Complexities of Horn Description Logics

  • Authors:

  • Markus Krötzsch;Sebastian Rudolph;Pascal Hitzler

  • Affiliations:

  • University of Oxford;Karlsruhe Institute of Technology;Wright State University

  • Venue:
  • ACM Transactions on Computational Logic (TOCL)
  • Year:
  • 2013

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Abstract

Description logics (DLs) have become a prominent paradigm for representing knowledge in a variety of application areas, partly due to their ability to achieve a favourable balance between expressivity of the logic and performance of reasoning. Horn description logics are obtained, roughly speaking, by disallowing all forms of disjunctions. They have attracted attention since their (worst-case) data complexities are in general lower than those of their non-Horn counterparts, which makes them attractive for reasoning with large sets of instance data (ABoxes). It is therefore natural to ask whether Horn DLs also provide advantages for schema (TBox) reasoning, that is, whether they also feature lower combined complexities. This article settles this question for a variety of Horn DLs. An example of a tractable Horn logic is the DL underlying the ontology language OWL RL, which we characterize as the Horn fragment of the description logic SROIQ without existential quantifiers. If existential quantifiers are allowed, however, many Horn DLs become intractable. We find that Horn-ALC already has the same worst-case complexity as ALC, that is, ExpTime, but we also identify various DLs for which reasoning is PSpace-complete. As a side effect, we derive simplified syntactic definitions of Horn DLs for which we exploit suitable normal form transformations.