Possibilistic and Standard Probabilistic Semantics of Conditional Knowledge Bases

The authors have proposed in their previous works to view a set of default pieces of information of the form, "generally, from αi deduce βi" as the family of possibility distributions satisfying constraints expressing that the situations where αiΛβi is true are more possible than the situations where αiΛ¬βi is true. A representation theorem in terms of this semantics, for default reasoning obeying the System P of postulates proposed by Kraus, Lehmann and Magidor, has been obtained. This paper offers a detailed analysis of the structure of this family of possibility distributions by exploiting two different orderings between them: Yager's specificity ordering and a new refinement ordering. It is shown that from a representation point of view, it is sufficient to consider the subset of linear possibility distributions which corresponds to all the possible completions of the default knowledge in agreement with the constraints. There also exists a semantics for system P in terms of infinitesimal probabilities. Surprisingly, it is also shown that a standard probabilistic semantics can be equivalently given to System P, without referring to infinitesimals, by using a special family of probability measures, that two of the authors have called acceptance functions, and that has been also recently considered by Snow in that perspective.