On the complexities of Horn description logics

Description Logics are currently advancing to become a very prominent paradigm for rep-resenting knowledge bases in a variety of application areas. Central to leveraging them for corresponding systems is the provision of a favourable balance between expressivity of the knowledge representation formalism on the one hand, and runtime performance of reason-ing algorithms on the other. Due to this, Horn description logics (Horn-DLs) have recently started to attract attention since their (worst-case) data complexities are in general lower than their overall (i.e. combined) complexities, which makes them attractive for reasoning with large sets of instance data (ABoxes). However, the natural question whether Horn-DLs also provide advantages for schema (TBox) reasoning has hardly been addressed so far. In this paper, we therefore provide a thorough and comprehensive analysis of the combined complexities of Horn-DLs. While the combined complexity for many Horn-DLs turns out to be the same as for their non-Horn counterparts, we identify subboolean DLs where Horn-ness simplifies reasoning. We also provide convenient normal forms for Horn-DLs