Two methods of constructing contractions and revisions of knowledge systems

The theory of nonmonotonic reasoning and the theory of belief revision share a very important subject. Both theories help to understand how it is possible rationally to pass from one knowledge system into another knowledge system that is in contradiction with the former one. In nonmonotonic reasoning this transition is accomplished by distinguishing between axioms ("explicit beliefs") and theorems ("implicit beliefs") and giving up the doctrine that more axioms must always yield more theorems. Thus my old (implicit) belief that Tweety can fly may well turn into disbelief after getting the information (acquiring the explicit belief) that Tweety is a penguin. A considerable limitation of this approach is that the new axiom must be (monotonica!ly) consistent with the previous axioms, or else we get an inconsistent knowledge base. The theory of belief revision on which I shall focus my attention is not subject to this restriction. In fact, it does not at all distinguish between axioms and theorems, or between beliefs and their reasons. The knowledge systems it takes into consideration are whole theories, where a theory, or knowledge set, is understood as a set of sentences closed under some appropriate logic Cn. We shall assume that this logic includes classical propositional logic, that it is compact (i.e., if