# description logics

Description logics are a family of formal knowledge representation languages. Usually, their expressivity is between propositional logic and first order logic.

## Description Logics

### Syntax

Below, we define the \(\mathcal{SROIQ}\) syntax.

#### RBox

RBox contains the roles axioms, roles are either the universal role or a role name \(r\) or its inverse \(r^{-}\).

The axioms are of the for \(r_{1} \circ r_{2} \circ \dots \circ r_{n} \sqsubseteq r\), si n > 1 it is called a transitivity statement.

#### TBox

Here, we list the possible *concept expressions* with \(C\) and \(D\) two concept expressions and \(r\) is a simple role:

*top*,*bottom**nominal concepts*, defined as finite set of individual name,*negation*or*complementary*, if \(C\) is a concept expression, then \(\neg C\) is a concept expression too,*intersection*, \(C \sqcap D\) is a concept expression,*union*, \(C \sqcup D\) is a concept expression,*existential quantification*, \(\exists r.C\) is a concept expression,*universal quantification*, \(\forall r.C\) is a concept expression,*self restriction*, if r is simple, \(\exists r.Self\) is a concept expression,*at-least restriction*, for \(n\) a natural number, \(\geq n r.C\) is a concept expression,*at-most restriction*, for \(n\) a natural number, \(\leq n r.C\) is a concept expression,Axioms are in a TBox are concept expressions inclusions, like \(C \sqsubseteq D\).

### AL-languages

\(\mathcal{AL}\) is an abbreviation for attributive language. It allows the following concept descriptions:

- atomic concepts, top and bottom
- negation of atomic concept
- intersection
- universal quantification
limited existential quantification of the form \(\exists r. \top\)

The AL language can be extended with union (\(\mathcal U\)), full existential quantification (\(\mathcal E\)), at-least and at-most restriction (\(\mathcal N\)) and negation (\(\mathcal C\)).

### The letters S, R, O, I, F, N, Q

The letters:

- \(\mathcal S\) denotes \(\mathcal{ALC}\), where we additionally allow transitivity statements,
- \(\mathcal H\) in the name of a DL indicates that role hierarchy are supported,
- \(\mathcal SR\) denotes \(\mathcal{ALC}\) with all kinds of RBox axioms as well as self concepts,
- \(\mathcal O\) in the name of a DL indicates that nominal concepts are supported,
- \(\mathcal I\) in the name of a DL indicates that inverse roles are supported,
- \(\mathcal F\) in the name of a DL indicates that role functionally statements are supported (\(\perp \sqsubseteq \leq1 r.\top\)),
- \(\mathcal N\) in the name of a DL indicates that unqualified at-least or at-most restrictions are supported, i.e. \(\leq n r.\top\) or \(\geq n r.\top\),
- \(\mathcal Q\) in the name of a DL indicates that qualified number restrictions are supported.

### The description logic EL

EL allows only concepts (no roles), which can be either the top concept, an atomic concept, the intersection of two concepts or the existential quantification of a concept (atomic or not). The TBox is a finite set of concept inclusions.

## Metamodeling

### OWL-Full

In DBLP:journals/logcom/Motik07, OWL-Full is represented as the most expressive of the Semantic Web ontology languages. Contrary to OWL-DL, OWL-Full does not impose the following restrictions:

- the sets of logical and metalogical symbols (e.g. rdf:type) are strictly separated,
- the sets of symbols used as concepts, roles and individuals are strictly separated,
restrictions required to yield a decidable logic, such as the one on simple roles in number restrictions.

Since 3. is not enforced in OWL-Full, OWL-Full is undecidable. The subset ALC-Full is undecidable if 1. and 2. are not enforced.

## Systems

### Reasoners

### Knowledge base systems

- RDFox is C++ in memory RDF triple store. It supports RDFS, datalog and OWL2 RL reasoning and SPARQL queries.

### Editors

- Protégé is a free, open-source ontology editor,
- WebVOWL is a web visualizer for OWL file

## References

- wikipedia page: https://en.wikipedia.org/wiki/Description_logic
- reference book on DLs: DBLP:conf/dlog/2003handbook
- about the DL-Lite family: ArtaleDLLitefamilyrelations2009
- introduction to descriptions logics: https://www.aifb.kit.edu/images/1/19/DL-Intro.pdf