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Reasoning with Expressive Description Logics Logical Foundations for the Semantic Web Ian Horrocks horrocks@cs.man.ac.uk University of Manchester Manchester, UK Reasoning with Expressive Description Logics p. 1/40 Talk Outline Reasoning

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SLIDE 1

Reasoning with Expressive Description Logics

Logical Foundations for the Semantic Web

Ian Horrocks

horrocks@cs.man.ac.uk

University of Manchester Manchester, UK

Reasoning with Expressive Description Logics – p. 1/40

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SLIDE 2

Talk Outline

Reasoning with Expressive Description Logics – p. 2/40

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SLIDE 3

Talk Outline

Introduction to Description Logics

Reasoning with Expressive Description Logics – p. 2/40

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SLIDE 4

Talk Outline

Introduction to Description Logics The Semantic Web: Killer App for (DL) Reasoning? Web Ontology Languages DAML+OIL Language

Reasoning with Expressive Description Logics – p. 2/40

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SLIDE 5

Talk Outline

Introduction to Description Logics The Semantic Web: Killer App for (DL) Reasoning? Web Ontology Languages DAML+OIL Language Reasoning with DAML+OIL OilEd Demo

Reasoning with Expressive Description Logics – p. 2/40

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SLIDE 6

Talk Outline

Introduction to Description Logics The Semantic Web: Killer App for (DL) Reasoning? Web Ontology Languages DAML+OIL Language Reasoning with DAML+OIL OilEd Demo Description Logic Reasoning

Reasoning with Expressive Description Logics – p. 2/40

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SLIDE 7

Talk Outline

Introduction to Description Logics The Semantic Web: Killer App for (DL) Reasoning? Web Ontology Languages DAML+OIL Language Reasoning with DAML+OIL OilEd Demo Description Logic Reasoning Research Challenges

Reasoning with Expressive Description Logics – p. 2/40

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SLIDE 8

Introduction to Description Logics

Reasoning with Expressive Description Logics – p. 3/40

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SLIDE 9

What are Description Logics?

Reasoning with Expressive Description Logics – p. 4/40

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SLIDE 10

What are Description Logics?

☞ A family of logic based Knowledge Representation formalisms

  • Descendants of semantic networks and KL-ONE
  • Describe domain in terms of concepts (classes), roles

(relationships) and individuals

Reasoning with Expressive Description Logics – p. 4/40

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SLIDE 11

What are Description Logics?

☞ A family of logic based Knowledge Representation formalisms

  • Descendants of semantic networks and KL-ONE
  • Describe domain in terms of concepts (classes), roles

(relationships) and individuals ☞ Distinguished by:

  • Formal semantics (model theoretic)

– Decidable fragments of FOL – Closely related to Propositional Modal & Dynamic Logics

  • Provision of inference services

– Sound and complete decision procedures for key problems – Implemented systems (highly optimised)

Reasoning with Expressive Description Logics – p. 4/40

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SLIDE 12

DL Architecture

Tbox (schema) Abox (data)

Knowledge Base

Inference System

Interface Man . = Human ⊓ Male Happy-Father . = Man ⊓ ∃has-child.Female ⊓ . . . . . . . . . John : Happy-Father John, Mary : has-child

Reasoning with Expressive Description Logics – p. 5/40

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SLIDE 13

Short History of Description Logics

Reasoning with Expressive Description Logics – p. 6/40

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SLIDE 14

Short History of Description Logics

Phase 1: ☞ Incomplete systems (Back, Classic, Loom, . . . ) ☞ Based on structural algorithms

Reasoning with Expressive Description Logics – p. 6/40

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SLIDE 15

Short History of Description Logics

Phase 1: ☞ Incomplete systems (Back, Classic, Loom, . . . ) ☞ Based on structural algorithms Phase 2: ☞ Development of tableau algorithms and complexity results ☞ Tableau-based systems for Pspace logics (e.g., Kris, Crack) ☞ Investigation of optimisation techniques

Reasoning with Expressive Description Logics – p. 6/40

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SLIDE 16

Short History of Description Logics

Phase 1: ☞ Incomplete systems (Back, Classic, Loom, . . . ) ☞ Based on structural algorithms Phase 2: ☞ Development of tableau algorithms and complexity results ☞ Tableau-based systems for Pspace logics (e.g., Kris, Crack) ☞ Investigation of optimisation techniques Phase 3: ☞ Tableau algorithms for very expressive DLs ☞ Highly optimised tableau systems for ExpTime logics (e.g., FaCT, DLP , Racer) ☞ Relationship to modal logic and decidable fragments of FOL

Reasoning with Expressive Description Logics – p. 6/40

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SLIDE 17

Latest Developments

Phase 4:

Reasoning with Expressive Description Logics – p. 7/40

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SLIDE 18

Latest Developments

Phase 4: ☞ Mature implementations

Reasoning with Expressive Description Logics – p. 7/40

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SLIDE 19

Latest Developments

Phase 4: ☞ Mature implementations ☞ Mainstream applications and Tools

  • Databases

– Consistency of conceptual schemata (EER, UML etc.) – Schema integration – Query subsumption (w.r.t. a conceptual schema)

  • Ontologies and Semantic Web (and Grid)

– Ontology engineering (design, maintenance, integration) – Reasoning with ontology-based markup (meta-data) – Service description and discovery

Reasoning with Expressive Description Logics – p. 7/40

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SLIDE 20

Latest Developments

Phase 4: ☞ Mature implementations ☞ Mainstream applications and Tools

  • Databases

– Consistency of conceptual schemata (EER, UML etc.) – Schema integration – Query subsumption (w.r.t. a conceptual schema)

  • Ontologies and Semantic Web (and Grid)

– Ontology engineering (design, maintenance, integration) – Reasoning with ontology-based markup (meta-data) – Service description and discovery ☞ Commercial implementations

  • Cerebra system from Network Inference Ltd

Reasoning with Expressive Description Logics – p. 7/40

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SLIDE 21

The Semantic Web

Reasoning with Expressive Description Logics – p. 8/40

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SLIDE 22

The Semantic Web Vision

Reasoning with Expressive Description Logics – p. 9/40

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SLIDE 23

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

Reasoning with Expressive Description Logics – p. 9/40

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SLIDE 24

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

Reasoning with Expressive Description Logics – p. 9/40

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SLIDE 25

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

Reasoning with Expressive Description Logics – p. 9/40

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SLIDE 26

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

Reasoning with Expressive Description Logics – p. 9/40

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SLIDE 27

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

Reasoning with Expressive Description Logics – p. 9/40

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SLIDE 28

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

Reasoning with Expressive Description Logics – p. 9/40

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SLIDE 29

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

Reasoning with Expressive Description Logics – p. 9/40

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SLIDE 30

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

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SLIDE 31

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

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SLIDE 32

Ontologies

Reasoning with Expressive Description Logics – p. 10/40

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SLIDE 33

Ontologies

☞ Semantic markup must be meaningful to automated processes

Reasoning with Expressive Description Logics – p. 10/40

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SLIDE 34

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

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SLIDE 35

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

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SLIDE 36

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

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SLIDE 37

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

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SLIDE 38

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

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SLIDE 39

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

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SLIDE 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

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SLIDE 41

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

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SLIDE 42

Web Ontology Languages

Reasoning with Expressive Description Logics – p. 11/40

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SLIDE 43

Web Ontology Languages

☞ OIL and DAML-ONT web ontology languages developed in European and DARPA projects

Reasoning with Expressive Description Logics – p. 11/40

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SLIDE 44

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

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SLIDE 45

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

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SLIDE 46

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

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SLIDE 47

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

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SLIDE 48

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

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SLIDE 49

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

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SLIDE 50

Logical Foundations of DAML+OIL

Reasoning with Expressive Description Logics – p. 12/40

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SLIDE 51

Logical Foundations of DAML+OIL

☞ DAML+OIL equivalent to very expressive Description Logic

Reasoning with Expressive Description Logics – p. 12/40

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SLIDE 52

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

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SLIDE 53

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

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SLIDE 54

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

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SLIDE 55

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

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SLIDE 56

DAML+OIL Class Constructors

Reasoning with Expressive Description Logics – p. 13/40

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SLIDE 57

DAML+OIL Class Constructors

Constructor DL Syntax Example (Modal Syntax) intersectionOf C1 ⊓ . . . ⊓ Cn Human ⊓ Male C1 ∧ . . . ∧ Cn unionOf C1 ⊔ . . . ⊔ Cn Doctor ⊔ Lawyer C1 ∨ . . . ∨ Cn complementOf ¬C ¬Male ¬C

  • neOf

{x1 . . . xn} {john, mary} x1 ∨ . . . ∨ xn toClass ∀P.C ∀hasChild.Doctor [P]C hasClass ∃P.C ∃hasChild.Lawyer PC maxCardinalityQ nP.C 1hasChild.Male [P]n+1C minCardinalityQ nP.C 2hasChild.Lawyer PnC

Reasoning with Expressive Description Logics – p. 13/40

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SLIDE 58

DAML+OIL Class Constructors

Constructor DL Syntax Example (Modal Syntax) intersectionOf C1 ⊓ . . . ⊓ Cn Human ⊓ Male C1 ∧ . . . ∧ Cn unionOf C1 ⊔ . . . ⊔ Cn Doctor ⊔ Lawyer C1 ∨ . . . ∨ Cn complementOf ¬C ¬Male ¬C

  • neOf

{x1 . . . xn} {john, mary} x1 ∨ . . . ∨ xn toClass ∀P.C ∀hasChild.Doctor [P]C hasClass ∃P.C ∃hasChild.Lawyer PC maxCardinalityQ nP.C 1hasChild.Male [P]n+1C minCardinalityQ nP.C 2hasChild.Lawyer PnC ☞ XMLS datatypes as well as classes in ∀P.C and ∃P.C

  • E.g., ∃hasAge.nonNegativeInteger

Reasoning with Expressive Description Logics – p. 13/40

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SLIDE 59

DAML+OIL Class Constructors

Constructor DL Syntax Example (Modal Syntax) intersectionOf C1 ⊓ . . . ⊓ Cn Human ⊓ Male C1 ∧ . . . ∧ Cn unionOf C1 ⊔ . . . ⊔ Cn Doctor ⊔ Lawyer C1 ∨ . . . ∨ Cn complementOf ¬C ¬Male ¬C

  • neOf

{x1 . . . xn} {john, mary} x1 ∨ . . . ∨ xn toClass ∀P.C ∀hasChild.Doctor [P]C hasClass ∃P.C ∃hasChild.Lawyer PC maxCardinalityQ nP.C 1hasChild.Male [P]n+1C minCardinalityQ nP.C 2hasChild.Lawyer PnC ☞ 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

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SLIDE 60

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

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SLIDE 61

Semantics

Reasoning with Expressive Description Logics – p. 15/40

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SLIDE 62

Semantics

☞ Semantics defined by interpretations: I = (∆I, ·I)

  • concepts −

→ subsets of ∆I

  • roles −

→ binary relations over ∆I (subsets of ∆I × ∆I)

  • individuals −

→ elements of ∆I

Reasoning with Expressive Description Logics – p. 15/40

slide-63
SLIDE 63

Semantics

☞ Semantics defined by interpretations: I = (∆I, ·I)

  • concepts −

→ subsets of ∆I

  • roles −

→ binary relations over ∆I (subsets of ∆I × ∆I)

  • individuals −

→ elements of ∆I ☞ Interpretation function ·I extended to concept expressions

  • (C ⊓ D)I = CI ∩ DI

(C ⊔ D)I = CI ∪ DI (¬C)I = ∆I \ CI

  • {xn, . . . , xn}I = {xI

n, . . . , xI n}

  • (∀R.C)I = {x | ∀y.(x, y) ∈ RI ⇒ y ∈ CI}
  • (∃R.C)I = {x | ∃y.x, y ∈ RI ∧ y ∈ CI}
  • (nR.C)I = {x | #{y | x, y ∈ RI ∧ y ∈ CI} n}
  • (nR.C)I = {x | #{y | x, y ∈ RI ∧ y ∈ CI} n}

Reasoning with Expressive Description Logics – p. 15/40

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SLIDE 64

DAML+OIL Axioms

Reasoning with Expressive Description Logics – p. 16/40

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SLIDE 65

DAML+OIL Axioms

Axiom DL Syntax Example subClassOf C1 ⊑ C2 Human ⊑ Animal ⊓ Biped sameClassAs C1 ≡ C2 Man ≡ Human ⊓ Male disjointWith C1 ⊑ ¬C2 Male ⊑ ¬Female sameIndividualAs {x1} ≡ {x2} {President_Bush} ≡ {G_W_Bush} differentIndividualFrom {x1} ⊑ ¬{x2} {john} ⊑ ¬{peter} subPropertyOf P1 ⊑ P2 hasDaughter ⊑ hasChild samePropertyAs P1 ≡ P2 cost ≡ price inverseOf P1 ≡ P −

2

hasChild ≡ hasParent− transitiveProperty P + ⊑ P ancestor+ ⊑ ancestor uniqueProperty ⊤ ⊑ 1P ⊤ ⊑ 1hasMother unambiguousProperty ⊤ ⊑ 1P − ⊤ ⊑ 1hasSSN−

Reasoning with Expressive Description Logics – p. 16/40

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SLIDE 66

DAML+OIL Axioms

Axiom DL Syntax Example subClassOf C1 ⊑ C2 Human ⊑ Animal ⊓ Biped sameClassAs C1 ≡ C2 Man ≡ Human ⊓ Male disjointWith C1 ⊑ ¬C2 Male ⊑ ¬Female sameIndividualAs {x1} ≡ {x2} {President_Bush} ≡ {G_W_Bush} differentIndividualFrom {x1} ⊑ ¬{x2} {john} ⊑ ¬{peter} subPropertyOf P1 ⊑ P2 hasDaughter ⊑ hasChild samePropertyAs P1 ≡ P2 cost ≡ price inverseOf P1 ≡ P −

2

hasChild ≡ hasParent− transitiveProperty P + ⊑ P ancestor+ ⊑ ancestor uniqueProperty ⊤ ⊑ 1P ⊤ ⊑ 1hasMother unambiguousProperty ⊤ ⊑ 1P − ⊤ ⊑ 1hasSSN− ☞ I satisfies C1 ⊑ C2 iff CI

1 ⊆ CI 2 ; satisfies P1 ⊑ P2 iff P I 1 ⊆ P I 2

Reasoning with Expressive Description Logics – p. 16/40

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SLIDE 67

DAML+OIL Axioms

Axiom DL Syntax Example subClassOf C1 ⊑ C2 Human ⊑ Animal ⊓ Biped sameClassAs C1 ≡ C2 Man ≡ Human ⊓ Male disjointWith C1 ⊑ ¬C2 Male ⊑ ¬Female sameIndividualAs {x1} ≡ {x2} {President_Bush} ≡ {G_W_Bush} differentIndividualFrom {x1} ⊑ ¬{x2} {john} ⊑ ¬{peter} subPropertyOf P1 ⊑ P2 hasDaughter ⊑ hasChild samePropertyAs P1 ≡ P2 cost ≡ price inverseOf P1 ≡ P −

2

hasChild ≡ hasParent− transitiveProperty P + ⊑ P ancestor+ ⊑ ancestor uniqueProperty ⊤ ⊑ 1P ⊤ ⊑ 1hasMother unambiguousProperty ⊤ ⊑ 1P − ⊤ ⊑ 1hasSSN− ☞ I satisfies C1 ⊑ C2 iff CI

1 ⊆ CI 2 ; satisfies P1 ⊑ P2 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

slide-68
SLIDE 68

XML Datatypes in DAML+OIL

Reasoning with Expressive Description Logics – p. 17/40

slide-69
SLIDE 69

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

slide-70
SLIDE 70

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: dI ⊆ ∆D, and ∆D ∩ ∆I = ∅
  • Disjoint datatype properties: P I

D ⊆ ∆I × ∆D

Reasoning with Expressive Description Logics – p. 17/40

slide-71
SLIDE 71

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: dI ⊆ ∆D, and ∆D ∩ ∆I = ∅
  • Disjoint datatype properties: P I

D ⊆ ∆I × ∆D

☞ 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

slide-72
SLIDE 72

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: dI ⊆ ∆D, and ∆D ∩ ∆I = ∅
  • Disjoint datatype properties: P I

D ⊆ ∆I × ∆D

☞ 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 dI

1 ∩ . . . ∩ dI n, where di is a (possibly negated) datatype

Reasoning with Expressive Description Logics – p. 17/40

slide-73
SLIDE 73

Reasoning with DAML+OIL

Reasoning with Expressive Description Logics – p. 18/40

slide-74
SLIDE 74

Reasoning

Reasoning with Expressive Description Logics – p. 19/40

slide-75
SLIDE 75

Reasoning

☞ Why do we want it?

Reasoning with Expressive Description Logics – p. 19/40

slide-76
SLIDE 76

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

slide-77
SLIDE 77

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

slide-78
SLIDE 78

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

slide-79
SLIDE 79

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

slide-80
SLIDE 80

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.

  • ntology

– . . .

Reasoning with Expressive Description Logics – p. 19/40

slide-81
SLIDE 81

Basic Inference Problems

Reasoning with Expressive Description Logics – p. 20/40

slide-82
SLIDE 82

Basic Inference Problems

☞ Consistency — check if knowledge is meaningful

  • Is O consistent?

There exists some model I of O

  • Is C consistent?

CI = ∅ in some model I of O

Reasoning with Expressive Description Logics – p. 20/40

slide-83
SLIDE 83

Basic Inference Problems

☞ Consistency — check if knowledge is meaningful

  • Is O consistent?

There exists some model I of O

  • Is C consistent?

CI = ∅ in some model I of O ☞ Subsumption — structure knowledge, compute taxonomy

  • C ⊑O D ?

CI ⊆ DI in all models I of O

Reasoning with Expressive Description Logics – p. 20/40

slide-84
SLIDE 84

Basic Inference Problems

☞ Consistency — check if knowledge is meaningful

  • Is O consistent?

There exists some model I of O

  • Is C consistent?

CI = ∅ in some model I of O ☞ Subsumption — structure knowledge, compute taxonomy

  • C ⊑O D ?

CI ⊆ DI in all models I of O ☞ Equivalence — check if two classes denote same set of instances

  • C ≡O D ?

CI = DI in all models I of O

Reasoning with Expressive Description Logics – p. 20/40

slide-85
SLIDE 85

Basic Inference Problems

☞ Consistency — check if knowledge is meaningful

  • Is O consistent?

There exists some model I of O

  • Is C consistent?

CI = ∅ in some model I of O ☞ Subsumption — structure knowledge, compute taxonomy

  • C ⊑O D ?

CI ⊆ DI in all models I of O ☞ Equivalence — check if two classes denote same set of instances

  • C ≡O D ?

CI = DI in all models I of O ☞ Instantiation — check if individual i instance of class C

  • i ∈O C?

i ∈ CI in all models I of O

Reasoning with Expressive Description Logics – p. 20/40

slide-86
SLIDE 86

Basic Inference Problems

☞ Consistency — check if knowledge is meaningful

  • Is O consistent?

There exists some model I of O

  • Is C consistent?

CI = ∅ in some model I of O ☞ Subsumption — structure knowledge, compute taxonomy

  • C ⊑O D ?

CI ⊆ DI in all models I of O ☞ Equivalence — check if two classes denote same set of instances

  • C ≡O D ?

CI = DI in all models I of O ☞ Instantiation — check if individual i instance of class C

  • i ∈O C?

i ∈ CI in all models I of O ☞ Retrieval — retrieve set of individuals that instantiate C

  • set of i s.t. i ∈ CI in all models I of O

Reasoning with Expressive Description Logics – p. 20/40

slide-87
SLIDE 87

Basic Inference Problems

☞ Consistency — check if knowledge is meaningful

  • Is O consistent?

There exists some model I of O

  • Is C consistent?

CI = ∅ in some model I of O ☞ Subsumption — structure knowledge, compute taxonomy

  • C ⊑O D ?

CI ⊆ DI in all models I of O ☞ Equivalence — check if two classes denote same set of instances

  • C ≡O D ?

CI = DI in all models I of O ☞ Instantiation — check if individual i instance of class C

  • i ∈O C?

i ∈ CI in all models I of O ☞ Retrieval — retrieve set of individuals that instantiate C

  • set of i s.t. i ∈ CI 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

slide-88
SLIDE 88

Reasoning Support for Ontology Design: OilEd

Reasoning with Expressive Description Logics – p. 21/40

slide-89
SLIDE 89

Description Logic Reasoning

Reasoning with Expressive Description Logics – p. 22/40

slide-90
SLIDE 90

Tableaux Algorithms — Basics

Reasoning with Expressive Description Logics – p. 23/40

slide-91
SLIDE 91

Tableaux Algorithms — Basics

☞ Tableaux algorithms used to test satisfiability

Reasoning with Expressive Description Logics – p. 23/40

slide-92
SLIDE 92

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

slide-93
SLIDE 93

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

slide-94
SLIDE 94

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

slide-95
SLIDE 95

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

slide-96
SLIDE 96

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

slide-97
SLIDE 97

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

slide-98
SLIDE 98

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

slide-99
SLIDE 99

Tableaux Algorithms — Details

Reasoning with Expressive Description Logics – p. 24/40

slide-100
SLIDE 100

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

slide-101
SLIDE 101

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

☞ T initialised with single root node labeled {C}

Reasoning with Expressive Description Logics – p. 24/40

slide-102
SLIDE 102

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

☞ T initialised with single root node labeled {C} ☞ Tableau rules repeatedly applied to node labels

  • Extend labels or extend/modify T structure
  • Rules can be blocked, e.g, if predecessor has superset label
  • Nondeterministic rules −

→ search possible extensions

Reasoning with Expressive Description Logics – p. 24/40

slide-103
SLIDE 103

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

☞ T initialised with single root node labeled {C} ☞ Tableau rules repeatedly applied to node labels

  • Extend labels or extend/modify T structure
  • Rules can be blocked, e.g, if predecessor has superset label
  • Nondeterministic rules −

→ search possible extensions ☞ T contains Clash if obvious contradiction in some node label

  • E.g., {A, ¬A} ⊆ L(x) for some concept A and node x

Reasoning with Expressive Description Logics – p. 24/40

slide-104
SLIDE 104

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

☞ T initialised with single root node labeled {C} ☞ Tableau rules repeatedly applied to node labels

  • Extend labels or extend/modify T structure
  • Rules can be blocked, e.g, if predecessor has superset label
  • Nondeterministic rules −

→ search possible extensions ☞ T contains Clash if obvious contradiction in some node label

  • E.g., {A, ¬A} ⊆ L(x) for some concept A and node x

☞ T fully expanded if no rules are applicable

Reasoning with Expressive Description Logics – p. 24/40

slide-105
SLIDE 105

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

☞ T initialised with single root node labeled {C} ☞ Tableau rules repeatedly applied to node labels

  • Extend labels or extend/modify T structure
  • Rules can be blocked, e.g, if predecessor has superset label
  • Nondeterministic rules −

→ search possible extensions ☞ T contains Clash if obvious contradiction in some node label

  • E.g., {A, ¬A} ⊆ L(x) for some concept A and node x

☞ T fully expanded if no rules are applicable ☞ C satisfiable iff fully expanded clash free T found

  • Trivial correspondence between such a T and a model of C

Reasoning with Expressive Description Logics – p. 24/40

slide-106
SLIDE 106

Tableaux Rules for ALC

Reasoning with Expressive Description Logics – p. 25/40

slide-107
SLIDE 107

Tableaux Rules for ALC

  • ✁
✁ ✂ ✂ ✄ ✄ ☎ ☎ ✆ ✆ ✝ ✝ ✞ ✞ ✟ ✟ ✟ ✟ ✠ ✠ ✡ ✡ ✡ ✡ ☛ ☛ ☞ ☞ ✌ ✌ ✍ ✍ ✎ ✎ ✏ ✏ ✑ ✑ ✒ ✒ ✓ ✓ ✔ ✔ ✕ ✕ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✙ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜ ✜

→⊓ x {∃R.C, . . .} x {C} {∃R.C, . . .} R y x R y {C, . . .} y R x {∀R.C, . . .} {. . .} {∀R.C, . . .} →∃ →∀ →⊔ for C ∈ {C1, C2} x {C1 ⊓ C2, C, . . .} x {C1 ⊓ C2, C1, C2, . . .} x {C1 ⊔ C2, . . .} x {C1 ⊓ C2, . . .}

Reasoning with Expressive Description Logics – p. 25/40

slide-108
SLIDE 108

Tableaux Rule for Transitive Roles

Reasoning with Expressive Description Logics – p. 26/40

slide-109
SLIDE 109

Tableaux Rule for Transitive Roles

✢ ✢ ✣ ✣ ✤ ✤ ✥ ✥ ✦ ✦ ✧ ✧ ★ ★ ✩ ✩ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✪ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✫ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✬ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✮ ✯ ✯ ✯ ✯ ✯

x R y y R x {∀R.C, . . .} {. . .} {∀R.C, . . .} {∀R.C, . . .} →∀+

Where R is a transitive role (i.e., (RI)+ = RI)

Reasoning with Expressive Description Logics – p. 26/40

slide-110
SLIDE 110

Tableaux Rule for Transitive Roles

✰ ✰ ✱ ✱ ✲ ✲ ✳ ✳ ✴ ✴ ✵ ✵ ✶ ✶ ✷ ✷ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✼ ✽ ✽ ✽ ✽ ✽

x R y y R x {∀R.C, . . .} {. . .} {∀R.C, . . .} {∀R.C, . . .} →∀+

Where R is a transitive role (i.e., (RI)+ = RI) ☞ No longer naturally terminating (e.g., if C = ∃R.⊤)

Reasoning with Expressive Description Logics – p. 26/40

slide-111
SLIDE 111

Tableaux Rule for Transitive Roles

✾ ✾ ✿ ✿ ❀ ❀ ❁ ❁ ❂ ❂ ❃ ❃ ❄ ❄ ❅ ❅ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❋ ❋ ❋ ❋ ❋

x R y y R x {∀R.C, . . .} {. . .} {∀R.C, . . .} {∀R.C, . . .} →∀+

Where R is a transitive role (i.e., (RI)+ = RI) ☞ No longer naturally terminating (e.g., if C = ∃R.⊤) ☞ Need blocking

  • Simple blocking suffices for ALC plus transitive roles
  • I.e., do not expand node label if ancestor has superset label
  • More expressive logics (e.g., with inverse roles) need more

sophisticated blocking strategies

Reasoning with Expressive Description Logics – p. 26/40

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SLIDE 112

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

Reasoning with Expressive Description Logics – p. 27/40

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SLIDE 113

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w L(w) = {∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

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SLIDE 114

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w L(w) = {∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

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SLIDE 115

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

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SLIDE 116

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

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SLIDE 117

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)} L(x) = {C} x S

Reasoning with Expressive Description Logics – p. 27/40

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SLIDE 118

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)} L(x) = {C} x S

Reasoning with Expressive Description Logics – p. 27/40

slide-119
SLIDE 119

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w L(x) = {C, ¬C ⊔ ¬D} x S L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

slide-120
SLIDE 120

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w L(x) = {C, ¬C ⊔ ¬D} x S L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

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SLIDE 121

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w x S L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)} L(x) = {C, (¬C ⊔ ¬D), ¬C}

Reasoning with Expressive Description Logics – p. 27/40

slide-122
SLIDE 122

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w x S L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)} clash L(x) = {C, (¬C ⊔ ¬D), ¬C}

Reasoning with Expressive Description Logics – p. 27/40

slide-123
SLIDE 123

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w L(x) = {C, ¬C ⊔ ¬D} x S L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

slide-124
SLIDE 124

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w x S L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)} L(x) = {C, (¬C ⊔ ¬D), ¬D}

Reasoning with Expressive Description Logics – p. 27/40

slide-125
SLIDE 125

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w x L(x) = {C, (¬C ⊔ ¬D), ¬D} S L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

slide-126
SLIDE 126

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w x y L(y) = {C} L(x) = {C, (¬C ⊔ ¬D), ¬D} R S L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

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SLIDE 127

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w x y L(y) = {C} L(x) = {C, (¬C ⊔ ¬D), ¬D} R S L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

slide-128
SLIDE 128

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w x y L(y) = {C, ∃R.C, ∀R.(∃R.C)} L(x) = {C, (¬C ⊔ ¬D), ¬D} R S L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

slide-129
SLIDE 129

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w x y L(y) = {C, ∃R.C, ∀R.(∃R.C)} L(x) = {C, (¬C ⊔ ¬D), ¬D} R S L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

slide-130
SLIDE 130

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w x y L(y) = {C, ∃R.C, ∀R.(∃R.C)} L(x) = {C, (¬C ⊔ ¬D), ¬D} z L(z) = {C} R S R L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

slide-131
SLIDE 131

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w x y L(y) = {C, ∃R.C, ∀R.(∃R.C)} L(x) = {C, (¬C ⊔ ¬D), ¬D} z L(z) = {C} R S R L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

slide-132
SLIDE 132

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w x y L(y) = {C, ∃R.C, ∀R.(∃R.C)} L(x) = {C, (¬C ⊔ ¬D), ¬D} z L(z) = {C, ∃R.C, ∀R.(∃R.C)} R S R L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)}

Reasoning with Expressive Description Logics – p. 27/40

slide-133
SLIDE 133

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w x y L(y) = {C, ∃R.C, ∀R.(∃R.C)} L(x) = {C, (¬C ⊔ ¬D), ¬D} z L(z) = {C, ∃R.C, ∀R.(∃R.C)} R S R L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)} blocked

Reasoning with Expressive Description Logics – p. 27/40

slide-134
SLIDE 134

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w x y L(y) = {C, ∃R.C, ∀R.(∃R.C)} L(x) = {C, (¬C ⊔ ¬D), ¬D} z L(z) = {C, ∃R.C, ∀R.(∃R.C)} R S R L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)} blocked

Concept is satisfiable: T corresponds to model

Reasoning with Expressive Description Logics – p. 27/40

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SLIDE 135

Tableaux Algorithm — Example

Test satisfiability of ∃S.C ⊓ ∀S.(¬C ⊔ ¬D) ⊓ ∃R.C ⊓ ∀R.(∃R.C)} where R is a transitive role

w x y L(y) = {C, ∃R.C, ∀R.(∃R.C)} L(x) = {C, (¬C ⊔ ¬D), ¬D} R S L(w) = {∃S.C, ∀S.(¬C ⊔ ¬D), ∃R.C, ∀R.(∃R.C)} R

Concept is satisfiable: T corresponds to model

Reasoning with Expressive Description Logics – p. 27/40

slide-136
SLIDE 136

More Advanced Techniques

Reasoning with Expressive Description Logics – p. 28/40

slide-137
SLIDE 137

More Advanced Techniques

Satisfiability w.r.t. a Terminology ☞ For each axiom C ⊑ D ∈ T , add ¬C ⊔ D to every node label

Reasoning with Expressive Description Logics – p. 28/40

slide-138
SLIDE 138

More Advanced Techniques

Satisfiability w.r.t. a Terminology ☞ For each axiom C ⊑ D ∈ T , add ¬C ⊔ D to every node label More expressive DLs

Reasoning with Expressive Description Logics – p. 28/40

slide-139
SLIDE 139

More Advanced Techniques

Satisfiability w.r.t. a Terminology ☞ For each axiom C ⊑ D ∈ T , add ¬C ⊔ D to every node label More expressive DLs ☞ Basic technique can be extended to deal with

  • Role inclusion axioms (role hierarchy)
  • Number restrictions
  • Inverse roles
  • Concrete domains and datatypes
  • Aboxes
  • etc.

Reasoning with Expressive Description Logics – p. 28/40

slide-140
SLIDE 140

More Advanced Techniques

Satisfiability w.r.t. a Terminology ☞ For each axiom C ⊑ D ∈ T , add ¬C ⊔ D to every node label More expressive DLs ☞ Basic technique can be extended to deal with

  • Role inclusion axioms (role hierarchy)
  • Number restrictions
  • Inverse roles
  • Concrete domains and datatypes
  • Aboxes
  • etc.

☞ Extend expansion rules and use more sophisticated blocking strategy

Reasoning with Expressive Description Logics – p. 28/40

slide-141
SLIDE 141

More Advanced Techniques

Satisfiability w.r.t. a Terminology ☞ For each axiom C ⊑ D ∈ T , add ¬C ⊔ D to every node label More expressive DLs ☞ Basic technique can be extended to deal with

  • Role inclusion axioms (role hierarchy)
  • Number restrictions
  • Inverse roles
  • Concrete domains and datatypes
  • Aboxes
  • etc.

☞ Extend expansion rules and use more sophisticated blocking strategy ☞ Forest instead of Tree (for Aboxes)

  • Root nodes correspond to individuals in Abox

Reasoning with Expressive Description Logics – p. 28/40

slide-142
SLIDE 142

Highly Optimised Implementation

Reasoning with Expressive Description Logics – p. 29/40

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SLIDE 143

Highly Optimised Implementation

☞ Naive implementation − → effective non-termination

Reasoning with Expressive Description Logics – p. 29/40

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SLIDE 144

Highly Optimised Implementation

☞ Naive implementation − → effective non-termination ☞ Modern systems include MANY optimisations

Reasoning with Expressive Description Logics – p. 29/40

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SLIDE 145

Highly Optimised Implementation

☞ Naive implementation − → effective non-termination ☞ Modern systems include MANY optimisations ☞ Optimised classification (compute partial ordering)

  • Use enhanced traversal (exploit information from previous tests)
  • Use structural information to select classification order

Reasoning with Expressive Description Logics – p. 29/40

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SLIDE 146

Highly Optimised Implementation

☞ Naive implementation − → effective non-termination ☞ Modern systems include MANY optimisations ☞ Optimised classification (compute partial ordering)

  • Use enhanced traversal (exploit information from previous tests)
  • Use structural information to select classification order

☞ Optimised subsumption testing (search for models)

  • Normalisation and simplification of concepts
  • Absorption (simplification) of general axioms
  • Davis-Putnam style semantic branching search
  • Dependency directed backtracking
  • Caching of satisfiability results and (partial) models
  • Heuristic ordering of propositional and modal expansion
  • . . .

Reasoning with Expressive Description Logics – p. 29/40

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SLIDE 147

Research Challenges

Reasoning with Expressive Description Logics – p. 30/40

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SLIDE 148

Challenges

Reasoning with Expressive Description Logics – p. 31/40

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SLIDE 149

Challenges

☞ Increased expressive power

  • Existing DL systems implement (at most) SHIQ
  • DAML+OIL extends SHIQ with datatypes and nominals

Reasoning with Expressive Description Logics – p. 31/40

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SLIDE 150

Challenges

☞ Increased expressive power

  • Existing DL systems implement (at most) SHIQ
  • DAML+OIL extends SHIQ with datatypes and nominals

☞ Scalability

  • Very large KBs
  • Reasoning with (very large numbers of) individuals

Reasoning with Expressive Description Logics – p. 31/40

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SLIDE 151

Challenges

☞ Increased expressive power

  • Existing DL systems implement (at most) SHIQ
  • DAML+OIL extends SHIQ with datatypes and nominals

☞ Scalability

  • Very large KBs
  • Reasoning with (very large numbers of) individuals

☞ Other reasoning tasks

  • Querying
  • Matching
  • Least common subsumer
  • . . .

Reasoning with Expressive Description Logics – p. 31/40

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SLIDE 152

Challenges

☞ Increased expressive power

  • Existing DL systems implement (at most) SHIQ
  • DAML+OIL extends SHIQ with datatypes and nominals

☞ Scalability

  • Very large KBs
  • Reasoning with (very large numbers of) individuals

☞ Other reasoning tasks

  • Querying
  • Matching
  • Least common subsumer
  • . . .

☞ Tools and Infrastructure

  • Support for large scale ontological engineering and deployment

Reasoning with Expressive Description Logics – p. 31/40

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SLIDE 153

Increased Expressive Power: Datatypes

Reasoning with Expressive Description Logics – p. 32/40

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SLIDE 154

Increased Expressive Power: Datatypes

☞ DAML+OIL has simple form of datatypes

  • Unary predicates plus disjoint object-class/datatype domains

Reasoning with Expressive Description Logics – p. 32/40

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SLIDE 155

Increased Expressive Power: Datatypes

☞ DAML+OIL has simple form of datatypes

  • Unary predicates plus disjoint object-class/datatype domains

☞ Well understood theoretically

  • Existing work on concrete domains [Baader & Hanschke, Lutz]
  • Algorithm already known for SHOQ(D) [Horrocks & Sattler]
  • Can use hybrid reasoning (DL reasoner + datatype “oracle”)

Reasoning with Expressive Description Logics – p. 32/40

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SLIDE 156

Increased Expressive Power: Datatypes

☞ DAML+OIL has simple form of datatypes

  • Unary predicates plus disjoint object-class/datatype domains

☞ Well understood theoretically

  • Existing work on concrete domains [Baader & Hanschke, Lutz]
  • Algorithm already known for SHOQ(D) [Horrocks & Sattler]
  • Can use hybrid reasoning (DL reasoner + datatype “oracle”)

☞ May be practically challenging

  • All XMLS datatypes supported (?)

Reasoning with Expressive Description Logics – p. 32/40

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SLIDE 157

Increased Expressive Power: Datatypes

☞ DAML+OIL has simple form of datatypes

  • Unary predicates plus disjoint object-class/datatype domains

☞ Well understood theoretically

  • Existing work on concrete domains [Baader & Hanschke, Lutz]
  • Algorithm already known for SHOQ(D) [Horrocks & Sattler]
  • Can use hybrid reasoning (DL reasoner + datatype “oracle”)

☞ May be practically challenging

  • All XMLS datatypes supported (?)

☞ Already seeing some (partial) implementations

  • Cerebra system (Network Inference), Racer system (Hamburg)

Reasoning with Expressive Description Logics – p. 32/40

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SLIDE 158

Increased Expressive Power: Nominals

Reasoning with Expressive Description Logics – p. 33/40

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SLIDE 159

Increased Expressive Power: Nominals

☞ DAML+OIL oneOf constructor equivalent to hybrid logic nominals

  • Extensionally defined concepts, e.g., EU ≡ {France, Italy, . . .}

Reasoning with Expressive Description Logics – p. 33/40

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SLIDE 160

Increased Expressive Power: Nominals

☞ DAML+OIL oneOf constructor equivalent to hybrid logic nominals

  • Extensionally defined concepts, e.g., EU ≡ {France, Italy, . . .}

☞ Theoretically very challenging

  • Resulting logic has known high complexity (NExpTime)
  • No known “practical” algorithm
  • Not obvious how to extend tableaux techniques in this direction

– Loss of tree model property – Spy-points: ⊤ ⊑ ∃R.{Spy} – Finite domains: {Spy} ⊑ nR−

Reasoning with Expressive Description Logics – p. 33/40

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SLIDE 161

Increased Expressive Power: Nominals

☞ DAML+OIL oneOf constructor equivalent to hybrid logic nominals

  • Extensionally defined concepts, e.g., EU ≡ {France, Italy, . . .}

☞ Theoretically very challenging

  • Resulting logic has known high complexity (NExpTime)
  • No known “practical” algorithm
  • Not obvious how to extend tableaux techniques in this direction

– Loss of tree model property – Spy-points: ⊤ ⊑ ∃R.{Spy} – Finite domains: {Spy} ⊑ nR−

  • ?? automata based algorithms ??

Reasoning with Expressive Description Logics – p. 33/40

slide-162
SLIDE 162

Increased Expressive Power: Nominals

☞ DAML+OIL oneOf constructor equivalent to hybrid logic nominals

  • Extensionally defined concepts, e.g., EU ≡ {France, Italy, . . .}

☞ Theoretically very challenging

  • Resulting logic has known high complexity (NExpTime)
  • No known “practical” algorithm
  • Not obvious how to extend tableaux techniques in this direction

– Loss of tree model property – Spy-points: ⊤ ⊑ ∃R.{Spy} – Finite domains: {Spy} ⊑ nR−

  • ?? automata based algorithms ??

☞ Standard solution is weaker semantics for nominals

  • Treat nominals as (disjoint) primitive classes
  • Loose some inferential power, e.g., w.r.t. max cardinality

Reasoning with Expressive Description Logics – p. 33/40

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SLIDE 163

Scalability

Reasoning with Expressive Description Logics – p. 34/40

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SLIDE 164

Scalability

☞ Reasoning hard (ExpTime) even without nominals (i.e., SHIQ)

Reasoning with Expressive Description Logics – p. 34/40

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SLIDE 165

Scalability

☞ Reasoning hard (ExpTime) even without nominals (i.e., SHIQ) ☞ Web ontologies may grow very large

Reasoning with Expressive Description Logics – p. 34/40

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SLIDE 166

Scalability

☞ Reasoning hard (ExpTime) even without nominals (i.e., SHIQ) ☞ Web ontologies may grow very large ☞ Good empirical evidence of scalability/tractability for DL systems

  • E.g., 5,000 (complex) classes; 100,000+ (simple) classes

Reasoning with Expressive Description Logics – p. 34/40

slide-167
SLIDE 167

Scalability

☞ Reasoning hard (ExpTime) even without nominals (i.e., SHIQ) ☞ Web ontologies may grow very large ☞ Good empirical evidence of scalability/tractability for DL systems

  • E.g., 5,000 (complex) classes; 100,000+ (simple) classes

☞ But evidence mostly w.r.t. SHF (no inverse)

Reasoning with Expressive Description Logics – p. 34/40

slide-168
SLIDE 168

Scalability

☞ Reasoning hard (ExpTime) even without nominals (i.e., SHIQ) ☞ Web ontologies may grow very large ☞ Good empirical evidence of scalability/tractability for DL systems

  • E.g., 5,000 (complex) classes; 100,000+ (simple) classes

☞ But evidence mostly w.r.t. SHF (no inverse) ☞ Problems can arise when SHF extended to SHIQ

  • Important optimisations no longer (fully) work

Reasoning with Expressive Description Logics – p. 34/40

slide-169
SLIDE 169

Scalability

☞ Reasoning hard (ExpTime) even without nominals (i.e., SHIQ) ☞ Web ontologies may grow very large ☞ Good empirical evidence of scalability/tractability for DL systems

  • E.g., 5,000 (complex) classes; 100,000+ (simple) classes

☞ But evidence mostly w.r.t. SHF (no inverse) ☞ Problems can arise when SHF extended to SHIQ

  • Important optimisations no longer (fully) work

☞ Reasoning with individuals

  • Deployment of web ontologies will mean reasoning with

(possibly very large numbers of) individuals/tuples

  • Unlikely that standard Abox techniques will be able to cope

Reasoning with Expressive Description Logics – p. 34/40

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SLIDE 170

Other Reasoning Tasks

Reasoning with Expressive Description Logics – p. 35/40

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SLIDE 171

Other Reasoning Tasks

☞ Querying

  • Retrieval and instantiation wont be sufficient
  • Minimum requirement will be DB style query language
  • May also need “what can I say about x?” style of query

Reasoning with Expressive Description Logics – p. 35/40

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SLIDE 172

Other Reasoning Tasks

☞ Querying

  • Retrieval and instantiation wont be sufficient
  • Minimum requirement will be DB style query language
  • May also need “what can I say about x?” style of query

☞ Explanation

  • To support ontology design
  • Justifications and proofs (e.g., of query results)

Reasoning with Expressive Description Logics – p. 35/40

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SLIDE 173

Other Reasoning Tasks

☞ Querying

  • Retrieval and instantiation wont be sufficient
  • Minimum requirement will be DB style query language
  • May also need “what can I say about x?” style of query

☞ Explanation

  • To support ontology design
  • Justifications and proofs (e.g., of query results)

☞ “Non-Standard Inferences”, e.g., LCS, matching

  • To support ontology integration
  • To support “bottom up” design of ontologies

Reasoning with Expressive Description Logics – p. 35/40

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SLIDE 174

Summary

Reasoning with Expressive Description Logics – p. 36/40

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SLIDE 175

Summary

☞ Description Logics are family of logical KR formalisms

Reasoning with Expressive Description Logics – p. 36/40

slide-176
SLIDE 176

Summary

☞ Description Logics are family of logical KR formalisms ☞ Applications of DLs include DataBases and Semantic Web

  • Ontologies will provide vocabulary for semantic markup
  • DAML+OIL web ontology language based on SHIQ DL
  • Set to become W3C standard (OWL) & already widely adopted
  • Use of DL provides formal foundations and reasoning support

Reasoning with Expressive Description Logics – p. 36/40

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SLIDE 177

Summary

☞ Description Logics are family of logical KR formalisms ☞ Applications of DLs include DataBases and Semantic Web

  • Ontologies will provide vocabulary for semantic markup
  • DAML+OIL web ontology language based on SHIQ DL
  • Set to become W3C standard (OWL) & already widely adopted
  • Use of DL provides formal foundations and reasoning support

☞ DL Reasoning based on tableau algorithms

Reasoning with Expressive Description Logics – p. 36/40

slide-178
SLIDE 178

Summary

☞ Description Logics are family of logical KR formalisms ☞ Applications of DLs include DataBases and Semantic Web

  • Ontologies will provide vocabulary for semantic markup
  • DAML+OIL web ontology language based on SHIQ DL
  • Set to become W3C standard (OWL) & already widely adopted
  • Use of DL provides formal foundations and reasoning support

☞ DL Reasoning based on tableau algorithms ☞ Highly Optimised implementations used in DL systems

Reasoning with Expressive Description Logics – p. 36/40

slide-179
SLIDE 179

Summary

☞ Description Logics are family of logical KR formalisms ☞ Applications of DLs include DataBases and Semantic Web

  • Ontologies will provide vocabulary for semantic markup
  • DAML+OIL web ontology language based on SHIQ DL
  • Set to become W3C standard (OWL) & already widely adopted
  • Use of DL provides formal foundations and reasoning support

☞ DL Reasoning based on tableau algorithms ☞ Highly Optimised implementations used in DL systems ☞ Challenges remain

  • Reasoning with full DAML+OIL/OWL language
  • (Convincing) demonstration(s) of scalability
  • New reasoning tasks
  • Development of (high quality) tools and infrastructure

Reasoning with Expressive Description Logics – p. 36/40

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SLIDE 180

Acknowledgements

Reasoning with Expressive Description Logics – p. 37/40

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SLIDE 181

Acknowledgements

☞ Members of the OIL and DAML+OIL development teams, in particular Dieter Fensel and Frank van Harmelen (Amsterdam) and Peter Patel-Schneider (Bell Labs)

Reasoning with Expressive Description Logics – p. 37/40

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SLIDE 182

Acknowledgements

☞ Members of the OIL and DAML+OIL development teams, in particular Dieter Fensel and Frank van Harmelen (Amsterdam) and Peter Patel-Schneider (Bell Labs) ☞ Franz Baader, Uli Sattler and Stefan Tobies (Dresden)

Reasoning with Expressive Description Logics – p. 37/40

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SLIDE 183

Acknowledgements

☞ Members of the OIL and DAML+OIL development teams, in particular Dieter Fensel and Frank van Harmelen (Amsterdam) and Peter Patel-Schneider (Bell Labs) ☞ Franz Baader, Uli Sattler and Stefan Tobies (Dresden) ☞ Members of the Information Management, Medical Informatics and Formal Methods Groups at the University of Manchester

Reasoning with Expressive Description Logics – p. 37/40

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SLIDE 184

Resources

Slides from this talk http://www.cs.man.ac.uk/~horrocks/Slides/ed02.pdf FaCT system (open source) http://www.cs.man.ac.uk/FaCT/ OilEd (open source) http://oiled.man.ac.uk/ OIL http://www.ontoknowledge.org/oil/ DAML+OIL http://www.w3c.org/Submission/2001/12/ W3C Web-Ontology (WebOnt) working group (OWL) http://www.w3.org/2001/sw/WebOnt/ DL Handbook — available autumn 2002 from Cambridge University Press

Reasoning with Expressive Description Logics – p. 38/40

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SLIDE 185

Select Bibliography

  • I. Horrocks. DAML+OIL: a reason-able web ontology language. In Proc. of

EDBT 2002, number 2287 in Lecture Notes in Computer Science, pages 2–13. Springer-Verlag, Mar. 2002.

  • I. Horrocks, P

. F. Patel-Schneider, and F. van Harmelen. Reviewing the design of DAML+OIL: An ontology language for the semantic web. In Proc.

  • f AAAI 2002, 2002. To appear.
  • I. Horrocks and S. Tessaris. Querying the semantic web: a formal
  • approach. In I. Horrocks and J. Hendler, editors, Proc. of the 2002

International Semantic Web Conference (ISWC 2002), number 2342 in Lecture Notes in Computer Science. Springer-Verlag, 2002.

  • C. Lutz. The Complexity of Reasoning with Concrete Domains. PhD

thesis, Teaching and Research Area for Theoretical Computer Science, RWTH Aachen, 2001.

Reasoning with Expressive Description Logics – p. 39/40

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SLIDE 186

Select Bibliography

  • I. Horrocks and U. Sattler. Ontology reasoning in the SHOQ(D)

description logic. In B. Nebel, editor, Proc. of IJCAI-01, pages 199–204. Morgan Kaufmann, 2001.

  • F. Baader, S. Brandt, and R. Küsters. Matching under side conditions in

description logics. In B. Nebel, editor, Proc. of IJCAI-01, pages 213–218, Seattle, Washington, 2001. Morgan Kaufmann.

  • A. Borgida, E. Franconi, and I. Horrocks. Explaining ALC subsumption. In
  • Proc. of ECAI 2000, pages 209–213. IOS Press, 2000.
  • D. Calvanese, G. De Giacomo, M. Lenzerini, D. Nardi, and R. Rosati. A

principled approach to data integration and reconciliation in data

  • warehousing. In Proceedings of the International Workshop on Design

and Management of Data Warehouses (DWDM’99), 1999.

Reasoning with Expressive Description Logics – p. 40/40