Consensus theories: an oriented survey

This article surveys seven directions of consensus theories: Arrowian results, federation consensus rules, metric consensus rules, tournament solutions, restricted domains, abstract consensus theories, algorithmic and complexity issues. This survey is oriented in the sense that it is mainly – but not exclusively – concentrated on the most significant results obtained, sometimes with other searchers, by a team of French searchers who are or were full or associate members of the Centre d'Analyse et de Mathématique Sociale (CAMS).Consensus theories ; Arrowian results ; aggregation rules ; metric consensus rules ; median ; tournament solutions ; restricted domains ; lower valuations ; median semilattice ; complexity

Consensus theories: an oriented survey

URL des Documents de travail : http://ces.univ-paris1.fr/cesdp/cesdp2010.htmlDocuments de travail du Centre d'Economie de la Sorbonne 2010.57 - ISSN : 1955-611XThis article surveys seven directions of consensus theories: Arrowian results, federation consensus rules, metric consensus rules, tournament solutions, restricted domains, abstract consensus theories, algorithmic and complexity issues. This survey is oriented in the sense that it is mainly – but not exclusively – concentrated on the most significant results obtained, sometimes with other searchers, by a team of French searchers who are or were full or associate members of the Centre d'Analyse et de Mathématique Sociale (CAMS).Cet article présente une vue d'ensemble de sept directions de recherche en théorie du consensus : résultats arrowiens, règles d'agrégation définies au moyen de fédérations, règles définies au moyen de distances, solutions de tournoi, domaines restreints, théories abstraites du consensus, questions de complexité et d'algorithmique. Ce panorama est orienté dans la mesure où il présente principalement – mais non exclusivement – les travaux les plus significatifs obtenus – quelquefois avec d'autres chercheurs – par une équipe de chercheurs français qui sont – ou ont été – membres pléniers ou associés du Centre d'Analyse et de Mathématique Sociale (CAMS)

On the degree of transitivity of a fuzzy relation

Considering a family generated from a t-norm T, the degree of T-transitivity of a fuzzy relation R is revisited and proved to coincide with the greatest c for which R is -transitive. This fact gives rise to the study of new families of t-norms to generate different degrees of transitivity with respect to them. The mappings transforming fuzzy relations into transitive fuzzy relations smaller than or equal to the given ones are studied.Peer ReviewedPostprint (author's final draft

Quantitative Graded Semantics and Spectra of Behavioural Metrics

Behavioural metrics provide a quantitative refinement of classical two-valued behavioural equivalences on systems with quantitative data, such as metric or probabilistic transition systems. In analogy to the classical linear-time/branching-time spectrum of two-valued behavioural equivalences on transition systems, behavioural metrics come in various degrees of granularity, depending on the observer's ability to interact with the system. Graded monads have been shown to provide a unifying framework for spectra of behavioural equivalences. Here, we transfer this principle to spectra of behavioural metrics, working at a coalgebraic level of generality, that is, parametrically in the system type. In the ensuing development of quantitative graded semantics, we discuss presentations of graded monads on the category of metric spaces in terms of graded quantitative equational theories. Moreover, we obtain a canonical generic notion of invariant real-valued modal logic, and provide criteria for such logics to be expressive in the sense that logical distance coincides with the respective behavioural distance. We thus recover recent expressiveness results for coalgebraic branching-time metrics and for trace distance in metric transition systems; moreover, we obtain a new expressiveness result for trace semantics of fuzzy transition systems. We also provide a number of salient negative results. In particular, we show that trace distance on probabilistic metric transition systems does not admit a characteristic real-valued modal logic at all

Fuzzy implication functions based on powers of continuous t-norms

The modification (relaxation or intensification) of the antecedent or the consequent in a fuzzy “If, Then” conditional is an important asset for an expert in order to agree with it. The usual method to modify fuzzy propositions is the use of Zadeh's quantifiers based on powers of t-norms. However, the invariance of the truth value of the fuzzy conditional would be a desirable property when both the antecedent and the consequent are modified using the same quantifier. In this paper, a novel family of fuzzy implication functions based on powers of continuous t-norms which ensure the aforementioned property is presented. Other important additional properties are analyzed and from this study, it is proved that they do not intersect the most well-known classes of fuzzy implication functions.Peer ReviewedPostprint (author's final draft

Fuzzy closure systems: Motivation, definition and properties

The aim of this paper is to extend closure systems from being crisp sets with certain fuzzy properties to proper fuzzy sets. The presentation of the paper shows a thorough discussion on the different alternatives that could be taken to define the desired fuzzy closure systems. These plausible alternatives are discarded if they are proven impossible to be in a bijective correspondence with closure operators. Finally, a definition of fuzzy closure system is established and a one-to-one relation with closure operators is proved.The aim of this paper is to extend closure systems from being crisp sets with certain fuzzy properties to proper fuzzy sets. The presentation of the paper shows a thorough discussion on the different alternatives that could be taken to define the desired fuzzy closure systems. These plausible alternatives are discarded if they are proven impossible to be in a bijective correspondence with closure operators. Finally, a definition of fuzzy closure system is established and a one-to-one relation with closure operators is proved. Funding for open access charge: Universidad de Málaga / CBU