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Description Logics (DLs) are a family of logic-based knowledge representation languages that form the basis of the Web Ontology Language (OWL) and are used to specify and reason about ontologies in application areas such as medicine or biology. Classical DLs do not offer to the user the ability to express certain application-relevant numerical constraints. To overcome this deficit, different kinds of numerical extensions of DLs have been proposed in the literature, such as:

- Description Logics with cardinality restrictions (which can e.g. express a certain circuit gate having more inputs than outputs)
- Description Logics with concrete domain restrictions (which can, e.g., represent the heart rate of a patient and compare it with a threshold value or other measurements)
- Probabilistic extensions of Description Logics (which can, e.g., express the probability that a protein with a binding to a certain type of molecule possesses some characteristic of interest)

After presenting these quantitative extensions of DLs, we will look at the problem of characterizing the expressive power of the resulting logics, that is, determining what formal properties they satisfy and see how these properties can be used to show e.g. how a certain logic is more expressive than another one.