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

A discrete duality between nonmonotonic consequence relations and convex geometries

In this paper we present a duality between nonmonotonic consequence relations and well-founded convex geometries. On one side of the duality we consider nonmonotonic consequence relations satisfying the axioms of an infinitary variant of System P, which is one of the most studied axiomatic systems for nonmonotonic reasoning, conditional logic and belief revision. On the other side of the duality we consider well-founded convex geometries, which are infinite convex geometries that generalize well-founded posets. Since there is a close correspondence between nonmonotonic consequence relations and path independent choice functions one can view our duality as an extension of an existing duality between path independent choice functions and convex geometries that has been developed independently by Koshevoy and by Johnson and Dean

Z-Numbers-Based Approach to Hotel Service Quality Assessment

In this study, we are analyzing the possibility of using Z-numbers for measuring the service quality and decision-making for quality improvement in the hotel industry. Techniques used for these purposes are based on consumer evalu- ations - expectations and perceptions. As a rule, these evaluations are expressed in crisp numbers (Likert scale) or fuzzy estimates. However, descriptions of the respondent opinions based on crisp or fuzzy numbers formalism not in all cases are relevant. The existing methods do not take into account the degree of con- fidence of respondents in their assessments. A fuzzy approach better describes the uncertainties associated with human perceptions and expectations. Linguis- tic values are more acceptable than crisp numbers. To consider the subjective natures of both service quality estimates and confidence degree in them, the two- component Z-numbers Z = (A, B) were used. Z-numbers express more adequately the opinion of consumers. The proposed and computationally efficient approach (Z-SERVQUAL, Z-IPA) allows to determine the quality of services and iden- tify the factors that required improvement and the areas for further development. The suggested method was applied to evaluate the service quality in small and medium-sized hotels in Turkey and Azerbaijan, illustrated by the example