quasi-possibilistic logic, that encompasses possibilistic logic and quasi-classical logic, and preserves the merits of both logics. Indeed, it can handle plain conflicts taking place at the same level of certainty (as in quasi-classical logic), and take advantage of the stratification of the knowledge base into certainty layers for introducing gradedness in conflict analysis (as in possibilistic logic). When querying knowledge bases, it may be of interest to evaluate the extent to which the relevant available information is precise and consistent. The paper review measures of (im)precision and inconsistency/conflict existing in possibilistic logic and quasi-classical logic, and proposes generalized measures in the unified framework.

6

A definition of subjective possibility

Dubois D., Prade H., Smets P.

Badania Operacyjne i Decyzje

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2003

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nr 4

7-22

EN

The problem of finding a suitable belief function consistent with a given possibility distribution is considered. It is proved that this function is unique and consonant thus representable by means of a possibility distribution. The possibility distribution is subjective and unique. The results obtained in the paper allow us to define subjective possibility degrees, hence the membership function of fuzzy number.

PL

W pracy proponuje się subiektywne spojrzenie na teorię możliwości, polegające na założeniu, że kiedy konstruuje się pewien rozkład prawdopodobieństwa jest on faktycznie indukowany przez pewną funkcję ufności (belief function) reprezentującą rzeczywisty stan wiedzy. Zakłada się również, że przejście pewnej funkcji ufności do rozkładu prawdopodobieństwa jest realizowane za pomocą transformacji ( pignistic transformation), znanej jako wartość Shapleya. Rozważa się problem znalezienia odpowiedniej funkcji ufności zgodnej z danym rozkładem prawdopodobieństwa. Dowodzi się, ze funkcja ta jest jednoznacznie określona i zgodna. Można ją zatem reprezentować za pomocą rozkładu możliwości. Rozkład ten jest subiektywny i jednoznaczny. Otrzymane w pracy wyniki pozwalają na definiowanie subiektywnych stopni możliwości, a co za tym idzie - funkcji przynależności liczby mnogiej.

7

Making Revision Reversible: an Approach Based on Polynomials

Benferhat S., Dubois D., Lagrue S., Papini O.

Fundamenta Informaticae

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2002

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Vol. 53, nr 3,4

251-280

EN

This paper deals with iterated belief change and proposes a drastic revision rule that modifies a plausibility ordering of interpretations in such a way that any world where the input observation holds is more plausible that any world where it does not. This change rule makes sense in a dynamic context where observations are received, and the newer observations are considered more plausible than older ones. It is shown how to encode an epistemic state using polynomials equipped with the lexicographic ordering. This encoding makes it very easy to implement and iterate the revision rule using simple operations on these polynomials. Moreover, polynomials allow to keep track of the sequence of observations. Lastly, it is shown how to efficiently compute the revision rule at the syntactical level, when the epistemic state is concisely represented by a prioritized belief base. Our revision rule is the most drastic one can think of, in accordance with Darwiche and Pearl's principles, and thus contrasts with the minimal change rule called natural belief revision. The paper also shows how to obtain the reversibility of Boutilier's natural belief revision and possibilistic revision using polynomials.

8

Using possibilistic logic for modeling qualitative decision: ATMS-based algorithms

Dubois D., Le Berre D., Prade H., Sabbadin R.

Fundamenta Informaticae

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1999

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Vol. 37, Nr 1,2

1-30

EN

This paper describes a logical machinery for computing decision, where the available knowledge on the state of the world is described by a possibilistic prepositional logic base (i.e., a collection of logical statements associated with qualitative certainty levels), and where the preferences of the user are also described by another possibilistic logic base whose formula weights are interpreted in terms of priorities. Two attitudes are allowed for the decision maker: a pessimistic risk-averse one and an optimistic one. The computed decisions are in agreement with a qualitative counterpart to the classical theory of expected utility, recently developed by three of the authors. A link is established between this logical view of qualitative decision making and an ATMS-based computation procedure. Efficient algorithms for computing pessimistic and optimistic optimal decisions are finally given in this logical setting (using some previous work of the fourth author).