Description du projet
De nouvelles approches algébriques pour décrire la probabilité de déclarations floues
Dans la théorie des probabilités, les événements réels diffèrent des événements mathématiques auxquels on ne peut attribuer qu’une valeur de 0 ou de 1 selon qu’ils sont vérifiés ou non à un moment donné. En termes logiques, ils sont soit vrais, soit faux. Le projet PROREAL, financé par l’UE, prévoit d’utiliser des outils de sémantique algébrique pour étudier la logique floue, qui attribue des «degrés de vérité» plutôt que les habituelles valeurs «vrai ou faux» pour traiter les déclarations réelles vagues. La valeur de vérité de ces déclarations peut varier de complètement vraie à complètement fausse. Une attention particulière sera accordée à l’étude des connectivités floues basées sur les t-normes.
Objectif
The goal of this project is to develop logico-algebraic and measure theoretical techniques to study the probability theory of real-valued events. The classical approach to probability only allows the study of events that at some moment will either be verified or not. In more logical terms, they will become either true or false. In order to be able to treat the intrinsic vagueness of real-life declarative statements, we will study events whose truth-value can be intermediate, varying in the real unit interval (where 0 represents absolute falseness and 1 absolute truth). Our focus will be on the state theory, i.e. probability theory, of continuous (or more generally left-continuous) t-norm based fuzzy logics. The study will be carried out by the point of view of the algebraic semantics of such logics, which has proven to be extremely fruitful in recent years, showing interesting connections to measure theory. Indeed, in different fuzzy logics it is possible to introduce a notion of probability map on the algebras of formulas in such a way that it axiomatizes the Lebesgue integral of the formulas (which, thanks to universal algebraic results, can be seen as real-valued functions). As a most challenging goal, we will investigate the minimal properties for a logic to allow the axiomatization of a probability theory axiomatizing the Lebesgue integral of formulas. Moreover, as in recent years the artificial intelligence perspective is emerging, we will develop logical frameworks for many-valued probabilistic reasoning, and study the satisfiability of the probabilistic assertions for such logics. The study will strongly ground on preliminary work of research the fellow has conducted in recent years. Moreover, it will benefit from the broad experience of the supervisor as a leading researcher in the field, and of the high level, multidisciplinary environment of the hosting institute.
Champ scientifique
Mots‑clés
Programme(s)
Régime de financement
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinateur
28006 Madrid
Espagne