An Evidence-based Framework for a Theory of Inheritance T. Krishnaprasad, Michael Kifer Department of Computer Science SUNY at Stony Brook Stony Brook, NY 11794. Abstract We present an approach to formalizing non- monotonic multiple inheritance networks by combining concepts from logic programming and multi-valued logics in a uniform frame- work. A Horn-clause logic language is used for specifying inheritance networks. This allows a natural representation of class-subclass hierar- chies and ambiguous inheritance networks. It also provides means for resolving ambiguities resulting from the network topology, but which are not inherent to the problem. We provide a model theory for the language and show how a unique intended model can be associated with every inheritance network. This model resem- bles the unique extension obtained in the skep- tical theory of inheritance [Hor-87], but is more general. Finally, we present an algorithm which realizes the aforementioned semantics. 1 Introduction The notion of nonmonotonic inheritance is fundamental to common-sense reasoning. For instance, knowing that Bateman is a mammal, one would conclude that it does not fly. This is because normally mammals do not fly and in the absence of any other information, we regard Bateman as inheriting its inability to fly from mammals. If one later learns that Bateman is a bat, then he will probably change his mind, concluding that it does fly. However, after learning that Bateman is a dead bat, he would again change his mind concluding that it cannot fly. In this example, knowing that Bateman is a bat is more informative than knowing that it is a mammal, and knowing that it is a dead bat is even more informa- tive, as far as Bateman's ability to fly is concerned. In essence, the knowledge that an individual belongs to a subclass provides more information about the individual than the knowledge that the individual belongs to its superclass. We also notice that the first two conclusions are defeasible, while the third one is not. Birds normally fly, while toys normally do not. If an item is a toy bird, then we conclude that it does not fly. This is because the item being a toy contributes more evidence in support of its inability to fly than does bird in support of its flying ability. Notice that there is no class-subclass relationship between toys and birds. Some approaches to inheritance (e.g., [Touretzky, 1986], [Horty et a/., 1987]) are proof-theoretic. They give algorithms for computing sets of acceptable paths supported by a network, rather than specifying the states of the world the network represents. Others (e.g., [Etherington, 1983], [Haugh, 1988], [Krishnaprasad et a/., 1988a], [Przymusinska and Gelfond, 1988], [Th oma- son ei a/., 1987]) present translations of inheritance net- works into some standard logical formalism. The seman- tics of the networks is then captured through the model theory for the respective logical formalism. In particular, a theory based on prioritized circumscription transforms a network into a set of first-order sentences augmented with meta-level minimality constraints embodying pref- erences [Krishnaprasad et a/., 1988a]. The first-order models of the resulting translation are the states of the world the network represents. The circumscriptive the- ory of [Haugh, 1988] formalizes the network at a meta- level. The set of models of the translation in [Haugh, 1988] encodes the meaning of the network. [Przymusin- ska and Gelfond, 1988] views a network as representing a set of beliefs of a rational agent, and captures this interpretation by translating the network into Moore's Autoepistemic logic [Moore, 1985]. The preference crite- ria is axiomatized in the translated theory. [Pearl, 1988] provides probabilistic semantics to networks, by assign- ing to them a set of possible worlds and an associated probability distribution. We propose to view networks as specifying a set of belief-evidence pairs. "Strength of evidence" is explicitly incorporated into the object language, and we have de- veloped a model theory for the resulting logic. The dom- inance of property inheritance from a subclass over that from a superclass is captured by making the evidence contributed by subclass membership stronger than that contributed by superclass membership. This provides an evidence based semantics to networks as a set of justified beliefs. In this paper, we present a logic to formalize inher- itance networks by combining concepts from logic pro- gramming and multi-valued logics in a uniform frame- work. The main ideas behind this approach are as fol- lows:  A naive formalization of inheritance networks in first-order logic leads to inconsistencies. As we do Krishnaprasad and Kifer 1093