Editing Description logic (section)

==Nomenclature==
===Terminology compared to FOL and OWL===
The description logic community uses different terminology than the [[first-order logic]] (FOL) community for operationally equivalent notions; some examples are given below. The [[Web Ontology Language]] (OWL) uses again a different terminology, also given in the table below.

{| class="wikitable sortable"
|+ Synonyms
|-
! FOL
! OWL
! DL
!Examples
|-
| constant
| individual
| individual
|Mickey Mouse, Walter Elias Mouse, Paris, France, etc. 
|-
| unary predicate
| class
| concept
|(Being a) person, a city, a country, etc.
|-
| binary predicate
| property
| role
|father of, located in, etc.
|}

===Naming convention===
There are many varieties of description logics and there is an informal naming convention, roughly describing the operators allowed. The [[Expressive power (computer science)|expressivity]] is encoded in the label for a logic starting with one of the following basic logics:
{|
|-
|<math>\mathcal{AL}</math> ||Attributive language. This is the base language which allows:
|-
| ||
* Atomic negation (negation of concept names that do not appear on the left-hand side of axioms)
* Concept intersection
* Universal restrictions
* Limited existential quantification
|-
| ||
|-
|<math>\mathcal{FL}</math>  ||Frame based description language,<ref>{{cite journal | last1 = Levesque | first1 = Hector J. | author1-link = Hector Levesque | last2 = Brachmann | first2 = Ronald J. | author2-link = Ronald Brachman | title = Expressiveness and tractability in knowledge representation and reasoning | journal = [[Computational Intelligence (journal)|Computational Intelligence]] | volume = 3 | pages = 78–93 | year = 1987 | number = 3| doi = 10.1111/j.1467-8640.1987.tb00176.x | s2cid = 30031046 }}</ref> allows:
|-
| ||
* Concept intersection
* Universal restrictions
* Limited existential quantification
* Role restriction
|-
| ||
|-
|<math>\mathcal{EL}</math>  ||Existential language, allows:
|-
| ||
* Concept intersection
* Existential restrictions (of full existential quantification)
|}
Followed by any of the following extensions:
{|
|-
|<math>\mathcal{F}</math>  ||Functional properties, a special case of [[uniqueness quantification]].
|-
| ||
|-
|<math>\mathcal{E}</math>  ||Full existential qualification (existential restrictions that have fillers other than <math>\top</math>).
|-
| ||
|-
|<math>\mathcal{U}</math>  ||Concept union.
|-
| ||
|-
|<math>\mathcal{C}</math>  ||Complex concept negation.
|-
| ||
|-
|<math>\mathcal{H}</math>  ||Role hierarchy (subproperties: <code>rdfs:subPropertyOf</code>).
|-
| ||
|-
|<math>\mathcal{R}</math>  ||Limited complex role inclusion axioms; reflexivity and irreflexivity; role disjointness.
|-
| ||
|-
|<math>\mathcal{O}</math>  ||Nominals. (Enumerated classes of object value restrictions: <code>owl:oneOf</code>, <code>owl:hasValue</code>).
|-
| ||
|-
|<math>\mathcal{I}</math>  ||Inverse properties.
|-
| ||
|-
|<math>\mathcal{N}</math>  ||Cardinality restrictions (<code>owl:cardinality</code>, <code>owl:maxCardinality</code>), a special case of [[counting quantification]]
|-
| ||
|-
|<math>\mathcal{Q}</math>  ||Qualified cardinality restrictions (available in OWL 2, cardinality restrictions that have fillers other than <math>\top</math>).
|-
| ||
|-
|<math>^\mathcal{(D)}</math>  ||Use of datatype properties, data values or data types.
|}

====Exceptions====
Some canonical DLs that do not exactly fit this convention are:
{|
|-
|<math>\mathcal{S}</math>  ||An abbreviation for <math>\mathcal{ALC}</math> with transitive roles.
|-
| ||
|-
|<math>\mathcal{FL^-}</math> ||A sub-language of <math>\mathcal{FL}</math>, which is obtained by disallowing role restriction. This is equivalent to <math>\mathcal{AL}</math> without atomic negation.
|-
| ||
|-
|<math>\mathcal{FL}_o</math>||A sub-language of <math>\mathcal{FL^-}</math>, which is obtained by disallowing limited existential quantification.
|-
| ||
|-
|<math>\mathcal{EL^{++}}</math>||Alias for <math>\mathcal{ELRO}</math>.<ref>{{cite journal | first1 = Frederick | last1 = Maier | first2 = Raghava | last2 = Mutharaju | first3 = Pascal | last3 = Hitzler |author-link3=Pascal Hitzler | title = Distributed Reasoning with EL++ Using MapReduce | journal = Computer Science and Engineering Faculty Publications | url = http://corescholar.libraries.wright.edu/cse/243/ | year = 2010 | publisher = Technical Report, Kno.e.sis Center, Wright State University, Dayton, Ohio | access-date = 2016-08-24}}</ref>
|}

====Examples====
As an example, <math>\mathcal{ALC}</math> is a centrally important description logic from which comparisons with other varieties can be made. <math>\mathcal{ALC}</math> is simply <math>\mathcal{AL}</math> with complement of any concept allowed, not just atomic concepts. <math>\mathcal{ALC}</math> is used instead of the equivalent <math>\mathcal{ALUE}</math>.

A further example, the description logic <math>\mathcal{SHIQ}</math> is the logic <math>\mathcal{ALC}</math> plus  extended cardinality restrictions, and transitive and inverse roles. The naming conventions aren't purely systematic so that the logic <math>\mathcal{ALCOIN}</math> might be referred to as <math>\mathcal{ALCNIO}</math> and other abbreviations are also made where possible.

The Protégé ontology editor supports <math>\mathcal{SHOIN}^\mathcal{(D)}</math>. Three major biomedical informatics terminology bases, [[SNOMED CT]], GALEN, and GO, are expressible in <math>\mathcal{EL}</math> (with additional role properties).

OWL 2 provides the expressiveness of  <math>\mathcal{SROIQ}^\mathcal{(D)}</math>, OWL-DL is based on <math>\mathcal{SHOIN}^\mathcal{(D)}</math>, and for OWL-Lite it is <math>\mathcal{SHIF}^\mathcal{(D)}</math>.