Complexity Results and Practical Algorithms for Logics in Knowledge Representation

TL;DR

This paper presents new complexity results and practical reasoning algorithms for expressive Description Logics, focusing on counting quantifiers, with implications for system optimization and reasoning efficiency.

Contribution

It establishes novel complexity bounds for counting restrictions in DLs and introduces practical tableau algorithms for SHIQ and clique guarded fragment.

Findings

Local counting does not increase inference complexity in many DLs.

Global counting restrictions can significantly raise complexity.

Practical tableau algorithms were developed for SHIQ and CGF.

Abstract

Description Logics (DLs) are used in knowledge-based systems to represent and reason about terminological knowledge of the application domain in a semantically well-defined manner. In this thesis, we establish a number of novel complexity results and give practical algorithms for expressive DLs that provide different forms of counting quantifiers. We show that, in many cases, adding local counting in the form of qualifying number restrictions to DLs does not increase the complexity of the inference problems, even if binary coding of numbers in the input is assumed. On the other hand, we show that adding different forms of global counting restrictions to a logic may increase the complexity of the inference problems dramatically. We provide exact complexity results and a practical, tableau based algorithm for the DL SHIQ, which forms the basis of the highly optimized DL system iFaCT. Finally, we describe a tableau algorithm for the clique guarded fragment (CGF), which we hope will serve as the basis for an efficient implementation of a CGF reasoner.