Preferential reasoning in the perspective of Poole default logic

摘要

The sceptical inference relation associated with a Poole system without constraints is known to have a simple semantic representation by means of a smooth order directly defined on the set of interpretations associated with the underlying language. Conversely, we prove in this paper that, on a finite prepositional language, any preferential inference relation defined by such a model is induced by a Poole system without constraints. In the particular case of rational relations, the associated set of defaults may be chosen to be minimal; it then consists of a set of formulae, totally ordered through classical implication, with cardinality equal to the height of the given relation. This result can be applied to knowledge representation theory and corresponds, in revision theory, to Grove's family of spheres. In the framework of conditional knowledge bases and default extensions, it implies that any rational inference relation may be considered as the rational closure of a minimal knowledge base. An immediate consequence of this is the possibility of replacing any conditional knowledge base by a minimal one that provides the same amount of information.