IBM T.J.Watson Research Center PAX Box 704 Yorktown Heights, NY 10598 We propose a theory of default reasoning satisfying a list of natural postulates. These postulates imply that knowledge bases containing defaults should be understood not as sets of formulas (rules and facts) but as collections of partially ordered theories. As a result of this shift of perspective we obtain a rather natural theory of default reasoning in which priorities in interpretation of predicates are the source of nonmonotonicity in reasoning. We also prove that our theory shares a number of desirable properties (completeness, soundness etc.) with the theory of normal defaults of R. Reiter. We limit our discussion to logical properties of the proposed system and prove some theorems about it. Modal operators or second order formulas do not appear in our formalization.
Possibility theory offers a framework where both Lehmann's "preferential inference" and the more productive (but less cautious) "rational closure inference" can be represented. However, there are situations where the second inference does not provide expected results either because it cannot produce them, or even provide counter-intuitive conclusions. This state of facts is not due to the principle of selecting a unique ordering of interpretations (which can be encoded by one possibility distribution), but rather to the absence of constraints expressing pieces of knowledge we have implicitly in mind. It is advocated in this paper that constraints induced by independence information can help finding the right ordering of interpretations. In particular, independence constraints can be systematically assumed with respect to formulas composed of literals which do not appear in the conditional knowledge base, or for default rules with respect to situations which are "normal" according to the other default rules in the base. The notion of independence which is used can be easily expressed in the qualitative setting of possibility theory. Moreover, when a counter-intuitive plausible conclusion of a set of defaults, is in its rational closure, but not in its preferential closure, it is always possible to repair the set of defaults so as to produce the desired conclusion.
Nonmonotonic logics are meant to be a formalization of nonmonotonic reasoning. However, for the most part they fail to capture in a perspicuous fashion two of the most important aspects of such reasoning: the explicit computational nature of nonmonotonic inference, and the assignment of preferences among competing inferences. We propose a method of nonmonotonic reasoning in which the notion of inference from specific bodies of evidence plays a fundamental role. The formalization is based on autoepistemic logic, but introduces additional structure, a hierarchy of evidential subtheories. The method offers a natural formalization of many different applications of nomnonotonic reasoning, including reasoning about action, speech acts, belief revision, and various situations involving competing defaults. The nonmonotonic character of commonsense reasoning in va.rious domains of concern to AI is well established. Recent evidence, especially the work connected with the Yale Shooting Problem (see [Hanks and McDermott, 19871) has illuminated the often profound mismatch between nonmonotonic reasoning in the abstract, and the logical systems proposed to formalize it. This is not to say that we should abandon the use of formal nonmonotonic systems; rather, it argues that we should seek ways to make them model our intuitive conception of nonmonotonic reasoning more closely. Generally speaking, current formal nonmonotonic systems suffer from two shortcomings: 1.
We describe a system that combines default reasoning with contexts. Contexts are arranged in a hierarchy where more specific contexts represent revisions of the state of belief in more general contexts. We describe our algorithm for default reasoning in a context hierarchy and provide a translation of our representation into a default logic theory whose inferences agree with our algorithm. We conclude with a discussion of different notions of context and give a justification of our default rules when contexts are understood as states of belief.