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# OpenHPI 4.3 - How DO I Define a Formal Model of an Ontology

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### OpenHPI 4.3 - How DO I Define a Formal Model of an Ontology

1. 1. Semantic Web Technologies Lecture 4: Knowledge Representations I03: How Do I Deﬁne a Formal Model of an Ontology Dr. Harald Sack Hasso Plattner Institute for IT Systems Engineering University of Potsdam Spring 2013 This ﬁle is licensed under the Creative Commons Attribution-NonCommercial 3.0 (CC BY-NC 3.0)
2. 2. 2 Lecture 4: Knowledge Representations I Open HPI - Course: Semantic Web Technologies Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
3. 3. 163 03 How do I define a formal model of an ontology?Open HPI - Course: SemanticHarald Sack, Hasso-Plattner-Institut, Universität Potsdam Semantic Web Technologies , Dr. Web Technologies - Lecture 4: Knowledge Representations I
4. 4. Propositional Logic164 • 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. Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
5. 5. First Order Logic • In First Order Logic (FOL) quantors allow assertions about sets165 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! Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
6. 6. Description Logics166 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. Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
7. 7. Description Logics166 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. TBox terminological knowledge Lecture Knowledge about concepts of a domain (classes, attributes, relations…) Lecture ABox assertional knowlegde „Semantic Web Technologies“ knowledge about instances / entities Knowledge Base Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
8. 8. Description Logics - A Brief Summary167 • Concepts (unary predicates), • represent entities / classes • e.g., Person, Course, Student, Lecturer, Seminar, ... Student: { x | Student(x)} • Roles (binary predicates, properties) • represent properties / relations • e.g., participatesAt, givesLecture, isGivenByLecturer, … participatesAt: {(x,y)|participatesAt(x,y)} Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
9. 9. Description Logics - A Brief Summary168 • 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) Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
10. 10. Description Logics169 • Fundamental operators: • Conjunction (⊓), • Disjunction (⊔), • Negation (⌐) • restricted form of Quantiﬁcation (∀,∃) • represents Basic Description Logic  ALC • Attributive Language with Complement Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
11. 11. Attributive Language with Complement - ALC1610 • 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 Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
12. 12. Attributive Language with Complement - ALC1610 • Atomic Types • concept names A, B, ... • special concepts • ⊤ - Top (universal concept) • ⊥ - Bottom concept Deﬁnes a Class • role names R,S, ... range restriction • Constructors property / role • Negation: ¬C • Conjunction: C ⊓ D ∃attends.Lecture • Disjunction: C ⊔ D • Existential Quantiﬁer: ∃R.C • Universal Quantiﬁer: ∀R.C Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
13. 13. Attributive Language with Complement - ALC1611 • 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) Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
14. 14. Attributive Language with Complement - ALC1612 • Terminological Knowledge (TBox) • Axioms describing the structure of the represented domain (conceptional schema) • Human ⊑ ∃hasParent.Human Orphan ≣ Human ⊓ ¬∃hasParents.Alive • Assertional Knowledge (ABox) • Axioms describing speciﬁc situations (data) • Orphan(harrypotter) hasParent(harrypotter, jamespotter) Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
15. 15. Description Logics1613 Operator / Constructor Syntax Language Conjunction A⊓B Value Restriction ∀ R .C FL Existential Quantification ∃R Top (Universal Concept) ⊤ Bottom (Most Special Concept) ⊥ S* Negation (C) ⌐A Disjunction C⊔D AL* Existential Restriction ∃ R .C Cardinality Restriction (N) (≤ n R) (≥ n R) Set of Individuals (O) {a1,…,an} Hierarchy of Relations R⊆S H Inverse Relation R-1 I Qualified Cardinality Restriction (≤ n R.C) (≥ n R.C) Q Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
16. 16. Semantics of Description Logics1614 • 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 ∈ C I} (∃R.\$)I # =# {a ∈ ΔI | ∃ b.<a,b>∈ RI}# Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
17. 17. How should we represent Ontologies?1615 • Ontologies can also be modelled via database or software modelling technologies, as e.g. • UML, ER-Model, … n n Seminar participatesAt Person - Titel: String - GivenName: String n 1 - Semester: String givesLecture - FamilyName: String - Begin: Date -… - End: Date -… Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
18. 18. 1616 04 Ontology TypesOpen HPI - Course: SemanticHarald Sack, Hasso-Plattner-Institut, Universität Potsdam Semantic Web Technologies , Dr. Web Technologies - Lecture 4: Knowledge Representations I
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