Book contents
- Frontmatter
- Contents
- List of Figures
- List of Tables
- Abbreviations
- Symbols
- Acknowledgements
- 1 Introduction
- I Logical Preliminaries - Hybrid Logics, Decidability, Deductive Systems
- 2 Modal logic, decidability and complexity
- 3 Deductive systems
- 4 Hybrid logic
- 5 Logic M(En)
- 6 Remarks on description logics contributions
- II Deductive Systems for Hybrid Logics
- Bibliography
- Index
6 - Remarks on description logics contributions
from I - Logical Preliminaries - Hybrid Logics, Decidability, Deductive Systems
Published online by Cambridge University Press: 05 January 2015
- Frontmatter
- Contents
- List of Figures
- List of Tables
- Abbreviations
- Symbols
- Acknowledgements
- 1 Introduction
- I Logical Preliminaries - Hybrid Logics, Decidability, Deductive Systems
- 2 Modal logic, decidability and complexity
- 3 Deductive systems
- 4 Hybrid logic
- 5 Logic M(En)
- 6 Remarks on description logics contributions
- II Deductive Systems for Hybrid Logics
- Bibliography
- Index
Summary
This chapter is aimed to provide a concise overview of basic notions used in description logics. Description and modal (and also hybrid) logics have a lot in common. In fact, the former can be considered as a notational variant of the latter. For that reason, references to works from the description logic field occur throughout the whole book. A brief explanation of a description logic terminology we refer to in our considerations is therefore in order.
First, let's recall that the basic modal logic is defined over a sig-nature 〈 prop, R 〉, where prop is a denumerable set of propositional variables p1, p2, … and R is a binary relation over the universe of a model. Description logics use the notion of atomic concepts rather than propositional formulas. Let C0 denote a set of all atomic concepts of (an abstract) description logic D. Again, in description logics we talk about roles rather than about accessibility relations. Let R denote a set of all roles of D. We have all at hand to define a syntax of D.
Definition6.1 (Syntax of D). Let C0 be the set of atomic concepts and R be the set of roles in D. We define the set C of all concepts of the language of the description logic D as follows:
C := ⊤ | A | ⌝C | C ⊓ D | C ⊓ D | ∃R.C | ∀R. C | n ≥ R.C | n ≤ R.C,
where A ∈ C0, C, D ∈ C, R ∈ R and n ∈ ℕ.
Before we draw a parallel between concrete modal and description notions, let's define a model for D which is called an interpretation.
Definition 6.2 (Semantics of D).
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- Publisher: Jagiellonian University PressPrint publication year: 2014