Functional relationships between objects, called `attributes', are of considerable importance in knowledge representation languages, including Description Logics (DLs). A study of the literature indicates that papers have made, often implicitly, different assumptions about the nature of attributes: whether they are always required to have a value, or whether they can be partial functions. The work presented here is the first explicit study of this difference for subclasses of the CLASSIC DL, involving the same-as concept constructor. It is shown that although determining subsumption between concept descriptions has the same complexity (though requiring different algorithms), the story is different in the case of determining the least common subsumer (lcs). For attributes interpreted as partial functions, the lcs exists and can be computed relatively easily; even in this case our results correct and extend three previous papers about the lcs of DLs. In the case where attributes must have a value, the lcs may not exist, and even if it exists it may be of exponential size. Interestingly, it is possible to decide in polynomial time if the lcs exists.

The notion of class is ubiquitous in computer science and is central in many formalisms for the representation of structured knowledge used both in knowledge representation and in databases. In this paper we study the basic issues underlying such representation formalisms and single out both their common characteristics and their distinguishing features. Such investigation leads us to propose a unifying framework in which we are able to capture the fundamental aspects of several representation languages used in different contexts. The proposed formalism is expressed in the style of description logics, which have been introduced in knowledge representation as a means to provide a semantically well-founded basis for the structural aspects of knowledge representation systems. The description logic considered in this paper is a subset of first order logic with nice computational characteristics. It is quite expressive and features a novel combination of constructs that has not been studied before. The distinguishing constructs are number restrictions, which generalize existence and functional dependencies, inverse roles, which allow one to refer to the inverse of a relationship, and possibly cyclic assertions, which are necessary for capturing real world domains. We are able to show that it is precisely such combination of constructs that makes our logic powerful enough to model the essential set of features for defining class structures that are common to frame systems, object-oriented database languages, and semantic data models. As a consequence of the established correspondences, several significant extensions of each of the above formalisms become available. The high expressiveness of the logic we propose and the need for capturing the reasoning in different contexts forces us to distinguish between unrestricted and finite model reasoning. A notable feature of our proposal is that reasoning in both cases is decidable. We argue that, by virtue of the high expressive power and of the associated reasoning capabilities on both unrestricted and finite models, our logic provides a common core for class-based representation formalisms.

Given a knowledge base KB containing first-order and statistical facts, we consider a principled method, called the random-worlds method, for computing a degree of belief that some formula Phi holds given KB. If we are reasoning about a world or system consisting of N individuals, then we can consider all possible worlds, or first-order models, withdomain {1,...,N} that satisfy KB, and compute thefraction of them in which Phi is true. We define the degree of belief to be the asymptotic value of this fraction as N grows large. We show that when the vocabulary underlying Phi andKB uses constants and unary predicates only, we can naturally associate an entropy with each world. As N grows larger,there are many more worlds with higher entropy. Therefore, we can usea maximum-entropy computation to compute the degree of belief. This result is in a similar spirit to previous work in physics and artificial intelligence, but is far more general. Of equal interest to the result itself are the limitations on its scope. Most importantly, the restriction to unary predicates seems necessary. Although the random-worlds method makes sense in general, the connection to maximum entropy seems to disappear in the non-unary case. These observations suggest unexpected limitations to the applicability of maximum-entropy methods.