# A Brief Roadmap into Uncertain Knowledge Representation via Probabilistic Description Logics
## 3 Representing Uncertain Knowledge  
Representing uncertain knowledge has a prerequisite representing knowledge. Knowledge representation, by itself, has a very long history, during which a plethora of variations, limitations, and features have been considered. A natural first step is to consider a known logic for representing knowledge; hence, one cannot avoid mentioning propositional and (first-order) predicate logic as the foundations of logic-based knowledge representation languages. However, from a practical point of view, propositional logic tends to be too inexpensive, and even the elements which can be expressed sometimes require a complex and difficult to grasp construction to be correctly handled. On the other spectrum, in full predicate logic, it is known that verifying the satisfiability of a formula (which in terms of knowledge representation translates into deciding whether a knowledge base is consistent) is an undecidable problem; therefore, there is no algorithm which can provide a correct answer in finite time for any possible formula. For this paper, we focused on a family of formalisms which mainly lies within these two formalisms. More specifically, most languages within this family--the family of description logics (DLs) [17]--are more expressive than propositional logic (thus, able to formalise more complex knowledge in a simpler manner) and at the same time less expressive than predicate logic guaranteeing decidable reasoning tasks (with consistency among them). There are a few exceptions to this statement, which only help in increasing the relevance of the family as knowledge representation formalisms. The very inexpressive DLs \(\mathcal{EL}\)[18] and DL-Lite [19], which are specifically targeted for tractable reasoning, do not contain the full power of propositional logic, although they allow for additional constructors. At the other end of the spectrum, expressive description logics such as \(\mathcal{SROIQ}\)[20] include constructors (such as transitive closure) which cannot be directly expressed in first-order logic. These are handled in a manner that prevents the undecidability of reasoning. The semantics of description logics, which is based on interpretations akin to first-order logic--that is, with a domain representing all the relevant objects, and an interpretation function which expresses the properties of those individuals in relation to each other--is especially useful for dealing with the various interpretations of uncertainty. We will see this in detail later, but in a nutshell and using [PERSON]'s classification, statistical probabilities are handled by adding uncertainty over the elements of the domain (i.e., the population_) while subjective probabilities are dealt with through several potential interpretations (_possible worlds_). As previously mentioned, the differences may be important. For all these reasons, we consider description logics as a basic formalism for representing uncertain knowledge. This is mainly meant as a prototypical representation: most of the ideas that we describe can similarly be applied to other formalisms without major modifications. We emphasise, however, that the classical family of description logics has some limitations which we will not consider further. Most notably, it cannot handle non-monotonic [21] nor temporal knowledge [22; 23] natively. Importantly, combining uncertainty with non-monotonicity and with temporal constructors is known to be especially problematic [24], both in terms of conceptual understanding and in the computational complexity of reasoning. Without going into too many details, the basic building blocks in description logic are _concepts_ (that is, sets of individuals) and _roles_, which represent relationships between individuals; slightly more formally, concepts are unary predicates, and roles are binary predicates of first-order logic. Hence, the Student is a concept that refers to all the students in the world of interest, while supervises expresses the relationship between a supervisor and their student.