Nonmonotonic consequences in default domain theory

Abstract

Default domain theory is a framework for representing and reasoning about commonsense knowledge. Although this theory is motivated by ideas in Reiter’s work on default logic, it is in some sense a dual framework. We make Reiter’s default extension operator into a constructive method of building models, not theories. Domain theory, which is a well established tool for representing partial information in the semantics of programming languages, is adopted as the basis for constructing partial models. This paper considers some of the laws of nonmonotonic consequence, due to Gabbay and to Kraus, Lehmann, and Magidor, in the light of default domain theory. We remark that in some cases Gabbay’s law of cautious monotony is open to question. We consider an axiomatization of the nonmonotonic consequence relation on prime open sets in the Scott topology – the natural logic – of a domain, which omits this law. We prove a representation theorem showing that such relations are in one to one correspondence with the consequence relations determined by extensions in Scott domains augmented with default sets. This means that defaults are very expressive: they can, in a sense, represent any reasonable nonmonotonic entailment. Results about what kind of defaults determine cautious monotony are also discussed. In particular, we show that the property of unique extensions guarantees cautious monotony, and we give several classes of default structures which determine unique extensions.