Identifiers

• URL (Uniform Resource Locator)
• It is a mechanism to specify and locate a web resource on a computer network.
• Eg: http://, ftp://
• URN (Uniform Resource Name)
• They are persistent location independent identifiers that use the urn scheme.
• Eg: ISBN of a book is a URN used to identify a book uniquely

Identifiers

• URI (Uniform Resource Identifier)
• It is an identifier that can uniquely and unambiguously identify resources
• ASCII encoding is used in the identifiers
• URL and URN are type of URIs. URIs and URNs need not be resolvable to a location, which is the case with URL
• Eg: http://www.w3.org/2002/07/owl#
• IRI (International Resource Identifier)
• IRI is similar to URI except that the character set for encoding the identifier is Universal Character Set (UTF-8)
• Eg: é, ö could be used in IRI

Naming Conventions

• Class: They are written in upper camel case (or Pascal case) where every first letter in a compound word is capitalized
  Eg: Father, Parent, MyBirthdayGuests
• Properties: They are written in lower camel case where every first letter in a compound word excluding the first word is capitalized
  Eg: hasDaughter, hasAge
• Individuals: same as Properties
  Eg: mary, john, julie

• Following syntaxes are supported by OWL
• Functional-Style syntax
• RDF/XML
• OWL/XML
• Manchester syntax
• We will follow the functional syntax since it is human friendly

• Declaration(Class(Person))
• Declaration(NamedIndividual(mary))
• ClassAssertion(Person mary)
• ClassAssertion(Women mary)
• ObjectPropertyAssertion(hasWife john mary)
  
  
• Axioms: Basic statements such as the above are called axioms of an OWL ontology
• Entities: Class, Properties, and Individuals
• Expression: combinations of entities to form complex descriptions from basic ones
• All these together capture the knowledge of a domain

Intersection

• Intersection of two classes consists of exactly those individuals which are instances of both the classes
• EquivalentClasses(Mother ObjectIntersectionOf(Woman Parent))
• Class Mother consists of exactly those instances that belong to both Woman and Parent

Union

• union of two classes contains every individual which is contained in at least one of these classes
• EquivalentClasses(Parent ObjectUnionOf(Mother Father))
• DisjointClasses(Mother Father)
• Instance of Parent is either an instance of Mother or Father

Complement

• Complement of a class consists of exactly those individuals that are not members of the other class
• EquivalentClasses(ChildlessPerson ObjectIntersectionOf(Person ObjectComplementOf(Parent))
• Class constructors can be nested
• SubClassOf(Grandfather ObjectIntersectionOf(Man Parent))
• ClassAssertion(ObjectIntersectionOf(Person ObjectComplementOf(Parent)) Jack)

• A quantifier specifies the quantity of specimen from a domain of discourse over which a property acts on.
• Domain of discourse is the universe or the set of objects that we are interested in. Eg: Car, Person, Mother, Father
• Two quantifiers are of interest to us
• Existential quantifier ($\exists$)
• Universal quantifier ($\forall$)

Existential Quantifier

• There should exist at least one instance from the domain of discourse (class) that participates in the given relation/property
• Eg: Any Exam must have at least one Examiner
• SubClassOf(Exam ObjectSomeValuesFrom(hasExaminer Examiner))
• Eg: Parent is someone who has at least one child
• EquivalentClasses(Parent ObjectSomeValuesFrom(hasChild Person))
• Phrases such as "at least one", "for some" are used for this quantifier

Universal Quantifier

• A given property participates with all the instances of a particular domain of discourse (class)
• Eg: All Examiners of an Exam must be Professors
• SubClassOf(Exam ObjectAllValuesFrom(hasExaminer Professor))
• Corner case: Exams which have no Examiner are also covered by this quantifier
• "for all" is used for this quantifier

Universal Quantifier

• Eg: A happy person is someone who has a happy child
• EquivalentClasses(HappyPerson ObjectAllValuesFrom(hasChild HappyPerson))
• The above quantification indicates that a childless person can also be happy
• Intuitively, we want to say that in order to be happy, a person must have at least one happy child
• EquivalentClasses(HappyPerson ObjectIntersectionOf(ObjectAllValuesFrom(hasChild HappyPerson) ObjectSomeValuesFrom(hasChild HappyPerson)))

Max cardinality

• John has at most 4 children
• ClassAssertion(ObjectMaxCardinality(4 hasChild Parent) John)
• This declares an upper bound on the number of children that John has

Min cardinality

• John has at least 2 children
• ClassAssertion(ObjectMinCardinality(2 hasChild Parent) John)
• This declares a lower bound on the number of children that John has

Exact cardinality

• John has exactly 3 children
• ClassAssertion(ObjectExactCardinality(3 hasChild Parent) John)
• This declares an exact number of children that John has

Inverse Property

• If one property can be obtained by inverting or changing the direction of the other then it is an inverse property
• InverseObjectProperties(hasParent hasChild)
• ObjectPropertyAssertion(hasParent A B)
• ObjectPropertyAssertion(hasChild B A)

Symmetric Property

• If a property and its inverse are the same then it is called a symmetric property
• SymmetricObjectProperty(hasSpouse)
• ObjectPropertyAssertion(hasSpouse A B)
• ObjectPropertyAssertion(hasSpouse B A)

Assymmetric Property

• It explicitly states that if A is connected to B through a certain property then B cannot be connected to A by the same property
• AsymmetricObjectProperty(hasChild)

Reflexive Property

• A reflexive property connects everything to itself
• ReflexiveObjectProperty(hasRelative)
• Everyone is related to himself/herself

Irreflexive Property

• If an individual cannot be connected to itself by the given property then such a property is irreflexive
• IrreflexiveObjectProperty(parentOf)
• Nobody can be one's own parent

Functional Property

• If a property connects A to B then there can be only one unique value for B
• FunctionalObjectProperty(hasHusband)
• ObjectPropertyAssertion(hasHusband Mary James)
• ObjectPropertyAssertion(hasHusband Mary Jim)
• Conclusion: Since hasHusband is functional, James and Jim should be the same

Inverse Functional Property

• If a property connects A to B then there can be only one unique value for A
• InverseFunctionalObjectProperty(hasHusband)
• An individual can be the husband of at most one another individual

Transitive Property

• A transitive property connects A with C if it connects A to B and B to C
• TransitiveObjectProperty(hasAncestor)
• Transitive properties can be used to connect whole-part relations
• TransitiveObjectProperty(subRegionOf)

Disjoint Property

• Two properties are disjoint if there are no two individuals that are connected by the two properties
• DisjointObjectProperties(hasParent hasSpouse)
• Parents and children cannot marry each other

• An OWL 2 profile is a fragment or a sublanguage of OWL 2
• Each profile is a trimmed down version of OWL 2 that trades expressivity with efficiency of reasoning
• OWL 2 Full, OWL 2 DL, OWL 2 EL, OWL 2 QL, and OWL 2 RL
• OWL 2 DL
• $SROIQ(D)$ is the description logic behind OWL 2 DL
• Reasoning complexity is N2ExpTime
• OWL 2 Full
• OWL 2 DL + RDF(S)
• Undecidable

• OWL 2 EL, OWL 2 QL, OWL 2 RL are the tractable profiles of OWL 2
• OWL 2 EL
• $EL^{++}$ is the formal description logic underpinning for OWL 2 EL
• Useful in applications that contain very large number of classes and properties
• Large biomedical ontologies such as SNOMED CT is in OWL 2 EL
• OWL 2 QL
• The description logic underpinning for OWL 2 QL is DL-Lite
• Useful in applications that have large instance data
• Query answering is the most important reasoning task in this profile
• OWL 2 RL
• Useful for applications that require scalable reasoning
• OWL 2 RL is inspired by Description Logic Programs (DLP) and pD$^{*}$
• Rule-based reasoning techniques are used in this profile