Reasoning with Expressive Description Logics Logical Foundations for - PowerPoint PPT Presentation

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 Reasoning with Expressive Description Logics – p. 9/40

Ontologies Reasoning with Expressive Description Logics – p. 10/40

Ontologies ☞ Semantic markup must be meaningful to automated processes Reasoning with Expressive Description Logics – p. 10/40

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) Reasoning with Expressive Description Logics – p. 10/40

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 Reasoning with Expressive Description Logics – p. 10/40

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) Reasoning with Expressive Description Logics – p. 10/40

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 Reasoning with Expressive Description Logics – p. 10/40

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 buyer–seller communication in e-commerce • In semantic based search • To provide richer service descriptions that can be more flexibly interpreted by intelligent agents Reasoning with Expressive Description Logics – p. 10/40

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 buyer–seller communication in e-commerce • In semantic based search • To provide richer service descriptions that can be more flexibly interpreted by intelligent agents Reasoning with Expressive Description Logics – p. 10/40

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 buyer–seller communication in e-commerce • In semantic based search • To provide richer service descriptions that can be more flexibly interpreted by intelligent agents Reasoning with Expressive Description Logics – p. 10/40

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 buyer–seller communication in e-commerce • In semantic based search • To provide richer service descriptions that can be more flexibly interpreted by intelligent agents Reasoning with Expressive Description Logics – p. 10/40

Web Ontology Languages Reasoning with Expressive Description Logics – p. 11/40

Web Ontology Languages ☞ OIL and DAML-ONT web ontology languages developed in European and DARPA projects Reasoning with Expressive Description Logics – p. 11/40

Web Ontology Languages ☞ OIL and DAML-ONT web ontology languages developed in European and DARPA projects ☞ Efforts merged to produce DAML+OIL Reasoning with Expressive Description Logics – p. 11/40

Web Ontology Languages ☞ OIL and DAML-ONT web ontology languages developed in European and DARPA projects ☞ Efforts merged to produce DAML+OIL • Submitted to W3C as basis for standardisation • WebOnt working group developing OWL language standard Reasoning with Expressive Description Logics – p. 11/40

Web Ontology Languages ☞ OIL and DAML-ONT web ontology languages developed in European and DARPA projects ☞ 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, . . . ) Reasoning with Expressive Description Logics – p. 11/40

Web Ontology Languages ☞ OIL and DAML-ONT web ontology languages developed in European and DARPA projects ☞ 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, . . . ) Reasoning with Expressive Description Logics – p. 11/40

Web Ontology Languages ☞ OIL and DAML-ONT web ontology languages developed in European and DARPA projects ☞ 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 class/property structure of domain (Tbox) • E.g., Person subclass of Animal whose parents are all Persons Reasoning with Expressive Description Logics – p. 11/40

Web Ontology Languages ☞ OIL and DAML-ONT web ontology languages developed in European and DARPA projects ☞ 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 class/property structure of domain (Tbox) • E.g., Person subclass of Animal whose parents are all Persons ☞ Uses RDF for class/property membership assertions (Abox) • E.g., john instance of Person; � john , mary � instance of parent Reasoning with Expressive Description Logics – p. 11/40

Logical Foundations of DAML+OIL Reasoning with Expressive Description Logics – p. 12/40

Logical Foundations of DAML+OIL ☞ DAML+OIL equivalent to very expressive Description Logic Reasoning with Expressive Description Logics – p. 12/40

Logical Foundations of DAML+OIL ☞ DAML+OIL equivalent to very expressive Description Logic ☞ More precisely, DAML+OIL is (extension of) SHIQ DL Reasoning with Expressive Description Logics – p. 12/40

Logical Foundations of DAML+OIL ☞ DAML+OIL equivalent to very expressive Description Logic ☞ More precisely, DAML+OIL is (extension of) SHIQ DL ☞ DAML+OIL benefits from many years of DL research • Well defined semantics • Formal properties well understood (complexity, decidability) • Known reasoning algorithms • Implemented systems (highly optimised) Reasoning with Expressive Description Logics – p. 12/40

Logical Foundations of DAML+OIL ☞ DAML+OIL equivalent to very expressive Description Logic ☞ More precisely, DAML+OIL is (extension of) SHIQ DL ☞ DAML+OIL benefits from many years of DL research • Well defined semantics • Formal properties well understood (complexity, decidability) • Known reasoning algorithms • Implemented systems (highly optimised) ☞ DAML+OIL classes can be names (URI’s) or expressions • Various constructors provided for building class expressions Reasoning with Expressive Description Logics – p. 12/40

Logical Foundations of DAML+OIL ☞ DAML+OIL equivalent to very expressive Description Logic ☞ More precisely, DAML+OIL is (extension of) SHIQ DL ☞ DAML+OIL benefits from many years of DL research • Well defined semantics • Formal properties well understood (complexity, decidability) • Known reasoning algorithms • Implemented systems (highly optimised) ☞ DAML+OIL classes can be names (URI’s) or expressions • Various constructors provided for building class expressions ☞ Expressive power determined by • Kinds of constructor provided • Kinds of axiom allowed Reasoning with Expressive Description Logics – p. 12/40

DAML+OIL Class Constructors Reasoning with Expressive Description Logics – p. 13/40

DAML+OIL 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 toClass ∀ P.C ∀ hasChild . Doctor [ P ] C ∃ P.C ∃ hasChild . Lawyer � P � C hasClass maxCardinalityQ � nP.C � 1 hasChild . Male [ P ] n +1 C minCardinalityQ � 2 hasChild . Lawyer � P � n C � nP.C Reasoning with Expressive Description Logics – p. 13/40

DAML+OIL 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 toClass ∀ P.C ∀ hasChild . Doctor [ P ] C ∃ P.C ∃ hasChild . Lawyer � P � C hasClass maxCardinalityQ � nP.C � 1 hasChild . Male [ P ] n +1 C minCardinalityQ � 2 hasChild . Lawyer � P � n C � nP.C ☞ XMLS datatypes as well as classes in ∀ P.C and ∃ P.C • E.g., ∃ hasAge . nonNegativeInteger Reasoning with Expressive Description Logics – p. 13/40

DAML+OIL 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 toClass ∀ P.C ∀ hasChild . Doctor [ P ] C ∃ P.C ∃ hasChild . Lawyer � P � C hasClass maxCardinalityQ � nP.C � 1 hasChild . Male [ P ] n +1 C minCardinalityQ � 2 hasChild . Lawyer � P � n C � nP.C ☞ 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 ) Reasoning with Expressive Description Logics – p. 13/40

RDFS Syntax <daml:Class> <daml:intersectionOf rdf:parseType="daml:collection"> <daml:Class rdf:about="#Person"/> <daml:Restriction> <daml:onProperty rdf:resource="#hasChild"/> <daml:toClass> <daml:unionOf rdf:parseType="daml:collection"> <daml:Class rdf:about="#Doctor"/> <daml:Restriction> <daml:onProperty rdf:resource="#hasChild"/> <daml:hasClass rdf:resource="#Doctor"/> </daml:Restriction> </daml:unionOf> </daml:toClass> </daml:Restriction> </daml:intersectionOf> </daml:Class> Reasoning with Expressive Description Logics – p. 14/40

Semantics Reasoning with Expressive Description Logics – p. 15/40

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 − Reasoning with Expressive Description Logics – p. 15/40

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.C ) I = { x | # { y | � x, y � ∈ R I ∧ y ∈ C I } � n } • ( � nR.C ) I = { x | # { y | � x, y � ∈ R I ∧ y ∈ C I } � n } Reasoning with Expressive Description Logics – p. 15/40

DAML+OIL Axioms Reasoning with Expressive Description Logics – p. 16/40

DAML+OIL Axioms Axiom DL Syntax Example C 1 ⊑ C 2 Human ⊑ Animal ⊓ Biped subClassOf sameClassAs 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 } differentIndividualFrom subPropertyOf P 1 ⊑ P 2 hasDaughter ⊑ hasChild P 1 ≡ P 2 cost ≡ price samePropertyAs hasChild ≡ hasParent − P 1 ≡ P − inverseOf 2 ancestor + ⊑ ancestor P + ⊑ P transitiveProperty uniqueProperty ⊤ ⊑ � 1 P ⊤ ⊑ � 1 hasMother ⊤ ⊑ � 1 hasSSN − ⊤ ⊑ � 1 P − unambiguousProperty Reasoning with Expressive Description Logics – p. 16/40

DAML+OIL Axioms Axiom DL Syntax Example C 1 ⊑ C 2 Human ⊑ Animal ⊓ Biped subClassOf sameClassAs 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 } differentIndividualFrom subPropertyOf P 1 ⊑ P 2 hasDaughter ⊑ hasChild P 1 ≡ P 2 cost ≡ price samePropertyAs hasChild ≡ hasParent − P 1 ≡ P − inverseOf 2 ancestor + ⊑ ancestor P + ⊑ P transitiveProperty uniqueProperty ⊤ ⊑ � 1 P ⊤ ⊑ � 1 hasMother ⊤ ⊑ � 1 hasSSN − ⊤ ⊑ � 1 P − unambiguousProperty ☞ 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 Reasoning with Expressive Description Logics – p. 16/40

DAML+OIL Axioms Axiom DL Syntax Example C 1 ⊑ C 2 Human ⊑ Animal ⊓ Biped subClassOf sameClassAs 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 } differentIndividualFrom subPropertyOf P 1 ⊑ P 2 hasDaughter ⊑ hasChild P 1 ≡ P 2 cost ≡ price samePropertyAs hasChild ≡ hasParent − P 1 ≡ P − inverseOf 2 ancestor + ⊑ ancestor P + ⊑ P transitiveProperty uniqueProperty ⊤ ⊑ � 1 P ⊤ ⊑ � 1 hasMother ⊤ ⊑ � 1 hasSSN − ⊤ ⊑ � 1 P − unambiguousProperty ☞ 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 Reasoning with Expressive Description Logics – p. 16/40

XML Datatypes in DAML+OIL Reasoning with Expressive Description Logics – p. 17/40

XML Datatypes in DAML+OIL ☞ DAML+OIL supports XML Schema datatypes • Primitive (e.g., decimal) and derived (e.g., integer sub-range) Reasoning with Expressive Description Logics – p. 17/40

XML Datatypes in DAML+OIL ☞ DAML+OIL supports XML Schema datatypes • Primitive (e.g., decimal) and derived (e.g., integer sub-range) ☞ Clean separation between “object” classes and datatypes • Disjoint interpretation domain: d I ⊆ ∆ D , and ∆ D ∩ ∆ I = ∅ D ⊆ ∆ I × ∆ D • Disjoint datatype properties: P I Reasoning with Expressive Description Logics – p. 17/40

XML Datatypes in DAML+OIL ☞ DAML+OIL supports XML Schema datatypes • Primitive (e.g., decimal) and derived (e.g., integer sub-range) ☞ 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 Reasoning with Expressive Description Logics – p. 17/40

XML Datatypes in DAML+OIL ☞ DAML+OIL supports XML Schema datatypes • Primitive (e.g., decimal) and derived (e.g., integer sub-range) ☞ 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 Reasoning with Expressive Description Logics – p. 17/40

Reasoning with DAML+OIL Reasoning with Expressive Description Logics – p. 18/40

Reasoning Reasoning with Expressive Description Logics – p. 19/40

Reasoning ☞ Why do we want it? Reasoning with Expressive Description Logics – p. 19/40

Reasoning ☞ Why do we want it? • Semantic Web aims at “machine understanding” • Understanding closely related to reasoning Reasoning with Expressive Description Logics – p. 19/40

Reasoning ☞ Why do we want it? • Semantic Web aims at “machine understanding” • Understanding closely related to reasoning ☞ What can we do with it? Reasoning with Expressive Description Logics – p. 19/40

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 Reasoning with Expressive Description Logics – p. 19/40

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 Reasoning with Expressive Description Logics – p. 19/40

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 description more general than another w.r.t. ontology – . . . Reasoning with Expressive Description Logics – p. 19/40

Basic Inference Problems Reasoning with Expressive Description Logics – p. 20/40

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? Reasoning with Expressive Description Logics – p. 20/40

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 ? Reasoning with Expressive Description Logics – p. 20/40

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 ? Reasoning with Expressive Description Logics – p. 20/40

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 ? Reasoning with Expressive Description Logics – p. 20/40

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 Reasoning with Expressive Description Logics – p. 20/40

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 Reasoning with Expressive Description Logics – p. 20/40

Reasoning Support for Ontology Design: OilEd Reasoning with Expressive Description Logics – p. 21/40

Description Logic Reasoning Reasoning with Expressive Description Logics – p. 22/40

Tableaux Algorithms — Basics Reasoning with Expressive Description Logics – p. 23/40

Tableaux Algorithms — Basics ☞ Tableaux algorithms used to test satisfiability Reasoning with Expressive Description Logics – p. 23/40

Tableaux Algorithms — Basics ☞ Tableaux algorithms used to test satisfiability ☞ Try to build tree-like model I of input concept C Reasoning with Expressive Description Logics – p. 23/40

Tableaux Algorithms — Basics ☞ Tableaux algorithms used to test satisfiability ☞ Try to build tree-like model I of input concept C ☞ Work on concepts in negation normal form • Push in negation using de Morgan’s, ¬∃ R.C � ∀ R. ¬ C etc. Reasoning with Expressive Description Logics – p. 23/40

Tableaux Algorithms — Basics ☞ Tableaux algorithms used to test satisfiability ☞ Try to build tree-like model I of input concept C ☞ Work on concepts in negation normal form • Push in negation using de Morgan’s, ¬∃ R.C � ∀ R. ¬ C etc. ☞ Break down C syntactically , inferring constraints on elements of I Reasoning with Expressive Description Logics – p. 23/40

Tableaux Algorithms — Basics ☞ Tableaux algorithms used to test satisfiability ☞ Try to build tree-like model I of input concept C ☞ Work on concepts in negation normal form • Push in negation using de Morgan’s, ¬∃ R.C � ∀ R. ¬ C etc. ☞ Break down C syntactically , inferring constraints on elements of I ☞ Decomposition uses tableau rules corresponding to constructors in logic (e.g., ⊓ , ∃ ) • Some rules are nondeterministic (e.g., ⊔ , � ) • In practice, this means search Reasoning with Expressive Description Logics – p. 23/40

Tableaux Algorithms — Basics ☞ Tableaux algorithms used to test satisfiability ☞ Try to build tree-like model I of input concept C ☞ Work on concepts in negation normal form • Push in negation using de Morgan’s, ¬∃ R.C � ∀ R. ¬ C etc. ☞ Break down C syntactically , inferring constraints on elements of I ☞ Decomposition uses tableau rules corresponding to constructors in logic (e.g., ⊓ , ∃ ) • Some rules are nondeterministic (e.g., ⊔ , � ) • In practice, this means search ☞ Stop when clash occurs or when no rules are applicable Reasoning with Expressive Description Logics – p. 23/40

Tableaux Algorithms — Basics ☞ Tableaux algorithms used to test satisfiability ☞ Try to build tree-like model I of input concept C ☞ Work on concepts in negation normal form • Push in negation using de Morgan’s, ¬∃ R.C � ∀ R. ¬ C etc. ☞ Break down C syntactically , inferring constraints on elements of I ☞ Decomposition uses tableau rules corresponding to constructors in logic (e.g., ⊓ , ∃ ) • Some rules are nondeterministic (e.g., ⊔ , � ) • In practice, this means search ☞ Stop when clash occurs or when no rules are applicable ☞ Blocking (cycle check) used to guarantee termination Reasoning with Expressive Description Logics – p. 23/40

Tableaux Algorithms — Basics ☞ Tableaux algorithms used to test satisfiability ☞ Try to build tree-like model I of input concept C ☞ Work on concepts in negation normal form • Push in negation using de Morgan’s, ¬∃ R.C � ∀ R. ¬ C etc. ☞ Break down C syntactically , inferring constraints on elements of I ☞ Decomposition uses tableau rules corresponding to constructors in logic (e.g., ⊓ , ∃ ) • Some rules are nondeterministic (e.g., ⊔ , � ) • In practice, this means search ☞ Stop when clash occurs or when no rules are applicable ☞ Blocking (cycle check) used to guarantee termination ☞ Return “ C is consistent” iff C is consistent • Tree model property Reasoning with Expressive Description Logics – p. 23/40

Tableaux Algorithms — Details Reasoning with Expressive Description Logics – p. 24/40

Tableaux Algorithms — Details ☞ Work on tree T representing model I of concept C • Nodes represent elements of ∆ I ; labeled with subconcepts of C • Edges represent role-successorships between elements of ∆ I Reasoning with Expressive Description Logics – p. 24/40