OpenHPI 5.1 - Description Logics - ALC

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OpenHPI 5.1 - Description Logics - ALC

  1. 1. Semantic Web TechnologiesLecture 5: Knowledge Representations II 01: Description Logics - ALC 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 5: Knowledge Representations II Open HPI - Course: Semantic Web Technologies Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
  3. 3. 3 01 Description Logics - ALCOpen HPI - Course: Semantic Web Technologies - Lecture Potsdam Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität 5: Knowledge Representations II
  4. 4. Knowledge Representations4 logic-based non logic-based formalisms formalisms • more complex and difficult • closer to human intuition to understand • therefore easier to • all based on first order logic understand • consistent semantics • usually don‘t have • FOL Syntax consistent semantics • FOL Semantics • FOL Entailment • E.g.: • Semantic Networks • E.g.: • Frame-based representations • Description Logics • Rule-based representations Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  5. 5. FO FOL as Semantic Web Language?5 • Why not simply take FOL for Ontologies? L • FOL can do everything... • compare higher programming languages to assemblers • FOL has • high expressivity • too bulky for modelling • not appropriate to find consensus in modelling • proof theoretically very complex (semi-decidable) • FOL is also not a Markup Language Look for an appropriate fragment of FOL Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  6. 6. Description Logics (DLs)6 • DLs are Fragments of FOL • In DL from simple descriptions more complex descriptions are created with the help of Constructors. • DLs differ in the applied constructors (Expressivity) • DLs have been developed from „semantic Networks“ • DLs are decidable (most times) • DLs possess sufficient expressivity (most times) • DLs are related to modal logics • e.g., W3C Standard OWL 2 DL is based on description logics SHROIQ(D) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  7. 7. Attributive Language with Complements ALC7 Building Blocks: • Classes • Roles / Properties • Individuals • Student(Christian) Individual Christian is of class Student • Lecture(SemanticWeb) Individual SemanticWeb is of class lecture • visitsLecture(Christian, SemanticWeb) Christian visits the lecture SemanticWeb Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  8. 8. ALC - Building Blocks8 • 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 Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  9. 9. ALC - Building Blocks9 • Class Inclusion • Professor ⊑ FacultyMember • every Professor is a Faculty Member • equals (∀x)(Professor(x) → FacultyMember(x)) • Class Equivalence • Professor ≡ FacultyMember • the Faculty Members are exactly the Professors • equals (∀x)(Professor(x) FacultyMember(x)) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  10. 10. ALC - Complex Class Relations10 • Conjunction ⊓ • Disjunction ⊔ • Negation ¬ Professor ⊑ (Person ⊓ UniversityEmployee) " " ⊔ (Person ⊓ ¬Student) (∀x)(Professor(x) → ((Person(x) Λ UniversityEmployee(x)) V (Person(x) Λ ¬Student(x))) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  11. 11. ALC - Quantifiers on Roles11 •Strict Binding of the Range of a Role to a Class •Examination ⊑ ∀hasSupervisor.Professor • An Examination must be supervised by a Professor • (∀x)(Examination(x) → (∀y)(hasSupervisor(x,y) → Professor(y))) •Open Binding of the Range of a Role to a Class •Examination ⊑ ∃hasSupervisor.Person • Every Examination has at least one supervisor (who is a person) • (∀x)(Examination(x) → (∃y)(hasSupervisor(x,y) Λ Person(y))) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  12. 12. ALC - Formal Syntax12 •Production rules for creating classes in ALC: (A is an atomic class, C and D are complex classes and R is a Role) •C,D::= A|⊤|!|¬C|C⊓D|C⊔D|∃R.C|∀R.C Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  13. 13. ALC - Formal Syntax12 •Production rules for creating classes in ALC: (A is an atomic class, C and D are complex classes and R is a Role) •C,D::= A|⊤|!|¬C|C⊓D|C⊔D|∃R.C|∀R.C •An ALC TBox contains assertions of the form C ⊑ D and C ≡ D, where C,D are complex classes. Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  14. 14. ALC - Formal Syntax12 •Production rules for creating classes in ALC: (A is an atomic class, C and D are complex classes and R is a Role) •C,D::= A|⊤|!|¬C|C⊓D|C⊔D|∃R.C|∀R.C •An ALC TBox contains assertions of the form C ⊑ D and C ≡ D, where C,D are complex classes. •An ALC ABox contains assertions of the form C(a) and R(a,b), where C is a complex Class, R a Role and a,b Individuals. Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  15. 15. ALC - Formal Syntax12 •Production rules for creating classes in ALC: (A is an atomic class, C and D are complex classes and R is a Role) •C,D::= A|⊤|!|¬C|C⊓D|C⊔D|∃R.C|∀R.C •An ALC TBox contains assertions of the form C ⊑ D and C ≡ D, where C,D are complex classes. •An ALC ABox contains assertions of the form C(a) and R(a,b), where C is a complex Class, R a Role and a,b Individuals. •An ALC-Knowledge Base contains an ABox and a TBox. Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  16. 16. ALC - Semantic (Interpretation) •we define a model-theoretic semantic for ALC13 (i.e. Entailment will be defined via Interpretations) •an Interpretation I=(ΔI,.I) contains •a set ΔI (Domain) of Individuals and •an interpretation function .I that maps •Individual names a to domain elements aI∈ΔI •Class names C to a set of domain elements CI⊆ΔI •Role names R to a set of pairs of domain elements RI⊆ΔI×ΔI Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  17. 17. ALC - Semantic (Interpretation)14 Individual Names Class Names Role Names Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  18. 18. ALC - Semantic (Interpretation)15 • Extension for complex classes: •⊤I = ΔI and ⊥I = ∅ Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  19. 19. ALC - Semantic (Interpretation)15 • Extension for complex classes: •⊤I = ΔI and ⊥I = ∅ •(C ⊔ D)I = CI ∪ DI and (C ⊓ D)I = CI ∩ DI Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  20. 20. ALC - Semantic (Interpretation)15 • Extension for complex classes: •⊤I = ΔI and ⊥I = ∅ •(C ⊔ D)I = CI ∪ DI and (C ⊓ D)I = CI ∩ DI •(¬C)I = ΔI CI Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  21. 21. ALC - Semantic (Interpretation)15 • Extension for complex classes: •⊤I = ΔI and ⊥I = ∅ •(C ⊔ D)I = CI ∪ DI and (C ⊓ D)I = CI ∩ DI •(¬C)I = ΔI CI •∀R.C={a∈ΔI|(∀b∈ΔI)((a,b)∈RI&b∈CI)} Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  22. 22. ALC - Semantic (Interpretation)15 • Extension for complex classes: •⊤I = ΔI and ⊥I = ∅ •(C ⊔ D)I = CI ∪ DI and (C ⊓ D)I = CI ∩ DI •(¬C)I = ΔI CI •∀R.C={a∈ΔI|(∀b∈ΔI)((a,b)∈RI&b∈CI)} •∃R.C={a∈ΔI|(∃b∈ΔI)((a,b)∈RI∧b∈CI)} Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  23. 23. ALC - Semantic (Interpretation)16 •...and Axioms: • C(a) holds, iff aI ∈ CI • R(a,b) holds, iff (aI,bI) ∈ RI • C ⊑ D holds, iff CI ⊆ DI • C ≡ D holds, iff CI = DI Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  24. 24. ALC - Knowledgebase17 • Terminological Knowledge (TBox) Axioms that describe the structure of the modeled domain (conceptional schema): • Human ⊑ ∃parentOf.Human • Orphan ≡ Human ⊓ ¬∃hasParent.Alive • Assertional Knowledge (ABox) Axioms that describe specific situations (data): • Orphan(harrypotter) • hasParent(harrypotter,jamespotter) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  25. 25. Description Logics Operator/Constructor Syntax Language18 Conjunction A⊓B Value Restriction ∀R.C FL Existential Quantifier ∃R Top ⊤ Bottom ⊥ S* Negation ¬A Disjunction A⊔B AL* Existential Restriction ∃R.C Number Restriction (≤nR) (≥nR) Set of Inividuals {a1,...,a2} Role Hierarchy R⊑S H inverse Role R-1 I Qualified Number Restriction (≤nR.C) (≥nR.C) Q Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  26. 26. Description Logics • ALC: Attribute Language with Complement19 • S: ALC + Transitivity of Roles • H: Role Hierarchies • O: Nominals • I: Inverse Roles • N: Number restrictions ≤n R etc. • Q: Qualified number restrictions ≤n R.C etc. • (D): Datatypes • F: Functional Roles • R: Role Constructors • OWL 2 DL is SHROIQ(D) Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
  27. 27. 20 02 DL Inference and ReasoningOpen HPI - Course: SemanticHarald Sack, Hasso-Plattner-Institut, Universität Potsdam Semantic Web Technologies , Dr. Web Technologies - Lecture 5: Knowledge Representations II

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