# 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:

1. the sets of logical and metalogical symbols (e.g. rdf:type) are strictly separated,
2. the sets of symbols used as concepts, roles and individuals are strictly separated,
3. 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

• KAON2 is Java-based reasoner for query answering supporting SHIQ description logics,
• FaCT++ is a free open-source C++-based reasoner

### 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

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