Abstract

We present lower bounds on the computational complexity of satisfiability and subsumption in several Description Logics. We interpret these lower bounds as coming from different “sources of complexity”, which we isolate one by one. We consider both reasoning with simple concept expressions and reasoning with an underlying TBox. We discuss also complexity of instance checking in simple ABoxes. We have tried to enhance clarity and ease of presentation, sometimes sacrificing exhaustiveness for lack of space.

Introduction

Complexity of reasoning has been one of the major issues in the development of Description Logics. This is because such logics are conceived [Brachman and Levesque, 1984] as the formal specification of subsystems for representing knowledge, to be used in larger knowledge-based systems. Since using knowledge also means deriving implicit facts from the given ones, the implementation of derivation procedures should take into account the optimality of reasoning algorithms. The study of optimal algorithms starts from the elicitation of the computational complexity of the problem the algorithm should solve. Initially, studies about the complexity of reasoning problems in Description Logics were more focused on polynomial-time versus intractable (NP- or coNP-hard) problems. The idea was that a knowledge representation system based on a Description Logic with polynomial-time inference problems would guarantee timely answers to the rest of the system. However, once systems based on very expressive Description Logics with exponential-time reasoning problems were implemented [Horrocks, 1998b], it was recognized that knowledge bases of realistic size could be processed in reasonable time.