Complexities of Horn Description Logics

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.

Topics in Knowledge Bases: Epistemic Ontologies and Secrecy-preserving Reasoning

“…This is of some interest as Horn theories are widely used in AI and Data Bases [97] and more recently KBs [98]. The main reason for this interest is that the satisfiability and inference problems for Horn theories can be solved very efficiently [99].…”

Efficient OWL Reasoning with Logic Programs – Evaluations

“…Among the many possible approaches to address scalability, one of them concerns the use of logic programming for this purpose. It was recently shown that reasoning in Horn-SHIQ [2][3][4] can be realised by invoking Prolog systems on the output of the KAON2-transformations [5]. Still, performance experiments had not been reported yet.…”

“…Convenient access to the KAON2 transformation algorithm is given by means of the KAON2 OWL Tool 3 dlpconvert, 4 which can also produce F-Logic serialisations which can be used with F-Logic engines like OntoBroker.…”

“…Horn [2,3]. A direct definition using a grammar is due to [4]. Horn-SHIQ's data complexity is polynomial, qualifying it as a tractable description logic [8].…”

“…Horn-SHIQ's data complexity is polynomial, qualifying it as a tractable description logic [8]. Its combined complexity is still exponential [4]. In [5] was shown that using off-the shelf Prolog implementations for reasoning with Horn-SHIQ after the KAON2-transformation is possible in principle by using Prolog with tabling, as implemented e.g.…”