Towards More Useful Description Logics of Time, Change and Context

Description Logics (DLs) are a family of logic-based formalisms for the representation of and reasoning about knowledge. Classical DLs are fragments of first-order logic and therefore aim at capturing static knowledge. Alas, the lack of means of DLs to capture dynamic aspects of knowledge has been often criticized because many important DL applications depend on this kind of knowledge. As a reaction to this shortcoming of DLs, two-dimensional extensions of DLs with capabilities to represent and reason about dynamic knowledge were introduced. Two-dimesional DLs are constructed by combining DLs with well-known logic-based formalisms (e.g., modal logics) in the style of multi-dimensional modal logics, that is, we apply modal-like operators to different pieces of DL syntax. Clearly, two-dimensional DLs can model different dynamic aspects of knowledge depending on whether the modal-like operators are temporal, epistemic, etc. Notably, several two-dimensional DLs with different expressive power and different computational properties can be designed depending on the level of interaction between the component logics, e.g., depending on which pieces of DL syntax are enriched with modallike operators. In this thesis, we further the understanding and utility of two-dimensional DLs. We particularly focus on identifying two-dimensional DLs providing the right expressive power to model more accurately temporal and contextual aspects of knowledge required by certain DL applications, or providing better computational properties than other possible alternatives. With this in mind, we pursue three lines of research: 1. we study branching-time temporal DLs that emerge from the combination of classical DLs with the classical temporal logics CTL∗ and CTL; 2. we study description logics of change that emerge from the combination of classical DLs with the modal logic S5; 3. we study description logics of context that emerge from the combination of classical DLs with multi-modal logics. We particularly investigate temporal and contextual DLs based on the classical DL ALC and on members of the EL-family of DLs. We moreover consider different levels of interaction between the component logics. Our main technical contributions are algorithms for satisfiability and subsumption, and (mostly) tight complexity bounds.