Weighted logics for artificial intelligence : an introductory discussion

International audienceBefore presenting the contents of the special issue, we propose a structured introductory overview of a landscape of the weighted logics (in a general sense) that can be found in the Artificial Intelligence literature, highlighting their fundamental differences and their application areas

Possibilistic reasoning with partially ordered beliefs

International audienceThis paper presents the extension of results on reasoning with totally ordered belief bases to the partially ordered case. The idea is to reason from logical bases equipped with a partial order expressing relative certainty and to construct a partially ordered deductive closure. The difficult point lies in the fact that equivalent definitions in the totally ordered case are no longer equivalent in the partially ordered one. At the syntactic level we can either use a language expressing pairs of related formulas and axioms describing the properties of the ordering, or use formulas with partially ordered symbolic weights attached to them in the spirit of possibilistic logic. A possible semantics consists in assuming the partial order on formulas stems from a partial order on interpretations. It requires the capability of inducing a partial order on subsets of a set from a partial order on its elements so as to extend possibility theory functions. Among different possible definitions of induced partial order relations, we select the one generalizing necessity orderings (closely related to epistemic entrenchments). We study such a semantic approach inspired from possibilistic logic, and show its limitations when relying on a unique partial order on interpretations. We propose a more general sound and complete approach to relative certainty, inspired by conditional modal logics, in order to get a partial order on the whole propositional language. Some links between several inference systems, namely conditional logic, modal epistemic logic and non-monotonic preferential inference are established. Possibilistic logic with partially ordered symbolic weights is also revisited and a comparison with the relative certainty approach is made via mutual translations

Other uncertainty theories based on capacities

International audienceThe two main uncertainty representations in the literature that tolerate imprecision are possibility distributions and random disjunctive sets. This chapter devotes special attention to the theories that have emerged from them. The first part of the chapter discusses epistemic logic and derives the need for capturing imprecision in information representations. It bridges the gap between uncertainty theories and epistemic logic showing that imprecise probabilities subsume modalities of possibility and necessity as much as probability. The second part presents possibility and evidence theories, their origins, assumptions and semantics, discusses the connections between them and the general framework of imprecise probability. Finally, chapter points out the remaining discrepancies between the different theories regarding various basic notions, such as conditioning, independence or information fusion and the existing bridges between them

Extending possibilistic logic over Gödel logic

In this paper we present several fuzzy logics trying to capture different notions of necessity (in the sense of possibility theory) for Gödel logic formulas. Based on different characterizations of necessity measures on fuzzy sets, a group of logics with Kripke style semantics is built over a restricted language, namely, a two-level language composed of non-modal and modal formulas, the latter, moreover, not allowing for nested applications of the modal operator N. Completeness and some computational complexity results are shown

Bipolar Possibility Theory as a Basis for a Logic of Desire and Beliefs

International audienceBipolar possibility theory relies on the use of four set functions. On the one hand, a weak possibility and a strong necessity measure are increasing set functions, which are respectively max-decomposable with respect to union and min-decomposable with respect to intersection. On the other hand, strong possibility and weak necessity measures are two decreasing set functions, which are respectively min-decomposable with respect to union and max-decomposable with respect to intersection. In the first part of the paper we advocate the use of the last two functions for modeling a notion of graded desire. Moreover, we show that the combination of weak possibility and strong possibility allows us to model a notion of realistic desire, i.e., a desire that does not only account for satisfactoriness but also for its epistemic possibility. In the second part of the paper we show that possibility theory offers a semantic basis for developing a modal logic of beliefs and desires

Possibilistic Boolean games: strategic reasoning under incomplete information

Boolean games offer a compact alternative to normal-form games, by encoding the goal of each agent as a propositional formula. In this paper, we show how this framework can be naturally extended to model situations in which agents are uncertain about other agents' goals. We first use uncertainty measures from possibility theory to semantically define (solution concepts to) Boolean games with incomplete information. Then we present a syntactic characterization of these semantics, which can readily be implemented, and we characterize the computational complexity

Symbolic Possibilistic Logic: Completeness and Inference Methods

International audienceThis paper studies the extension of possibilistic logic to the case when weights attached to formulas are symbolic and stand for variables that lie in a totally ordered scale, and only partial knowledge is available on the relative strength of these weights. A proof of the soundness and the completeness of this logic according to the relative certainty semantics in the sense of necessity measures is provided. Based on this result, two syntactic inference methods are presented. The first one calculates the necessity degree of a possibilistic formula using the notion of minimal inconsistent sub-base. A second method is proposed that takes inspiration from the concept of ATMS. Notions introduced in that area, such as nogoods and labels, are used to calculate the necessity degree of a possibilistic formula. A comparison of the two methods is provided, as well as a comparison with the original version of symbolic possibilistic logic

The legacy of 50 years of fuzzy sets: A discussion

International audienceThis note provides a brief overview of the main ideas and notions underlying fifty years of research in fuzzy set and possibility theory, two important settings introduced by L.A. Zadeh for representing sets with unsharp boundaries and uncertainty induced by granules of information expressed with words. The discussion is organized on the basis of three potential understanding of the grades of membership to a fuzzy set, depending on what the fuzzy set intends to represent: a group of elements with borderline members, a plausibility distribution, or a preference profile. It also questions the motivations for some existing generalized fuzzy sets. This note clearly reflects the shared personal views of its authors

Qualitative Capacities as Imprecise Possibilities.

National audienceThis paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite totally ordered scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multicriteria decision-making. More specifically, we investigate the question whether these qualitative set-functions can be viewed as classes of simpler set-functions, typically possibility measures, paralleling the situation of quantitative capacities with respect to imprecise probability theory. We show that any capacity is characterized by a non-empty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and their number is characteristic of its complexity. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular multi-source modal logics and their neighborhood semantics can be described in terms of qualitative Moebius transforms