A collection of formulae, regarded as a set of prerequisite-free normal defaults, generates a nonmonotonic inference relation through its Reiter skeptical extension. The structure of the initial set totally determines the behavior of the associated inference relation, and the aim of this paper is to investigate in two directions the link that exists between a set of defaults and its induced inference relation. First, we determine the structural conditions corresponding to the important property of rationality. For this purpose, we introduce the notion of stratification for a set of defaults, and prove that stratified sets are exactly those that induce a rational inference relation. This result is shown to have interesting consequences in belief revision theory, as it can be used to define a nontrivial full meet revision operator for belief bases. Then, we adopt a dynamic point of view and study the effects, on the induced inference relation, of a change in the set of defaults. In this perspective, the set of defaults, considered as a knowledge base together with its induced inference relation is treated as an expert system. We show how to modify the original set of defaults in order to obtain as output a rational relation. We propose a revision procedure that enables the user to incorporate a new data in the knowledge base, and we finally show what changes can be performed on the original set of defaults in order to take into account a particular conditional that has to retracted from or added to the primitive induced inference relation.
Freund, M. (1999). Statics and dynamics of induced systems. Artificial Intelligence, 110(1), 103–134. https://doi.org/10.1016/S0004-3702(99)00020-X