Qualitative possibility functions and integrals

ksInternational audienceMeasure theory relies on numerical set-functions. Here we consider relations induced by set-functions on a set of events. Such relations considered as primitive objects are more natural than set-functions for the purpose of modelling subjective uncertainty. Starting from a set of events and a preordering relation among events, some properties are described, that such relations should obey in order to be natural candidates for representing uncertainty. Well-known families of confidence relations on finite sets are surveyed, such as comparative probabilities, first proposed by De Finetti. An important particular case is studied in more details: comparative possibility measures. Their importance stems from the fact that they can be simply induced from a complete preordering between elements of the refèrential. Such orderings play an important role in Artificial intelligence, in theory revision and nonmonotonic reasoning. Several issues are discussed like information contents and conditioning. It is shown that possibility orderings are representable by means of idempotent measures or maxitive set-functions, called possibility measures, taking values on finite chains. The compared expressive powers between such set-functions and possibility orderings are discussed. Notions of conditioning and independence and integrals on finite ordinal scales are presented