openhpi 4.3 - how do i define a formal model of an ontology

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Dr. Harald Sack

Hasso Plattner Institute for IT Systems Engineering

University of Potsdam

Spring 2013

Semantic Web Technologies

Lecture 4: Knowledge Representations I03: How Do I Define a Formal Model of an Ontology

Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam

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Lecture 4: Knowledge Representations I

Open HPI - Course: Semantic Web Technologies


316

03 How do I define a formal model of an ontology?

Open HPI - Course: Semantic Web Technologies - Lecture 4: Knowledge Representations I


416• in propositional logic the world consists simply of facts and

nothing else (statements of assertions)

• Example for propositional logic assertions and deductions:

• If it rains, the road will get wet.• If the moon is made out of green cheese, then cows can fly.

• If Oliver is in love, then he will be happy.

• The world consists out of objects and properties that distinguish one object from another.

• Between objects are relations. Some relations are unique, i.e. functions.

Propositional Logic


516• In First Order Logic (FOL) quantors allow assertions about sets

of objects, without naming the objects explicitely.

• All humans are mortal.• Socrates is a human.• Socrates is mortal.

• FOL is perfectly suited for the description of ontologies, but...

• FOL is rather expressive,

• therefore also rather bulky for modelling,

• difficult to achieve consense in modelling and

• rather complex to proof (correctness and completeness of assertions)

• Therefore: look for some well suited fragment of FOL!

First Order Logic


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Description Logics

Description Logics are a family of languages for knowledge representation. Most description logics are a subset of First Order Logic, but in difference to FOL most description logics are decidable. Therefore, it is possible to make logical deductions based on description logics, i.e. to create new knowledge from existing knowledge.


616

Description Logics

Description Logics are a family of languages for knowledge representation. Most description logics are a subset of First Order Logic, but in difference to FOL most description logics are decidable. Therefore, it is possible to make logical deductions based on description logics, i.e. to create new knowledge from existing knowledge.

Lecture

Lecture„Semantic Web Technologies“

TBox terminological knowledge Knowledge about concepts of a domain (classes, attributes, relations…)

ABox assertional knowlegde knowledge about instances / entities

Knowledge Base


716 • Concepts (unary predicates),

• represent entities / classes

• e.g., Person, Course, Student, Lecturer, Seminar, ...

• Roles (binary predicates, properties)

• represent properties / relations

• e.g., participatesAt, givesLecture, isGivenByLecturer, …

Student: { x | Student(x)}

participatesAt: {(x,y)|participatesAt(x,y)}

Description Logics - A Brief Summary


816 • Individuals (constants, individual entities, concept assertion)

• e.g., Alice, Bob, SemanticWeb

• Syntax: Student(Alice)

• Operators / Constructors (to construct complex representations of concepts / roles)

• Expressivity is limited:

• Satisfiability and Subsumption is decidable and

• (preferably) of low complexity

• Syntax: participatesAt(Alice, SemanticWeb)

Description Logics - A Brief Summary


916• Fundamental operators:

• Conjunction (⊓),

• Disjunction (⊔),

• Negation (⌐)

• restricted form of Quantification (∀,∃)

• represents Basic Description Logic ALC• Attributive Language with Complement

Description Logics


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Attributive Language with Complement - ALC• Atomic Types

• concept names A, B, ...

• special concepts

• ⊤ - Top (universal concept)

• ⊥ - Bottom concept

• role names R,S, ...

• Constructors

• Negation: ¬C

• Conjunction: C ⊓ D

• Disjunction: C ⊔ D

• Existential Quantifier: ∃R.C

• Universal Quantifier: ∀R.C


1016

Attributive Language with Complement - ALC• Atomic Types

• concept names A, B, ...

• special concepts

• ⊤ - Top (universal concept)

• ⊥ - Bottom concept

• role names R,S, ...

• Constructors

• Negation: ¬C

• Conjunction: C ⊓ D

• Disjunction: C ⊔ D

• Existential Quantifier: ∃R.C

• Universal Quantifier: ∀R.C

∃attends.Lecture

Defines a Class

property / role

range restriction


1116 • Class Relations

• Inclusion C ⊑ D

• E.g., Man ⊑ Human

• Equality C ≣ D

• E.g., Frau ≣ Woman

• Class Constructors

• E.g., Seminarist ≡ Person ⊓ (∃participatesAt.Seminar ⊔ ∃manages.Seminar)

Attributive Language with Complement - ALC


1216 • Terminological Knowledge (TBox)

• Axioms describing the structure of the represented domain(conceptional schema)

• Human ⊑ ∃hasParent.HumanOrphan ≣ Human ⊓ ¬∃hasParents.Alive

• Assertional Knowledge (ABox)

• Axioms describing specific situations (data)

• Orphan(harrypotter)hasParent(harrypotter, jamespotter)

Attributive Language with Complement - ALC


1316 Operator / Constructor Syntax LanguageLanguage

Conjunction A ⊓ B

FL

S*

Value Restriction ∀ R .C FL

S*

Existential Quantification ∃ R

FL

S*

Top (Universal Concept) ⊤

AL*

S*Bottom (Most Special Concept) ⊥

AL*

S*Negation (C) ⌐ A

AL*

S*

Disjunction C ⊔ D AL*

S*

Existential Restriction ∃ R .C

AL*

S*

Cardinality Restriction (N) (≤ n R) (≥ n R)

AL*

S*

Set of Individuals (O) {a1,…,an}

AL*

S*

Hierarchy of Relations R ⊆ S HH

Inverse Relation R-1 II

Qualified Cardinality Restriction (≤ n R.C) (≥ n R.C) QQ

Description Logics


1416 • Semantics is determined via Interpretation (ΔI, I)

• ΔI … Domain of Discourse, ΔI ≠ ∅

• Interpretation Function:

• I :A " AI ⊆ ΔI , A ... atomic concept

• I :R " RI ⊆ ΔI x ΔI , R … atomic role/property

$I # # = # ΔI⊥I # # =# ∅(¬A)I # # =# ΔI \ AI # # #(C ⊓ D)I # =# CI ∩ DI # # #(∀R.C)I # =# {a ∈ ΔI | ∀ b.<a,b> ∈ RI ⇒ b ∈ CI}

(∃R.$)I # =# {a ∈ ΔI | ∃ b.<a,b>∈ RI}#

Semantics of Description Logics


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• Ontologies can also be modelled via database or softwaremodelling technologies, as e.g.

• UML, ER-Model, …

Seminar

- Titel: String - Semester: String - Begin: Date - End: Date - …

Person

- GivenName: String - FamilyName: String - …

participatesAtnn

givesLecture1n

How should we represent Ontologies?


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04 Ontology TypesOpen HPI - Course: Semantic Web Technologies - Lecture 4: Knowledge Representations I

openhpi 4.3 - how do i define a formal model of an ontology

Education

c semantic web technologies

web technologies lecture

c conjunction

c d disjunction

description of ontologies

c universal quantier

c d attends

c d existential quantier