Abstract
Many description logics (DLs) combine knowledge representation on an abstract, logical level with an interface to 'concrete' domains like numbers and strings with built-in predicates such as >, +, and prefix-of. These hybrid DLs have turned out to be useful in several application areas, such as reasoning about conceptual database models. We propose to further extend such DLs with key constraints that allow the expression of statements like 'US citizens are uniquely identified by their social security number'. Based on this idea, we introduce a number of natural description logics and perform a detailed analysis of their decidability and computational complexity. It turns out that naive extensions with key constraints easily lead to undecidability, whereas more careful extensions yield NExpTime-complete DLs for a variety of useful concrete domains.
Highlights
3.1 Undecidability of ALCK(D) with General Key Boxes We prove that satisfiability of ALCK(D)-concepts w.r.t. key boxes is undecidable for a large class of concrete domains if we allow complex ALCK(D)-concepts to occur in key assertions
Starting from this observation, we introduced a number of natural description logics and provided a comprehensive analysis of the decidability and complexity of reasoning
The main observation of our investigations is that key boxes can have dramatic consequences on the complexity of reasoning: for example, the PSpacecomplete description logics (DLs) ALC(D) becomes NExpTime-complete if extended with path-free, unary, Boolean key boxes and undecidable if extended with path-free, unary, non-Boolean key boxes
Summary
Description logics (DLs) are a family of formalisms that allow the representation of and reasoning about conceptual knowledge in a structured and semantically well-understood manner (Baader, Calvanese, McGuinness, Nardi, & Patel-Schneider, 2003). While the basic DL with concrete domains ALC(D) has already been discussed above, SHOQ(D) was proposed as an ontology language in (Horrocks & Sattler, 2001) It provides a wealth of expressive possibilities such as general concept inclusion axioms (GCIs), transitive roles, role hierarchies, nominals, and qualifying number restrictions. Our main result concerning nominals is that, in general being of lower expressive power than key boxes, they already make reasoning NExpTime-hard if combined with concrete domains: there exist concrete domains D such that ALCO(D)-concept satisfiability is NExpTime-complete. Our algorithm implies the following upper complexity bound: if D is a key-admissible concrete domain for which a non-deterministic polynomial time D-tester exists, ALCO(D)concept satisfiability w.r.t. Boolean key boxes is in NExpTime.
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