# A Brief Roadmap into Uncertain Knowledge Representation via Probabilistic Description Logics
## 3 Representing Uncertain Knowledge
These symbols receive an interpretation by setting a (potentially infinite) _domain_, which contains all the objects of interest, and an _interpretation function_ expressing which objects belong to which concepts, and which pairs are related via roles. What differentiates one description logic from another is the class of _constructors_ used to build more complex concepts--e.g., conjunction, negation, and number constraints--and how they are interpreted. The goal of description logics is not only to express different kinds of concepts, but to actually represent the _knowledge_ of a domain. This is achieved through a _knowledge base_ which is a finite set of _axioms_ that serve as constraints for the interpretations. That is, each axiom excludes some potential interpretations as not representing the domain knowledge. For example, an axiom could express that _\"every student must have at least one supervisor.\"_ In this case, any interpretation including a supervisor-free student will be excluded as a violation of the constraint. In general, given a knowledge base, there are still many different (actually, infinitely many) interpretations which satisfy all the constraints imposed. These so-called _models_ are the only interpretations of interest in the context of the knowledge base. When we use the term _reasoning_, we refer to the task of extracting consequences which logically follow from the knowledge expressed in the knowledge base. Recalling that the axioms within the knowledge base are simply constraints in the possible interpretations, reasoning then refers to finding other pieces of knowledge which are guaranteed by these constraints. In other words, the logical consequences of a knowledge base are those which follow in all possible models of this set of axioms. We usually say that reasoning is the task of making knowledge which is _implicitly_ encoded by the _explicit_ knowledge base. The motivation behind using several models for reasoning is that we consider that a knowledge base is always (necessarily) incomplete. This means that we believe that a knowledge base will always exclude some information, either because it is irrelevant, or because it is not yet known. In these cases, we want to leave open the possibility of such an assertion being true or false until it is known. This approach is commonly known as the _open world assumption_ in the literature. Once again, a knowledge base defines a class of interpretations, each of which introduces a set of individuals. When knowledge is uncertain, we thus have two natural choices to introduce a probability distribution: it can be defined over the class of interpretations, expressing the likelihood that each of them represents the actual state of the world; or it can be defined over the individuals of the interpretation domain, differentiating the characteristics of the individuals. These two choices easily transfer to the two kinds of probabilistic logics in the classification by [PERSON]. This correspondence has given rise to several probabilistic description logics.