Quantification in description logics of typicality - Câmara, Igor de Camargo e Souza (2023)
This thesis delves into the intersection of Description Logics (DLs) and prototype theory in the context of knowledge representation and reasoning. DLs are formalisms widely used in knowledge representation and serve as the backbone of the Semantic Web. The thesis proposes that integrating aspects from prototype theory into DLs would be a desirable upgrade, enabling the introduction of nonmonotonic reasoning. This augmentation would expand reasoning based on DLs to include regularities that are typically verified but not always true. For instance, it would include knowledge such as birds typically fly, which is generally but not always true. Inferences of this kind are fundamental for modeling human-inspired reasoning and tackling problems like reasoning under incomplete information. The approach taken in this thesis follows the tradition of combining DLs and typicality through defeasible reasoning by using defeasible concept inclusions (DCIs). Materialization-based semantics is one of the most successful techniques for dealing with defeasible knowledge in DLs. This technique reduces checking defeasible entailments such as concept subsumption and instance checking to an enriched classical query, in which concepts representing defeasible axioms are added to the left-hand side of the inclusion. These concepts are called the materialization of the axioms they represent. Distinct materialization-based semantics are characterized by their techniques to select the axioms to materialize with any given concept. Although materialization-based semantics are undeniably successful, they suffer from some serious drawbacks. In particular, they share a propositional nature and, therefore, cannot extend defeasible information through quantifiers, a problem known as quantification neglect. Hence, birds typically fly and robins are birds allow concluding that robins typically fly. However, it is impossible to conclude from cats eat birds that cats typically eat flying animals. The thesis builds on the recently-introduced typicality models to address these limitations to define a semantical framework that improves existing semantics and includes first-order properties. It expands the existing framework for typicality models for the logic EL, which is a semantics parametrized along strengths and coverages with six variations covering existing semantics. Additionally, the thesis proposes a new framework for the logic ELI, which includes a propositional semantics equivalent to materialization-based reasoning and a nested semantics that solves quantification neglect for existing materialization-based semantics.