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Introduction to Web Ontology Language (OWL)

Bangladesh Short-Term Course (Artificial Intelligence)

Plaban Kumar Bhowmick

Assistant Professor

Centre of Excellence in Artificial Intelligence

Indian Institute of Technology Kharagpur

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Ontology is a formal Specification of a shared conceptualization of a domain of interest [Gruber 93]

Formal Specification

of

Conceptualization

Concepts

…

Domain of

Interest

of

Reasoning

+

Processable

Group of

shared

People

Web agents

Applications

Ontology

Services

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What’s there in an ontology?

Concepts:

Classes + class-hierarchy

Properties:

often also called “Roles” or “Slots”

Axioms/Relations:

hasAncestor is a transitive property
A person can have at most one father or mother

Cardinality Restriction

Definition of Grandfather class

Reasoning tasks:

Classification: Does in individual belong to the Grandfather class?
Subsumption: Is a class subsumed by another class?
Consistency Checking: Whether the definitions/axioms are consistent or not.

Person

Man

Woman

Father

Grandfather

Daughter

Aunt

hasAncenstor

hasParent

isWifeof

hasWife

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Ontology Language Requirements

Requirements

Well defined syntax
Well defined semantics
Efficient reasoning support
Sufficient expressive power
Convenience of expression

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Formal Semantics and Reasoning

Class membership

If x is an instance of C and C is a subclass of D, then x is an instance of D

Equivalence of classes

If A is equivalent class of B and B is an equivalent class of C then A is equivalent to C

Consistency

x is an instance of class A and B. A and B are disjoint classes

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Limitation of Expressive Power of RDF Schema

Local scope of properties

No localized domain or range restriction
In RDFS we cannot represent

Cows eat only plants while other animals eat meat too

Disjointness of classes

In RDFS we cannot represent

Male and Female classes are disjoint

Boolean combination of classes

In RDFS we cannot say

Person class is disjoint union of Male and Female class

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Limitation of Expressive Power of RDF Schema

Cardinality restriction

In RDFS we cannot represent

A person has exactly two biological parents

Special characteristics of properties

In RDFS we cannot represent

A property may be transitive, symmetric
A property may be unique
A property is inverse of another property

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Origins of OWL

DAML

DAML+OIL

OIL

OWL

RDF

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Ontology in Layer Cake

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OWL-RDFS Connection

Same triple-based data model

Extends RDFS vocabulary

OWL is more expressive by including more complex relations of classes and properties

SUBJECT

PREDICATE

OBJECT

Same Data Model

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OWL-RDFS Connection

rdfs:Resource

rdfs:Class

rdf:Property

owl:Class

owl:ObjectProperty

owl:DatatypeProperty

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Three Species of OWL

OWL Full

Uses all OWL primitives
Combines primitives with RDF and RDFS
Pros:

Very expressive
Fully compatible with RDF and RDFS

Cons:

Inference is undecidable without any guarantee on complete and efficient reasoning

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Three Species of OWL

OWL DL

Uses a subset of OWL primitives
Restrictions on the RDF and RDFS primitives to be used
Pros:

Efficient reasoning support

Cons:

Loose full compatibility with RDF
Every RDF document is not an OWL document

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Three Species of OWL

OWL Lite

More restricted and easy to use OWL constructs
Pros:

Efficient reasoning, easier to grasp and implement

Cons:

Restricted expressivity

Full

DL

Lite

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The OWL Syntax

Class Elements

Classes are defined with owl:Class

<owl:Class rdf:ID="associateProfessor">

<rdfs:subClassOf rdf:resource="#academicStaffMember"/>

</owl:Class>

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Class Elements

<owl:Class rdf:about="#associateProfessor">

<owl:disjointWith rdf:resource="#professor"/>

<owl:disjointWith rdf:resource="#assistantProfessor"/>

</owl:Class>

Someone who is an associate professor is not either a professor or an assistant professor

<owl:Class rdf:ID="faculty">

<owl:equivalentClass rdf:resource="#academicStaffMember"/>

</owl:Class>

Define Faculty class to be equivalent to academicStaffMember class

Disjoint

Equivalence

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Two Special Classes

owl:Thing

owl:Nothing

Set of all individuals

Empty set

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Property Elements

Data Properties (Concrete Roles)

owl:DataTypeProperty

<owl:DatatypeProperty rdf:ID="age">

<rdfs:range rdf:resource=

"http://www.w3.org/2001/XMLSchema#nonNegativeInteger"/>

</owl:DatatypeProperty>

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Property Elements

Object Properties (Abstract Roles)

owl:ObjectProperty

<owl:ObjectProperty rdf:ID="isTaughtBy">

<rdfs:domain rdf:resource="#course"/>

<rdfs:range rdf:resource="#academicStaffMember"/>

<rdfs:subPropertyOf rdf:resource="#involves"/>

</owl:ObjectProperty>

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Property Elements

<owl:ObjectProperty rdf:ID="teaches">

<owl:inverseOf rdf:resource="#isTaughtBy"/>

#<rdfs:range rdf:resource="#course"/>

#<rdfs:domain rdf:resource="#academicStaffMember"/>

</owl:ObjectProperty>

Inverse Properties

owl:inverseOf

<owl:ObjectProperty rdf:ID="lecturesIn">

<owl:equivalentProperty rdf:resource="#teaches"/>

</owl:ObjectProperty>

Equivalent Properties

owl:equivalentProperty

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Property Restrictions

<owl:Class rdf:about="#firstYearCourse">

<rdfs:subClassOf>

<owl:Restriction>

<owl:onProperty rdf:resource="#isTaughtBy"/>

<owl:allValuesFrom rdf:resource="#Professor"/>

</owl:Restriction>

</rdfs:subClassOf>

</owl:Class>

Defining the class of all possible values that a property can take

owl:allValuesFrom

first year courses are taught by only professors

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Property Restrictions

<owl:Class rdf:about="#academicStaffMember">

<rdfs:subClassOf>

<owl:Restriction>

<owl:onProperty rdf:resource="#teaches"/>

<owl:someValuesFrom rdf:resource="#undergraduateCourse"/>

</owl:Restriction>

</rdfs:subClassOf>

</owl:Class>

Defining the class of some possible values that a property can take

owl:someValuesFrom

all academic staff members must teach at least one undergraduate course

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Property Restrictions

<owl:Class rdf:about="#mathCourse">

<rdfs:subClassOf>

<owl:Restriction>

<owl:onProperty rdf:resource="#isTaughtBy"/>

<owl:hasValue rdf:resource="#949352"/>

</owl:Restriction>

</rdfs:subClassOf>

</owl:Class>

A specific value that a property can take

owl:hasValue

mathematics courses are taught by staff member with

ID 949358

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Property Restrictions

<owl:Class rdf:about="#course">

<rdfs:subClassOf>

<owl:Restriction>

<owl:onProperty rdf:resource="#isTaughtBy"/>

<owl:minCardinality>

1

</owl:minCardinality>

</owl:Restriction>

</rdfs:subClassOf>

</owl:Class>

Maximum and minimum cardinality

owl:maxCardinality owl:minCardinality

every course is taught by at least someone

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OWL Special Properties

OWL Symmetric Property

Statements

exchangedFlagWith isA SymmetricProperty
Messi exchangedFlagWith Ronaldo

Inferences

Ronaldo exchangedFlagWith Messi

OWL Transitive Property

Statements

heavierThan isA TransitiveProperty
Jupiter heavierThan Earth
Earth heavierThan Venus

Inferences

Jupiter heavierThan Venus

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OWL Functional Property

To express uniqueness property

property with min cardinality 0 and max cardinality 1

Statements

hasBirthdate isA FunctionalProperty
TimBernersLee hasBirthdate “1955-06-08”
TimBernersLee hasBirthdate “1955-01-08”

Inferences

Either “195-06-08” and “1955-01-08” are same dates (which is not the case)

Error in ontology: an individual may have only one unique value for the hasBirthdate property

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OWL Inverse Functional Property

Inverse of the property is functional

Statements

bioMotherOf isA InverseFunctionalProperty
Kunti is bioMotherOf Arjuna
Pritha bioMotherOf Arjuna

Inferences

Only one individual may have one mother
Either Kunti and Pritha are same individuals or
Inconsistent ontology

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OWL Inverse Functional Property

bioMotherOf (invFunc)

Arjuna

Pritha

Kunti

bioMotherOf (invFunc)

hasBioMother (func)

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Boolean Combinations

<owl:Class rdf:about="#course">

<rdfs:subClassOf>

<owl:complementOf rdf:resource="#staffMember"/>

</rdfs:subClassOf>

</owl:Class>

Set complement operation

owl:complementOf

Courses and staff members are complement

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Boolean Combinations

<owl:Class rdf:ID="peopleAtUni">

<owl:equivalentClass>

<owl:unionOf rdf:parseType="Collection">

<owl:Class rdf:about="#staffMember"/>

<owl:Class rdf:about="#student"/>

</owl:unionOf>

</owl:equivalentClass>

</owl:Class>

Set union operation

owl:unionOf

People at university consist of staff members and students

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Boolean Combinations

<owl:Class rdf:ID="facultyInCS">

<owl:equivalentClass>

<owl:intersectionOf rdf:parseType="Collection">

<owl:Class rdf:about="#faculty"/>

<owl:Restriction>

<owl:onProperty rdf:resource="#belongsTo"/>

<owl:hasValue rdf:resource="#CSDepartment"/>

</owl:Restriction>

</owl:intersectionOf>

</owl:equivalentClass>

</owl:Class>

Set intersection operation

owl:inersectionOf

CS faculty is set equal to the intersection of faculty set and set of all objects from CS department

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Boolean Enumeration

<owl:Class rdf:ID=“WineColor">

<owl:oneOf rdf:parseType="Collection">

<owl:Thing rdf:about="#White"/>

<owl:Thing rdf:about="#Red"/>

<owl:Thing rdf:about="#Rose"/>

</owl:oneOf>

</owl:Class>

Enumerate a set of individuals

owl:oneOf

Winecolor = {White, Red, Rose}

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Defining Individuals

<rdf:Description rdf:ID=“949352”>

<rdf:type> rdf:resource=“#academicStaffMember”

</rdf:Description>

<owl:NamedIndividual rdf:about=“949352”/>

<rdf:type rdf:resource=“#academicStaffMember”/>

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Open World Assumption and Individuals

<owl:ObjectProperty rdf:ID=“isTaughtBy”>

<rdf:type rdf:resource=“&owl;FunctionalProperty” />

</rdf:ObjectProperty>

</course>

<owl:differentFrom rdf:resource=“#949352”/>

</course>

<owl:sameAs rdf:resource=“#949352”/>

</course>

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Defining Distinct Individuals

<owl:AllDifferent>

<owl:distinctMembers rdf:parseType=“Collection”>

</owl:distinctMembers>

</owl:AllDifferent>

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OWL Elements

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OWL Elements

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OWL 2 Extension

Type separation

Class names, role names and individual names must be distinct in OWL 1
Class name may be used as role name in OWL 2

<owl:Class rdf:about=“Professor“ />

<owl:Class rdf:about=“Institute” />

<owl:ObjectProperty rdf:about=“Professor” />

</Institute>

A name must not stand for both an abstract and an concrete role

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OWL 2 Extension

<owl:AllDisjointClasses>

<owl:members rdf:parseType="Collection">

<owl:Class rdf:about="#UndergraduateStudent"/>

<owl:Class rdf:about="#GraguateStudent"/>

<owl:Class rdf:about="#OtherStudent"/>

</owl:members>

</owl:AlllDisjointClasses>

Disjoint Classes more than two

owl:disjointWith owl:AllDisjointClasses

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OWL 2 Extension

Role characteristics and relationships

Asymmetric property

If A is related to B via such a role, B is never related to A via this role
owl:AsymmetricProperty

Reflexive property

Every individual A is related to itself via such a role
owl:reflexiveProperty

IrreflexiveProperty

No individual is related to itself via such a role
owl:IrreflexiveProperty

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OWL 2 Extension

Extensions on properties

Disjoint properties

owl:propertyDisjointWith

More than two disjoint properties

owl:AllDisjointProperties

Top and Bottom abstract property

owl:topObjectProperty
owl:bottomObjectProperty

Top and Bottom concrete property

owl:topDataProperty
owl:bottomDataProperty

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OWL 2 Extension

<owl:ObjectProperty rdf:about=“hasExaminer”>

<rdfs:subPropertyOf>

<owl:ObjectProperty>

<owl:inverseOf rdf:resource=“participatedIn”>

</owl:ObjectProperty>

</rdfs:subPropertyOf>

<owl:ObjectProperty>

Inverse property

owl:inverseOf

If exam A has the person B as examiner, then B participates in A; Not every B participating in exam A is an examiner in A

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OWL 2 Extension

A

hasExaminer

participtedIn

hasExaminer

participatedIn

subPropertyOf

inverseOf

B1

B2

hasParticipation

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OWL 2 Extension

<owl:ObjectProperty rdf:about=“hasUncle”>

<owl:propertyChainAxiom rdf:parseType=“Collection”>

<owl:ObjectProperty rdf:resource=“hasParent”>

<owl:ObjectProperty rdf:resource=“hasBrother”>

<owl:propertyChainAxiom>

<owl:ObjectProperty>

Role Chains

owl:propertyChainAxiom

Concrete roles must not be used

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OWL 2 Extension

<owl:Class rdf:about=“Exam”>

<rdfs:subClassOf>

<owl:Restriction>

<owl:onProperty rdf:resource=“hasExaminer”/>

<owl:maxQualifiedCardinality>

2

<owl:maxQualifiedCardinality>

<owl:onClass rdf:resource=“Professor”>

</owl:Restriction>

</rdfs:subClassOf>

</owl:Class>

Qualified Cardinality Restrictions

owl:qualifiedCardinality, owl:minQualifiedCardinality owl:maxQualifiedCardinality

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OWL 2 Extension

<owl:Class rdf:about=“SelfTaughtStudent”>

<owl:equivalentClass>

<owl:Restriction>

<owl:onProperty rdf:resource=“teaches”/>

<owl:hasSelf rdf:datatype=“&xsd;boolean”>true<owl:hasSelf />

</owl:Restriction>

</owl:equivalentClass>

</owl:Class>

Self Construct

owl:hasSelf

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OWL 2 Extension

<owl:NegativePropertyAssertion>

<owl:sourceIndividual rdf:about=“John Nash” />

<owl:assertionProperty rdf:about=“hasColleague” />

<owl:targetIndividual rdf:about=“Alan Turing”>

</owl:NegativePropertyAssertion>

Negated Role Assignments

owl:NegativePropertyAssertion

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OWL 2: Functional Style Syntax Specification

http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Functional-Style_Syntax

http://www.cs.man.ac.uk/~rector/papers/common_errors_ekaw_2004.pdf

Primary Resource

Rector A. et al. (2004) OWL Pizzas: Practical Experience of Teaching OWL-DL: Common Errors & Common Patterns. Engineering Knowledge in the Age of the Semantic Web. EKAW 2004. Springer, Berlin, Heidelberg

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Frequent Modelling Errors

Whether equivalent or subclass
Nothing is implicit
Class disjointness is taken for granted
Notoriety of Open World Assumption
Boolean combination with quantifiers

Some not or not some

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Defined Vs. Primitive Class

Defined Class

Necessary and Sufficiency condition
Modelled with Equivalent class

Primitive Class

Necessary Condition
Modelled with Subclass

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Equivalent Class

<owl:Class rdf:about="CheeseyPizza">

<owl:equivalentClass>

<owl:Class>

<owl:intersectionOf rdf:parseType="Collection">

<rdf:Description rdf:about="Pizza"/>

<owl:Restriction>

<owl:onProperty rdf:resource="hasTopping"/>

<owl:someValuesFrom rdf:resource="CheeseTopping"/>

</owl:Restriction>

</owl:intersectionOf>

</owl:Class>

</owl:equivalentClass>

</owl:Class>

A cheesy pizza is any pizza that has, amongst other things,

some cheese topping

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Equivalent Class

A cheesy pizza is any pizza that has, amongst other things,

some cheese topping

Pizza

rdf:type

:hasTopping

rdf:type

Cheese

:hasTopping

rdf:type

Meat

CheesyPizza

rdf:type

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Equivalent Class

EquivalentClasses(CheesyPizza

ObjectIntersectionOf(

Pizza

ObjectSomeValuesFrom

(hasTopping Cheese)))

A cheesy pizza is any pizza that has, amongst other things,

some cheese topping

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Sub Class

<owl:Class rdf:about=“MargheritaPizza”>

<rdfs:subClassOf>

<owl:Class>

<owl:intersectionOf rdf:parseType=“Collection”>

<owl:Restriction>

<owl:onProperty rdf:resource=“hasTopping”/>

<owl:someValuesFrom rdf:about="MozzarellaTopping“/>

</owl:Restriction>

<owl:Restriction>

<owl:onProperty rdf:resource=“hasTopping”/>

<owl:someValuesFrom rdf:about=" TomatoTopping”/>

</owl:Restriction>

</ owl:intersectionOf>

</owl:Class>

<rdfs:subClassOf>

</owl:Class>

All Margherita pizzas have, amongst other things, some mozzarella topping and also some tomato topping

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Sub Class

SubClassOf(MargheritaPizza

ObjectIntersectionOf(

ObjectSomeValuesFrom

(hasTopping Mozzarella)

ObjectSomeValuesFrom

(hasTopping Tomato)))

All Margherita pizzas have, amongst other things, some mozzarella topping and also some tomato topping

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Sub Class

SubClassOf(MargheritaPizza

ObjectSomeValuesFrom (hasTopping

ObjectIntersectionOf( Mozzarella Tomato)))

All Margherita pizzas have, amongst other things, some mozzarella topping and also some tomato topping

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Sub Class

All Margherita pizzas have, amongst other things, some mozzarella topping and also some tomato topping

:hasTopping

rdf:type

Mozzarella

:hasTopping

rdf:type

Tomato

MargheritaPizza

rdf:type

:hasTopping

rdf:type

Mozzarella

rdf:type

Tomato

MargheritaPizza

rdf:type

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Stating Disjointness

<owl:AllDisjointClasses>

<owl:members rdf:parseType="Collection">

<owl:Class rdf:about="#Vegetable"/>

<owl:Class rdf:about="#Meat"/>

<owl:Class rdf:about="#Seafood"/>

<owl:Class rdf:about="#Cheese"/>

</owl:members>

</owl:AlllDisjointClasses>

Vegetable, Meat, Seafood Cheese are all Disjoint

DisjointClasses(Vegetable Meat Seafood Cheese)

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<owl:Class rdf:about=“DogOwner”>

<owl:equivalentClass>

<owl:Class>

<owl:intersectionOf rdf:parseType=“Collection”>

<rdf:Description rdf:about=“Person”/>

<owl:Restriction>

<owl:onProperty rdf:resource=“hasPet”/>

<owl:someValuesFrom rdf:resource=“Dog”/>

</owl:Restriction>

</owl:intersectionOf>

</owl:Class>

</owl:equivalentClass>

</owl:Class>

A dog owner is any person who has as a pet some dog

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SomeValuesFrom: Existensial

EquivalentClasses(DogOwner

ObjectIntersectionOf(

Person

ObjectSomeValuesFrom

(hasPet Dog)))

A dog owner is any person who has as a pet some dog

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AllValuesFrom: Universal

<owl:Class rdf:about=“FirstClassLounge”>

<owl:equivalentClass>

<owl:Class>

<owl:intersectionOf rdf:parseType=“Collection”>

<rdf:Description rdf:about=“Lounge”/>

<owl:Restriction>

<owl:onProperty rdf:resource=“hasOccupant”/>

<owl:allValuesFrom rdf:resource=“FirstClassPassenger”/>

</owl:Restriction>

</owl:intersectionOf>

</owl:Class>

</owl:equivalentClass>

</owl:Class>

A first class lounge is any lounge where the occupants are

only first class passengers

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AllValuesFrom: Universal

EquivalentClasses(FirstClassLounge

ObjectIntersectionOf(

Lounge

ObjectAllValuesFrom

(hasOccupant FirstClassPassenger)))

A first class lounge is any lounge where the occupants are

only first class passengers

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AllValuesFrom: Universal

:occupant

rdf:type

FCP

:occupant

rdf:type

FCP

FirstClassLounge

rdf:type

A first class lounge is any lounge where the occupants are

only first class passengers

rdf:type

Lounge

:occupant

rdf:type

FCP

:occupant

rdf:type

SCP

FirstClassLounge

rdf:type

Lounge

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<owl:Class rdf:about=“VegPizza”>

<owl:equivalentClass>

<owl:Class>

<owl:intersectionOf rdf:parseType=“Collection”>

<rdf:Description rdf:about=“Pizza”/>

<owl:Restriction>

<owl:onProperty rdf:resource=“hasTopping”/>

<owl:allValuesFrom>

<owl:unionOf rdf:parseType=“Collection”>

<rdf:Description rdf:about=“Vegetable”/>

<rdf:Description rdf:about=“Cheese”/>

</owl:unionOf>

</owl:allValuesFrom>

</owl:Restriction>

</owl:intersectionOf>

</owl:Class>

</owl:equivalentClass>

</owl:Class>

A vegetarian pizza is any pizza which, amongst other things, has only vegetable and/or cheese toppings

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EquivalentClasses(VegPizza

ObjectIntersectionOf(

Pizza

ObjectAllValuesFrom(hasTopping

ObjectUnionOf

(Vegetable Cheese))))

A vegetarian pizza is any pizza which, amongst other things, has

only vegetable and/or cheese toppings

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A vegetarian pizza is any pizza which, amongst other things, has only vegetable and/or cheese toppings

Pizza

rdf:type

:hasTopping

rdf:type

Veg

:hasTopping

rdf:type

Cheese

VegPizza

rdf:type

Pizza

rdf:type

:hasTopping

rdf:type

Veg

VegPizza

rdf:type

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<owl:Class rdf:about=“ProtienLoversPizza”>

<owl:equivalentClass>

<owl:Class>

<owl:intersectionOf rdf:parseType=“Collection”>

<rdf:Description rdf:about=“Pizza”/>

<owl:Restriction>

<owl:onProperty rdf:resource=“hasTopping”/>

<owl:allValuesFrom>

<owl:intersectionOf rdf:parseType=“Collection”>

<rdf:Description rdf:about=“Meat”/>

<rdf:Description rdf:about=“SeaFood”/>

</owl:intersectionOf>

</owl:allValuesFrom>

</owl:Restriction>

</owl:intersectionOf>

</owl:Class>

</owl:equivalentClass>

</owl:Class>

A protein lover’s pizza is any pizza that, amongst other things, has only toppings that are both meat and also seafood

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EquivalentClasses(ProtienLoversPizza

ObjectIntersectionOf(

Pizza

ObjectAllValuesFrom(hasTopping

ObjectIntersectionOf

(Meat SeaFood))))

A protein lover’s pizza is any pizza that, amongst other things, has only toppings that are both meat and also seafood

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A protein lover’s pizza is any pizza that, amongst other things, has only toppings that are both meat and also seafood

Pizza

rdf:type

:hasTopping

rdf:type

Meat

:hasTopping

rdf:type

Seafood

ProtienLoverPizza

rdf:type

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Complement

ObjectComplementOf(ObjectIntersectionOf(class1 class2))

‘not both class1 and also class2’

ObjectComplementOf(ObjectUnionOf(class1 class2))

‘neither class1 nor class2’

ObjectSomeValuesFrom(prop ObjectComplementOf(class))

‘has some prop that are not class’
some(hasParticipation not(Professor))

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Complement

ObjectComplementOf(ObjectSomeValuesFrom(prop class))

does not have any prop that are class
not(some(hasExaminer ExamSubjectStudent))

ObjectAllValuesFrom(prop ObjectComplementOf(class))

has only prop that are not class
only(hasExaminer not(ExamSubjectStudent))

ObjectComplementOf(ObjectAllValuesFrom(prop class))

does not have only prop that are class
not(only(hasParticipation Faculty ))

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‘Some’ does not imply ‘Only’

SubClassOf(MargheritaPizza

ObjectIntersectionOf(

ObjectSomeValuesFrom

(hasTopping Mozzarella)

ObjectSomeValuesFrom

(hasTopping Tomato)))

SubClassOf(MargheritaPizza

ObjectIntersectionOf( ObjectSomeValuesFrom

(hasTopping Mozzarella)

ObjectSomeValuesFrom

(hasTopping Tomato)

ObjectAllValuesFrom

(hasTopping ObjectUnionOf

(Mozzarella Tomato))))

All Margherita pizzas have, amongst

other things, some mozzarella topping

and also some tomato topping

But, these “Margherita pizzas” might also have, say, spicy beef topping

All Margherita pizzas have, amongst

other things, some mozzarella topping

and also some tomato topping

and also have only mozzarella and/or tomato topping

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Satisfied

Not Satisfied

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‘Only’ does not imply ‘Some’

SubClassOf(EmptyPizza

ObjectComplementOf(

ObjectSomeValuesFrom

(hasTopping owl:Thing)))

EquivalentClasses(VegPizza

ObjectIntersectionOf(

Pizza

ObjectAllValuesFrom (hasTopping

ObjectComplementOf (Meat)))

ObjectAllValuesFrom (hasTopping

ObjectComplementOf (Fish)))))

All empty pizzas, amongst other things, do not have anything as topping

Empty pizza satisfies the definition of vegetarian pizza

A vegetarian pizza is any pizza which, amongst other things, has only topping that is not meat and also has only topping that is not fish.

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Confusion between “some not” and “not some”

EquivalentClasses(VegetarianPizzaW

ObjectIntersectionOf(

Pizza

ObjectSomeValuesFrom

(hasTopping (

ObjectComplementOf(Meat)))

ObjectSomeValuesFrom

(hasTopping (

ObjectComplementOf(Seafood))))

A vegetarian pizza (w) is any pizza that, amongst other things, both has

some topping which is not meat and also some topping which is not seafood

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Confusion between “some not” and “not some”

EquivalentClasses(VegetarianPizzaC

ObjectIntersectionOf(

Pizza

ObjectComplementOf(

ObjectSomeValuesFrom

(hasTopping Meat))

ObjectComplementOf(

ObjectSomeValuesFrom

(hasTopping Seafood))))

A vegetarian pizza (c) is any pizza that, amongst other things, both

does not have some meat topping and also does not have some seafood topping

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A vegetarian pizza (w) is any pizza that, amongst other things, both has

some topping which is not meat and also some topping which is not seafood

A vegetarian pizza (c) is any pizza that, amongst other things, both

does not have some meat topping and also does not have some seafood topping

Pizza

rdf:type

:hasTopping

rdf:type

:hasTopping

rdf:type

VegPizzaW

rdf:type

Pizza

rdf:type

:hasTopping

rdf:type

Meat

:hasTopping

rdf:type

Seafood

VegPizzaC

rdf:type

:hasTopping

Meat

rdf:type

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Cardinality Constraints

EquivalentClasses(InterestingPizza

ObjectIntersectionOf(

Pizza

ObjectMinCardinality

(3 hasTopping)))

An interesting pizza is any pizza that, amongst other things, has

at least 3 (distinct) toppings

SubClassOf(Pizza

ObjectMaxCardinality(1 hasBase))

Any pizza, amongst other things, has at most 1 pizza base

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An interesting pizza is any pizza that, amongst other things, has at least 3 (distinct) toppings

Pizza

rdf:type

:hasTopping

InterestingPizza

rdf:type

:hasTopping

Owl:differentFrom

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Modest are those people who do not praise themselves

EquivalentClasses(ModestPerson

ObjectIntersectionOf(

Person

ObjectComplementOf(

ObjectHasSelf

(praise))))

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All old ladies must have a pet and the pets can only be cats

SubClassOf(OldLady

ObjectIntersectionOf(

ObjectSomeValuesFrom

(hasPet owl:Thing)

ObjectAllValuesFrom

(hasPet Cat)))