Question: What are the axioms in the OWL?
Evidence: OWL BIBREF15 is the de-facto standard for machine processable and interoperable ontologies on the SW. In its second version, OWL is equivalent to the description logic $\mathcal {SROIQ}(D)$. Such expressiveness has a higher computational cost but allows the development of interesting applications such as automated reasoning BIBREF16. OWL 2 ontologies consist of the following three different syntactic categories:
Entities, such as classes, properties, and individuals, are identified by IRIs. They form the primitive terms and constitute the basic elements of an ontology. Classes denote sets of individuals and properties link two individuals or an individual and a data value along a property. For example, a class :Animal can be used to represent the set of all animals. Similarly, the object property :childOf can be used to represent the parent-child relationship and the data property :birthDate assigns a particular birth date to an individual. Finally, the individual :Alice can be used to represent a particular person called "Alice".
Expressions represent complex notions in the domain being described. For example, a class expression describes a set of individuals in terms of the restrictions on the individuals' characteristics. OWL offers existential (SOME) or universal (ONLY) qualifiers and a variety of typical logical constructs, such as negation (NOT), other Boolean operators (OR, AND), and more constructs such as cardinality restriction (MIN, MAX, EXACTLY) and value restriction (VALUE), to create class expressions. Such constructs can be combined in arbitrarily complex class expressions CE according to the following grammar
where A is an atomic class, C and D are class expressions, R is an object property, a as well as a$_1$ to a$_m$ with $\texttt {m} \ge 1$ are individuals, and $\texttt {n} \ge 0$ is an integer.
Axioms are statements that are asserted to be true in the domain being described. Usually, one distinguish between (1) terminological and (2) assertional axioms. (1) terminological axioms are used to describe the structure of the domain, i.e., the relationships between classes resp. class expressions. For example, using a subclass axiom (SubClassOf:), one can state that the class :Koala is a subclass of the class :Animal. Classes can be subclasses of other classes, thus creating a taxonomy. In addition, axioms can arrange properties in hierarchies (SubPropertyOf:) and can assign various characteristics (Characteristics:) such as transitivity or reflexivity to them. (2) Assertional axioms formulate facts about individuals, especially the classes they belong to and their mutual relationships. OWL can be expressed in various syntaxes with the most common computer readable syntax being RDF/XMLA more human-readable format is the MOS BIBREF17. For example, the class expression that models people who work at a university that is located in Spain could be as follows in MOS:
Likewise, expressing that every professor works at a university would read as
OWL 2 ontologies consist of Entities, Expressions and Axioms as introduced in subsec:owl. While both expressions and axioms can be mapped to RDF, i.e. into a set of RDF triples, using this mapping and applying the triple-based verbalization on it would lead to a non-human understandable text in many cases. For example, the intersection of two classes :A and :B can be represented in RDF by the six triples
The verbalization of these triples would result in Something that is a class and the intersection of something whose first is A and whose rest is something whose first is B and whose rest ist nil., which is obviously far away from how a human would express it in NL. Therefore, generating NL from OWL requires a different procedure based on its syntactic categories, OWL expressions and OWL axioms. We show the general rules for each of them in the following.
In theory, class expressions can be arbitrarily complex, but as it turned out in some previous analysis BIBREF22, in practice they seldom arise and can be seen as some corner cases. For example, an ontology could contain the following class expression about people and their birth place:
Class expressions do have a tree-like structure and can simply be parsed into a tree by means of the binary OWL class expressions constructors contained in it. For our example, this would result in the following tree:
every tree node/.style=align=center,anchor=base,font=, edge from parent/.style= thick, draw, edge from parent path=(.south) – +(0,-8pt) -| () frontier/.style=distance from root=9 [.AND Person [.SOME birthPlace [.AND City [.VALUE locatedIn France ] ] ] ]
Such a tree can be traversed in post-order, i.e. sub-trees are processed before their parent nodes recursively. For the sake of simplicity, we only process sub-trees that represent proper class expression in our example, i.e. we omit birthPlace, locatedIn, and France. Moreover and again for simplicity, we'll explain the transformation process by starting from the right-hand side of the tree. Thus, in our example we begin with the class expression City which is transformed to everything that is a city and locatedIn VALUE France resulting in everything that is located in France by application of a rule. Both class expressions are used in the conjunction City AND locatedIn VALUE France. Thus, the next step would be to merge both phrases. An easy way is to use the coordinating conjunction and, i.e. everything that is a city and everything that is located in France. Although the output of this transformation is correct, it still contains unnecessarily redundant information. Therefore, we apply the aggregation procedure described in subsec:grouping, i.e. we get everything that is a city and located in France. Yet, the aggregation can still be improved: if there is any atomic class in the conjunction, we know that this is more specific than the placeholder everything. Thus, we can replace it by the plural form of the class, finally resulting in cities that are located in France. The same procedure is applied for its parent class expression being the existential restriction
This will be transformed to everything whose birth place is a city that is located in France. Note, that we used the singular form here, assuming that the property birthPlace is supposed to be functional in the ontology. In the last step, we process the class expression Person, which gives us everything that is a person. Again, due to the conjunction we merge this result with with the previous one, such that in the end we get people whose birth place is a city that is located in France.
As we described in sec:owl, OWL axioms can roughly be categorized into terminological and assertional axioms. Therefore, we have different procedures for processing each category:
Assertional Axioms (ABox Axioms) - Most assertional axioms assert individuals to atomic classes or relate individuals to another individual resp. literal value. For example, axioms about the type as well as birth place and birth date of Albert Einstein can be expressed by
Those axioms can simply be rewritten as triples, thus, we can use the same procedure as we do for triples (sec:singletriple). Converting them into NL gives us Albert Einstein is a person whose birth place is Ulm and whose birth date is 14 March 1879. OWL also allows for assigning an individual to a complex class expression. In that case we'll use our conversion of OWL class expressions as described in subsec:owlce.
Terminological Axioms (TBox Axioms) - According to power2010, most of the terminological axioms used in ontologies are subclass axioms. By definition, subclass and superclass can be arbitrarily complex class expressions $\texttt {CE}_1$ and $\texttt {CE}_2$, i.e. [language=manchester,mathescape=true]CE$_1$ SubClassOf CE$_2$, but in praxis it is quite often only used with atomic classes as subclass or even more simple with the superclass also beeing an atomic class. Nevertheless, we support any kind of subclass axiom and all other logical OWL axioms in LD2NL. For simplicity, we outline here how we verbalize subclass axioms in LD2NL. The semantics of a subclass axiom denotes that every individual of the subclass also belongs to the superclass. Thus, the verbalization seems to be relatively straightforward, i.e. we verbalize both class expressions and follow the template : every $\rho (\texttt {\texttt {CE}$1$})$ is a $\rho (\texttt {\texttt {CE}$2$})$. Obviously, this works pretty well for subclass axioms with atomic classes only. For example, the axiom
is verbalized as every scientist is a person.
This work was supported by the German Federal Ministry of Transport and Digital Infrastructure (BMVI) through the projects LIMBO (no. 19F2029I) and OPAL (no. 19F2028A). This work was supported by the German Federal Ministry of Economics and Technology (BMWI) in the projects RAKI (no. 01MD19012D) as well as by the BMBF project SOLIDE (no. 13N14456).
Axioms are statements that are asserted to be true in the domain being described.
Answer: Axioms are statements that are asserted to be true in the domain being described.