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Logical Foundations for the Semantic Web Reasoning with Expressive Description Logics: Theory and Practice Ian Horrocks horrocks@cs.man.ac.uk University of Manchester Manchester, UK Logical Foundations for the Semantic Web p. 1/37 Talk

  1. The Semantic Web Vision ☞ Web made possible through established standards • TCP/IP for transporting bits down a wire • HTTP & HTML for transporting and rendering hyperlinked text ☞ Applications able to exploit this common infrastructure • Result is the WWW as we know it ☞ 1st generation web mostly handwritten HTML pages ☞ 2nd generation (current) web often machine generated/active ☞ Both intended for direct human processing/interaction ☞ In next generation web, resources should be more accessible to automated processes • To be achieved via semantic markup • Metadata annotations that describe content/function ☞ Coincides with Tim Berners-Lee’s vision of a Semantic Web Logical Foundations for the Semantic Web – p. 8/37

  2. Ontologies Logical Foundations for the Semantic Web – p. 9/37

  3. Ontologies ☞ Semantic markup must be meaningful to automated processes Logical Foundations for the Semantic Web – p. 9/37

  4. Ontologies ☞ Semantic markup must be meaningful to automated processes ☞ Ontologies will play a key role • Source of precisely defined terms (vocabulary) • Can be shared across applications (and humans) Logical Foundations for the Semantic Web – p. 9/37

  5. Ontologies ☞ Semantic markup must be meaningful to automated processes ☞ Ontologies will play a key role • Source of precisely defined terms (vocabulary) • Can be shared across applications (and humans) ☞ Ontology typically consists of: • Hierarchical description of important concepts in domain • Descriptions of properties of instances of each concept Logical Foundations for the Semantic Web – p. 9/37

  6. Ontologies ☞ Semantic markup must be meaningful to automated processes ☞ Ontologies will play a key role • Source of precisely defined terms (vocabulary) • Can be shared across applications (and humans) ☞ Ontology typically consists of: • Hierarchical description of important concepts in domain • Descriptions of properties of instances of each concept ☞ Degree of formality can be quite variable (NL–logic) Logical Foundations for the Semantic Web – p. 9/37

  7. Ontologies ☞ Semantic markup must be meaningful to automated processes ☞ Ontologies will play a key role • Source of precisely defined terms (vocabulary) • Can be shared across applications (and humans) ☞ Ontology typically consists of: • Hierarchical description of important concepts in domain • Descriptions of properties of instances of each concept ☞ Degree of formality can be quite variable (NL–logic) ☞ Increased formality and regularity facilitates machine understanding Logical Foundations for the Semantic Web – p. 9/37

  8. Ontologies ☞ Semantic markup must be meaningful to automated processes ☞ Ontologies will play a key role • Source of precisely defined terms (vocabulary) • Can be shared across applications (and humans) ☞ Ontology typically consists of: • Hierarchical description of important concepts in domain • Descriptions of properties of instances of each concept ☞ Degree of formality can be quite variable (NL–logic) ☞ Increased formality and regularity facilitates machine understanding ☞ Ontologies can be used, e.g.: • To facilitate agent-agent communication in e-commerce • In semantic based search • To provide richer service descriptions that can be more flexibly interpreted by intelligent agents Logical Foundations for the Semantic Web – p. 9/37

  9. Ontologies ☞ Semantic markup must be meaningful to automated processes ☞ Ontologies will play a key role • Source of precisely defined terms (vocabulary) • Can be shared across applications (and humans) ☞ Ontology typically consists of: • Hierarchical description of important concepts in domain • Descriptions of properties of instances of each concept ☞ Degree of formality can be quite variable (NL–logic) ☞ Increased formality and regularity facilitates machine understanding ☞ Ontologies can be used, e.g.: • To facilitate agent-agent communication in e-commerce • In semantic based search • To provide richer service descriptions that can be more flexibly interpreted by intelligent agents Logical Foundations for the Semantic Web – p. 9/37

  10. Ontologies ☞ Semantic markup must be meaningful to automated processes ☞ Ontologies will play a key role • Source of precisely defined terms (vocabulary) • Can be shared across applications (and humans) ☞ Ontology typically consists of: • Hierarchical description of important concepts in domain • Descriptions of properties of instances of each concept ☞ Degree of formality can be quite variable (NL–logic) ☞ Increased formality and regularity facilitates machine understanding ☞ Ontologies can be used, e.g.: • To facilitate agent-agent communication in e-commerce • In semantic based search • To provide richer service descriptions that can be more flexibly interpreted by intelligent agents Logical Foundations for the Semantic Web – p. 9/37

  11. Ontologies ☞ Semantic markup must be meaningful to automated processes ☞ Ontologies will play a key role • Source of precisely defined terms (vocabulary) • Can be shared across applications (and humans) ☞ Ontology typically consists of: • Hierarchical description of important concepts in domain • Descriptions of properties of instances of each concept ☞ Degree of formality can be quite variable (NL–logic) ☞ Increased formality and regularity facilitates machine understanding ☞ Ontologies can be used, e.g.: • To facilitate agent-agent communication in e-commerce • In semantic based search • To provide richer service descriptions that can be more flexibly interpreted by intelligent agents Logical Foundations for the Semantic Web – p. 9/37

  12. Web Ontology Languages Logical Foundations for the Semantic Web – p. 10/37

  13. Web Languages Logical Foundations for the Semantic Web – p. 11/37

  14. Web Languages ☞ Web languages already extended to facilitate content description • XML Schema (XMLS) • RDF and RDF Schema (RDFS) Logical Foundations for the Semantic Web – p. 11/37

  15. Web Languages ☞ Web languages already extended to facilitate content description • XML Schema (XMLS) • RDF and RDF Schema (RDFS) ☞ RDFS recognisable as an ontology language • Classes and properties • Range and domain • Sub/super-classes (and properties) Logical Foundations for the Semantic Web – p. 11/37

  16. Web Languages ☞ Web languages already extended to facilitate content description • XML Schema (XMLS) • RDF and RDF Schema (RDFS) ☞ RDFS recognisable as an ontology language • Classes and properties • Range and domain • Sub/super-classes (and properties) ☞ But RDFS not a suitable foundation for Semantic Web • Too weak to describe resources in sufficient detail Logical Foundations for the Semantic Web – p. 11/37

  17. Web Languages ☞ Web languages already extended to facilitate content description • XML Schema (XMLS) • RDF and RDF Schema (RDFS) ☞ RDFS recognisable as an ontology language • Classes and properties • Range and domain • Sub/super-classes (and properties) ☞ But RDFS not a suitable foundation for Semantic Web • Too weak to describe resources in sufficient detail ☞ Requirements for web ontology language: • Compatible with existing Web standards (XML, RDF, RDFS) • Easy to understand and use (based on familiar KR idioms) • Formally specified and of “adequate” expressive power • Possible to provide automated reasoning support Logical Foundations for the Semantic Web – p. 11/37

  18. OIL, DAML-ONT, DAML+OIL and OWL Logical Foundations for the Semantic Web – p. 12/37

  19. OIL, DAML-ONT, DAML+OIL and OWL ☞ Two languages developed to satisfy above requirements • OIL : developed by group of (largely) European researchers • DAML-ONT : developed in DARPA DAML programme Logical Foundations for the Semantic Web – p. 12/37

  20. OIL, DAML-ONT, DAML+OIL and OWL ☞ Two languages developed to satisfy above requirements • OIL : developed by group of (largely) European researchers • DAML-ONT : developed in DARPA DAML programme ☞ Efforts merged to produce DAML+OIL Logical Foundations for the Semantic Web – p. 12/37

  21. OIL, DAML-ONT, DAML+OIL and OWL ☞ Two languages developed to satisfy above requirements • OIL : developed by group of (largely) European researchers • DAML-ONT : developed in DARPA DAML programme ☞ Efforts merged to produce DAML+OIL ☞ Submitted to W3C as basis for standardisation • WebOnt working group developing OWL language standard Logical Foundations for the Semantic Web – p. 12/37

  22. OIL, DAML-ONT, DAML+OIL and OWL ☞ Two languages developed to satisfy above requirements • OIL : developed by group of (largely) European researchers • DAML-ONT : developed in DARPA DAML programme ☞ Efforts merged to produce DAML+OIL ☞ Submitted to W3C as basis for standardisation • WebOnt working group developing OWL language standard ☞ DAML+OIL/OWL “layered” on top of RDFS • RDFS based syntax and ontological primitives (subclass etc.) • Adds much richer set of primitives (transitivity, cardinality, . . . ) Logical Foundations for the Semantic Web – p. 12/37

  23. OIL, DAML-ONT, DAML+OIL and OWL ☞ Two languages developed to satisfy above requirements • OIL : developed by group of (largely) European researchers • DAML-ONT : developed in DARPA DAML programme ☞ Efforts merged to produce DAML+OIL ☞ Submitted to W3C as basis for standardisation • WebOnt working group developing OWL language standard ☞ DAML+OIL/OWL “layered” on top of RDFS • RDFS based syntax and ontological primitives (subclass etc.) • Adds much richer set of primitives (transitivity, cardinality, . . . ) Logical Foundations for the Semantic Web – p. 12/37

  24. OIL, DAML-ONT, DAML+OIL and OWL ☞ Two languages developed to satisfy above requirements • OIL : developed by group of (largely) European researchers • DAML-ONT : developed in DARPA DAML programme ☞ Efforts merged to produce DAML+OIL ☞ Submitted to W3C as basis for standardisation • WebOnt working group developing OWL language standard ☞ DAML+OIL/OWL “layered” on top of RDFS • RDFS based syntax and ontological primitives (subclass etc.) • Adds much richer set of primitives (transitivity, cardinality, . . . ) ☞ Describes structure of domain in terms of Classes and Properties • Ontology is set of axioms describing classes and properties • E.g., Person subclass of Animal whose parents are all Persons Logical Foundations for the Semantic Web – p. 12/37

  25. OIL, DAML-ONT, DAML+OIL and OWL ☞ Two languages developed to satisfy above requirements • OIL : developed by group of (largely) European researchers • DAML-ONT : developed in DARPA DAML programme ☞ Efforts merged to produce DAML+OIL ☞ Submitted to W3C as basis for standardisation • WebOnt working group developing OWL language standard ☞ DAML+OIL/OWL “layered” on top of RDFS • RDFS based syntax and ontological primitives (subclass etc.) • Adds much richer set of primitives (transitivity, cardinality, . . . ) ☞ Describes structure of domain in terms of Classes and Properties • Ontology is set of axioms describing classes and properties • E.g., Person subclass of Animal whose parents are all Persons ☞ Uses RDF for class/property membership assertions (ground facts) • E.g., john instance of Person; � john , mary � instance of parent Logical Foundations for the Semantic Web – p. 12/37

  26. OWL Language Logical Foundations for the Semantic Web – p. 13/37

  27. Foundations Logical Foundations for the Semantic Web – p. 14/37

  28. Foundations ☞ Three species of OWL • OWL full is union of OWL syntax and RDF • OWL DL restricted to FOL fragment ( ≈ DAML+OIL) • OWL Lite is “easier to implement” subset of OWL DL Logical Foundations for the Semantic Web – p. 14/37

  29. Foundations ☞ Three species of OWL • OWL full is union of OWL syntax and RDF • OWL DL restricted to FOL fragment ( ≈ DAML+OIL) • OWL Lite is “easier to implement” subset of OWL DL ☞ Semantic layering • OWL DL ≡ OWL full within DL fragment • DL semantics officially definitive Logical Foundations for the Semantic Web – p. 14/37

  30. Foundations ☞ Three species of OWL • OWL full is union of OWL syntax and RDF • OWL DL restricted to FOL fragment ( ≈ DAML+OIL) • OWL Lite is “easier to implement” subset of OWL DL ☞ Semantic layering • OWL DL ≡ OWL full within DL fragment • DL semantics officially definitive ☞ OWL DL based on SHIQ Description Logic Logical Foundations for the Semantic Web – p. 14/37

  31. Foundations ☞ Three species of OWL • OWL full is union of OWL syntax and RDF • OWL DL restricted to FOL fragment ( ≈ DAML+OIL) • OWL Lite is “easier to implement” subset of OWL DL ☞ Semantic layering • OWL DL ≡ OWL full within DL fragment • DL semantics officially definitive ☞ OWL DL based on SHIQ Description Logic ☞ Benefits from many years of DL research • Well defined semantics • Formal properties well understood (complexity, decidability) • Known reasoning algorithms • Implemented systems (highly optimised) Logical Foundations for the Semantic Web – p. 14/37

  32. OWL Class Constructors Constructor DL Syntax Example (Modal Syntax) intersectionOf C 1 ⊓ . . . ⊓ C n Human ⊓ Male C 1 ∧ . . . ∧ C n C 1 ⊔ . . . ⊔ C n Doctor ⊔ Lawyer C 1 ∨ . . . ∨ C n unionOf ¬ C ¬ Male ¬ C complementOf oneOf { x 1 . . . x n } { john , mary } x 1 ∨ . . . ∨ x n allValuesFrom ∀ P.C ∀ hasChild . Doctor [ P ] C ∃ P.C ∃ hasChild . Lawyer � P � C someValuesFrom maxCardinality � 1 hasChild [ P ] n +1 � nP minCardinality � 2 hasChild � P � n � nP Logical Foundations for the Semantic Web – p. 15/37

  33. OWL Class Constructors Constructor DL Syntax Example (Modal Syntax) intersectionOf C 1 ⊓ . . . ⊓ C n Human ⊓ Male C 1 ∧ . . . ∧ C n C 1 ⊔ . . . ⊔ C n Doctor ⊔ Lawyer C 1 ∨ . . . ∨ C n unionOf ¬ C ¬ Male ¬ C complementOf oneOf { x 1 . . . x n } { john , mary } x 1 ∨ . . . ∨ x n allValuesFrom ∀ P.C ∀ hasChild . Doctor [ P ] C ∃ P.C ∃ hasChild . Lawyer � P � C someValuesFrom maxCardinality � 1 hasChild [ P ] n +1 � nP minCardinality � 2 hasChild � P � n � nP ☞ XMLS datatypes as well as classes in ∀ P.C and ∃ P.C • E.g., ∃ hasAge . nonNegativeInteger Logical Foundations for the Semantic Web – p. 15/37

  34. OWL Class Constructors Constructor DL Syntax Example (Modal Syntax) intersectionOf C 1 ⊓ . . . ⊓ C n Human ⊓ Male C 1 ∧ . . . ∧ C n C 1 ⊔ . . . ⊔ C n Doctor ⊔ Lawyer C 1 ∨ . . . ∨ C n unionOf ¬ C ¬ Male ¬ C complementOf oneOf { x 1 . . . x n } { john , mary } x 1 ∨ . . . ∨ x n allValuesFrom ∀ P.C ∀ hasChild . Doctor [ P ] C ∃ P.C ∃ hasChild . Lawyer � P � C someValuesFrom maxCardinality � 1 hasChild [ P ] n +1 � nP minCardinality � 2 hasChild � P � n � nP ☞ XMLS datatypes as well as classes in ∀ P.C and ∃ P.C • E.g., ∃ hasAge . nonNegativeInteger ☞ Arbitrarily complex nesting of constructors • E.g., Person ⊓ ∀ hasChild . ( Doctor ⊔ ∃ hasChild . Doctor ) Logical Foundations for the Semantic Web – p. 15/37

  35. RDFS Syntax <owl:Class> <owl:intersectionOf rdf:parseType="collection"> <owl:Class rdf:about="#Person"/> <owl:Restriction> <owl:onProperty rdf:resource="#hasChild"/> <owl:toClass> <owl:unionOf rdf:parseType="collection"> <owl:Class rdf:about="#Doctor"/> <owl:Restriction> <owl:onProperty rdf:resource="#hasChild"/> <owl:hasClass rdf:resource="#Doctor"/> </owl:Restriction> </owl:unionOf> </owl:toClass> </owl:Restriction> </owl:intersectionOf> </owl:Class> Logical Foundations for the Semantic Web – p. 16/37

  36. OWL DL Semantics Logical Foundations for the Semantic Web – p. 17/37

  37. OWL DL Semantics ☞ Semantics defined by interpretations : I = (∆ I , · I ) → subsets of ∆ I • concepts − → binary relations over ∆ I (subsets of ∆ I × ∆ I ) • roles − → elements of ∆ I • individuals − Logical Foundations for the Semantic Web – p. 17/37

  38. OWL DL Semantics ☞ Semantics defined by interpretations : I = (∆ I , · I ) → subsets of ∆ I • concepts − → binary relations over ∆ I (subsets of ∆ I × ∆ I ) • roles − → elements of ∆ I • individuals − ☞ Interpretation function · I extended to concept expressions • ( C ⊓ D ) I = C I ∩ D I ( C ⊔ D ) I = C I ∪ D I ( ¬ C ) I = ∆ I \ C I • { x n , . . . , x n } I = { x I n , . . . , x I n } • ( ∃ R.C ) I = { x | ∃ y. � x, y � ∈ R I ∧ y ∈ C I } • ( ∀ R.C ) I = { x | ∀ y. ( x, y ) ∈ R I ⇒ y ∈ C I } • ( � nR ) I = { x | # { y | � x, y � ∈ R I } � n } • ( � nR ) I = { x | # { y | � x, y � ∈ R I } � n } Logical Foundations for the Semantic Web – p. 17/37

  39. OWL Axioms Axiom DL Syntax Example C 1 ⊑ C 2 Human ⊑ Animal ⊓ Biped subClassOf equivalentClass C 1 ≡ C 2 Man ≡ Human ⊓ Male C 1 ⊑ ¬ C 2 Male ⊑ ¬ Female disjointWith { x 1 } ≡ { x 2 } { President_Bush } ≡ { G_W_Bush } sameIndividualAs { x 1 } ⊑ ¬{ x 2 } { john } ⊑ ¬{ peter } differentFrom subPropertyOf P 1 ⊑ P 2 hasDaughter ⊑ hasChild P 1 ≡ P 2 cost ≡ price equivalentProperty hasChild ≡ hasParent − P 1 ≡ P − inverseOf 2 ancestor + ⊑ ancestor P + ⊑ P transitiveProperty functionalProperty ⊤ ⊑ � 1 P ⊤ ⊑ � 1 hasMother ⊤ ⊑ � 1 hasSSN − ⊤ ⊑ � 1 P − inverseFunctionalProperty Logical Foundations for the Semantic Web – p. 18/37

  40. OWL Axioms Axiom DL Syntax Example C 1 ⊑ C 2 Human ⊑ Animal ⊓ Biped subClassOf equivalentClass C 1 ≡ C 2 Man ≡ Human ⊓ Male C 1 ⊑ ¬ C 2 Male ⊑ ¬ Female disjointWith { x 1 } ≡ { x 2 } { President_Bush } ≡ { G_W_Bush } sameIndividualAs { x 1 } ⊑ ¬{ x 2 } { john } ⊑ ¬{ peter } differentFrom subPropertyOf P 1 ⊑ P 2 hasDaughter ⊑ hasChild P 1 ≡ P 2 cost ≡ price equivalentProperty hasChild ≡ hasParent − P 1 ≡ P − inverseOf 2 ancestor + ⊑ ancestor P + ⊑ P transitiveProperty functionalProperty ⊤ ⊑ � 1 P ⊤ ⊑ � 1 hasMother ⊤ ⊑ � 1 hasSSN − ⊤ ⊑ � 1 P − inverseFunctionalProperty ☞ I satisfies C 1 ⊑ C 2 iff C I 1 ⊆ C I 2 ; satisfies P 1 ⊑ P 2 iff P I 1 ⊆ P I 2 Logical Foundations for the Semantic Web – p. 18/37

  41. OWL Axioms Axiom DL Syntax Example C 1 ⊑ C 2 Human ⊑ Animal ⊓ Biped subClassOf equivalentClass C 1 ≡ C 2 Man ≡ Human ⊓ Male C 1 ⊑ ¬ C 2 Male ⊑ ¬ Female disjointWith { x 1 } ≡ { x 2 } { President_Bush } ≡ { G_W_Bush } sameIndividualAs { x 1 } ⊑ ¬{ x 2 } { john } ⊑ ¬{ peter } differentFrom subPropertyOf P 1 ⊑ P 2 hasDaughter ⊑ hasChild P 1 ≡ P 2 cost ≡ price equivalentProperty hasChild ≡ hasParent − P 1 ≡ P − inverseOf 2 ancestor + ⊑ ancestor P + ⊑ P transitiveProperty functionalProperty ⊤ ⊑ � 1 P ⊤ ⊑ � 1 hasMother ⊤ ⊑ � 1 hasSSN − ⊤ ⊑ � 1 P − inverseFunctionalProperty ☞ I satisfies C 1 ⊑ C 2 iff C I 1 ⊆ C I 2 ; satisfies P 1 ⊑ P 2 iff P I 1 ⊆ P I 2 ☞ I satisfies ontology O (is a model of O ) iff satisfies every axiom in O Logical Foundations for the Semantic Web – p. 18/37

  42. XML Datatypes in OWL Logical Foundations for the Semantic Web – p. 19/37

  43. XML Datatypes in OWL ☞ OWL supports XML Schema primitive datatypes Logical Foundations for the Semantic Web – p. 19/37

  44. XML Datatypes in OWL ☞ OWL supports XML Schema primitive datatypes ☞ Clean separation between “object” classes and datatypes • Disjoint interpretation domain: d I ⊆ ∆ D , and ∆ D ∩ ∆ I = ∅ D ⊆ ∆ I × ∆ D • Disjoint datatype properties: P I Logical Foundations for the Semantic Web – p. 19/37

  45. XML Datatypes in OWL ☞ OWL supports XML Schema primitive datatypes ☞ Clean separation between “object” classes and datatypes • Disjoint interpretation domain: d I ⊆ ∆ D , and ∆ D ∩ ∆ I = ∅ D ⊆ ∆ I × ∆ D • Disjoint datatype properties: P I ☞ Philosophical reasons: • Datatypes structured by built-in predicates • Not appropriate to form new datatypes using ontology language Logical Foundations for the Semantic Web – p. 19/37

  46. XML Datatypes in OWL ☞ OWL supports XML Schema primitive datatypes ☞ Clean separation between “object” classes and datatypes • Disjoint interpretation domain: d I ⊆ ∆ D , and ∆ D ∩ ∆ I = ∅ D ⊆ ∆ I × ∆ D • Disjoint datatype properties: P I ☞ Philosophical reasons: • Datatypes structured by built-in predicates • Not appropriate to form new datatypes using ontology language ☞ Practical reasons: • Ontology language remains simple and compact • Semantic integrity of ontology language not compromised • Implementability not compromised — can use hybrid reasoner – Only need sound and complete decision procedure for d I 1 ∩ . . . ∩ d I n , where d i is a (possibly negated) datatype Logical Foundations for the Semantic Web – p. 19/37

  47. Reasoning with OWL DL Logical Foundations for the Semantic Web – p. 20/37

  48. Reasoning Logical Foundations for the Semantic Web – p. 21/37

  49. Reasoning ☞ Why do we want it? Logical Foundations for the Semantic Web – p. 21/37

  50. Reasoning ☞ Why do we want it? • Semantic Web aims at “machine understanding” • Understanding closely related to reasoning Logical Foundations for the Semantic Web – p. 21/37

  51. Reasoning ☞ Why do we want it? • Semantic Web aims at “machine understanding” • Understanding closely related to reasoning ☞ What can we do with it? Logical Foundations for the Semantic Web – p. 21/37

  52. Reasoning ☞ Why do we want it? • Semantic Web aims at “machine understanding” • Understanding closely related to reasoning ☞ What can we do with it? • Design and maintenance of ontologies – Check class consistency and compute class hierarchy – Particularly important with large ontologies/multiple authors Logical Foundations for the Semantic Web – p. 21/37

  53. Reasoning ☞ Why do we want it? • Semantic Web aims at “machine understanding” • Understanding closely related to reasoning ☞ What can we do with it? • Design and maintenance of ontologies – Check class consistency and compute class hierarchy – Particularly important with large ontologies/multiple authors • Integration of ontologies – Assert inter-ontology relationships – Reasoner computes integrated class hierarchy/consistency Logical Foundations for the Semantic Web – p. 21/37

  54. Reasoning ☞ Why do we want it? • Semantic Web aims at “machine understanding” • Understanding closely related to reasoning ☞ What can we do with it? • Design and maintenance of ontologies – Check class consistency and compute class hierarchy – Particularly important with large ontologies/multiple authors • Integration of ontologies – Assert inter-ontology relationships – Reasoner computes integrated class hierarchy/consistency • Querying class and instance data w.r.t. ontologies – Determine if set of facts are consistent w.r.t. ontologies – Determine if individuals are instances of ontology classes – Retrieve individuals/tuples satisfying a query expression – Check if one class subsumes (is more general than) another w.r.t. ontology – . . . Logical Foundations for the Semantic Web – p. 21/37

  55. Why Decidable Reasoning? Logical Foundations for the Semantic Web – p. 22/37

  56. Why Decidable Reasoning? ☞ OWL DL constructors/axioms restricted so reasoning is decidable Logical Foundations for the Semantic Web – p. 22/37

  57. Why Decidable Reasoning? ☞ OWL DL constructors/axioms restricted so reasoning is decidable ☞ Consistent with Semantic Web’s layered architecture • XML provides syntax transport layer • RDF(S) provides basic relational language and simple ontological primitives • OWL DL provides powerful but still decidable ontology language • Further layers may (will) extend OWL – Will almost certainly be undecidable Logical Foundations for the Semantic Web – p. 22/37

  58. Why Decidable Reasoning? ☞ OWL DL constructors/axioms restricted so reasoning is decidable ☞ Consistent with Semantic Web’s layered architecture • XML provides syntax transport layer • RDF(S) provides basic relational language and simple ontological primitives • OWL DL provides powerful but still decidable ontology language • Further layers may (will) extend OWL – Will almost certainly be undecidable ☞ Facilitates provision of reasoning services • Known “practical” algorithms • Several implemented systems • Evidence of empirical tractability Logical Foundations for the Semantic Web – p. 22/37

  59. Why Decidable Reasoning? ☞ OWL DL constructors/axioms restricted so reasoning is decidable ☞ Consistent with Semantic Web’s layered architecture • XML provides syntax transport layer • RDF(S) provides basic relational language and simple ontological primitives • OWL DL provides powerful but still decidable ontology language • Further layers may (will) extend OWL – Will almost certainly be undecidable ☞ Facilitates provision of reasoning services • Known “practical” algorithms • Several implemented systems • Evidence of empirical tractability ☞ Understanding dependent on reliable & consistent reasoning Logical Foundations for the Semantic Web – p. 22/37

  60. Basic Inference Problems Logical Foundations for the Semantic Web – p. 23/37

  61. Basic Inference Problems ☞ Consistency — check if knowledge is meaningful • Is O consistent? There exists some model I of O C I � = ∅ in some model I of O • Is C consistent? Logical Foundations for the Semantic Web – p. 23/37

  62. Basic Inference Problems ☞ Consistency — check if knowledge is meaningful • Is O consistent? There exists some model I of O C I � = ∅ in some model I of O • Is C consistent? ☞ Subsumption — structure knowledge, compute taxonomy C I ⊆ D I in all models I of O • C ⊑ O D ? Logical Foundations for the Semantic Web – p. 23/37

  63. Basic Inference Problems ☞ Consistency — check if knowledge is meaningful • Is O consistent? There exists some model I of O C I � = ∅ in some model I of O • Is C consistent? ☞ Subsumption — structure knowledge, compute taxonomy C I ⊆ D I in all models I of O • C ⊑ O D ? ☞ Equivalence — check if two classes denote same set of instances C I = D I in all models I of O • C ≡ O D ? Logical Foundations for the Semantic Web – p. 23/37

  64. Basic Inference Problems ☞ Consistency — check if knowledge is meaningful • Is O consistent? There exists some model I of O C I � = ∅ in some model I of O • Is C consistent? ☞ Subsumption — structure knowledge, compute taxonomy C I ⊆ D I in all models I of O • C ⊑ O D ? ☞ Equivalence — check if two classes denote same set of instances C I = D I in all models I of O • C ≡ O D ? ☞ Instantiation — check if individual i instance of class C i ∈ C I in all models I of O • i ∈ O C ? Logical Foundations for the Semantic Web – p. 23/37

  65. Basic Inference Problems ☞ Consistency — check if knowledge is meaningful • Is O consistent? There exists some model I of O C I � = ∅ in some model I of O • Is C consistent? ☞ Subsumption — structure knowledge, compute taxonomy C I ⊆ D I in all models I of O • C ⊑ O D ? ☞ Equivalence — check if two classes denote same set of instances C I = D I in all models I of O • C ≡ O D ? ☞ Instantiation — check if individual i instance of class C i ∈ C I in all models I of O • i ∈ O C ? ☞ Retrieval — retrieve set of individuals that instantiate C • set of i s.t. i ∈ C I in all models I of O Logical Foundations for the Semantic Web – p. 23/37

  66. Basic Inference Problems ☞ Consistency — check if knowledge is meaningful • Is O consistent? There exists some model I of O C I � = ∅ in some model I of O • Is C consistent? ☞ Subsumption — structure knowledge, compute taxonomy C I ⊆ D I in all models I of O • C ⊑ O D ? ☞ Equivalence — check if two classes denote same set of instances C I = D I in all models I of O • C ≡ O D ? ☞ Instantiation — check if individual i instance of class C i ∈ C I in all models I of O • i ∈ O C ? ☞ Retrieval — retrieve set of individuals that instantiate C • set of i s.t. i ∈ C I in all models I of O ☞ Problems all reducible to consistency (satisfiability): • C ⊑ O D iff C ⊓ ¬ D not consistent w.r.t. O • i ∈ O C iff O ∪ { i ∈ ¬ C } is not consistent Logical Foundations for the Semantic Web – p. 23/37

  67. Reasoning Support for Ontology Design: OilEd Logical Foundations for the Semantic Web – p. 24/37

  68. Description Logic Reasoning Logical Foundations for the Semantic Web – p. 25/37

  69. Highly Optimised Implementation Logical Foundations for the Semantic Web – p. 26/37

  70. Highly Optimised Implementation ☞ DL reasoning based on tableaux algorithms Logical Foundations for the Semantic Web – p. 26/37

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