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Formal rmal Foundations oundations of of Ontologies Ontologies - - PowerPoint PPT Presentation

Feb 04, 2023 155 likes •897 views

Making Statements DL Knowledge Bases Entailment in DLs Formal rmal Foundations oundations of of Ontologies Ontologies and and Reasoning Reasoning Ivan Ivan Varzinczak rzinczak Universit dArtois & CNRS, France

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SLIDE 1

Making Statements DL Knowledge Bases Entailment in DLs

Formal rmal Foundations

  • undations of
  • f Ontologies

Ontologies and and Reasoning Reasoning Ivan Ivan Varzinczak rzinczak

Université d’Artois & CNRS, France http://www.ijv.ovh

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 1

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SLIDE 2

Making Statements DL Knowledge Bases Entailment in DLs

Outline of Part 2

Making Statements DL Knowledge Bases Entailment in DLs

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 2

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SLIDE 3

Making Statements DL Knowledge Bases Entailment in DLs

Outline of Part 2

Making Statements DL Knowledge Bases Entailment in DLs

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 3

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SLIDE 4

Making Statements DL Knowledge Bases Entailment in DLs

Motivation

Concept language of ALC

⊤, ⊥ (constants) A (atomic concept) ¬C (complement of C) C ⊓ D (intersection of C and D) C ⊔ D (union of C and D) ∃r.C (existential restriction) ∀r.C (value restriction)

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 4

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SLIDE 5

Making Statements DL Knowledge Bases Entailment in DLs

Motivation

Concept language of ALC

⊤, ⊥ (constants) A (atomic concept) ¬C (complement of C) C ⊓ D (intersection of C and D) C ⊔ D (union of C and D) ∃r.C (existential restriction) ∀r.C (value restriction)

Something is missing

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 4

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SLIDE 6

Making Statements DL Knowledge Bases Entailment in DLs

Motivation

Concept language of ALC

⊤, ⊥ (constants) A (atomic concept) ¬C (complement of C) C ⊓ D (intersection of C and D) C ⊔ D (union of C and D) ∃r.C (existential restriction) ∀r.C (value restriction)

Something is missing

  • The central notion in logic: C ‘→’ D

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 4

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SLIDE 7

Making Statements DL Knowledge Bases Entailment in DLs

Motivation

Concept language of ALC

⊤, ⊥ (constants) A (atomic concept) ¬C (complement of C) C ⊓ D (intersection of C and D) C ⊔ D (union of C and D) ∃r.C (existential restriction) ∀r.C (value restriction)

Something is missing

  • The central notion in logic: C ‘→’ D
  • What would C ‘→’ D mean here?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 4

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SLIDE 8

Making Statements DL Knowledge Bases Entailment in DLs

Motivation

Concept language of ALC

⊤, ⊥ (constants) A (atomic concept) ¬C (complement of C) C ⊓ D (intersection of C and D) C ⊔ D (union of C and D) ∃r.C (existential restriction) ∀r.C (value restriction)

Something is missing

  • The central notion in logic: C ‘→’ D
  • What would C ‘→’ D mean here? (We already have ¬C ⊔ D)

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 4

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SLIDE 9

Making Statements DL Knowledge Bases Entailment in DLs

Motivation

Concept language of ALC

⊤, ⊥ (constants) A (atomic concept) ¬C (complement of C) C ⊓ D (intersection of C and D) C ⊔ D (union of C and D) ∃r.C (existential restriction) ∀r.C (value restriction)

Something is missing

  • The central notion in logic: C ‘→’ D
  • What would C ‘→’ D mean here? (We already have ¬C ⊔ D)
  • DLs have a version of ‘→’ that is very special

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 4

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SLIDE 10

Making Statements DL Knowledge Bases Entailment in DLs

Statements

In many logics

meta-language (entailment, etc)

  • bject language

(formulae)

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 5

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SLIDE 11

Making Statements DL Knowledge Bases Entailment in DLs

Statements

In many logics

meta-language (entailment, etc)

  • bject language

(formulae)

In DLs

meta-language statements concept language

  • Two levels of language
  • Two notions of ‘entailment’
  • Two notions of ‘satisfaction’

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 5

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SLIDE 12

Making Statements DL Knowledge Bases Entailment in DLs

Making statements

Subsumption

  • Concept inclusion
  • Employed students are students
  • Employed students are employees

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 6

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SLIDE 13

Making Statements DL Knowledge Bases Entailment in DLs

Making statements

Subsumption

  • Concept inclusion
  • Employed students are students
  • Employed students are employees

Instantiation or assertions

  • Concept and role membership
  • John is an employed student

(John instantiates employed student)

  • John works for IBM

(John and IBM instantiate works for)

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 6

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SLIDE 14

Making Statements DL Knowledge Bases Entailment in DLs

Making statements

Subsumption

  • Concept inclusion
  • Employed students are students
  • Employed students are employees

Instantiation or assertions

  • Concept and role membership
  • John is an employed student

(John instantiates employed student)

  • John works for IBM

(John and IBM instantiate works for) Statements talk about concepts, roles and individuals

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 6

slide-15
SLIDE 15

Making Statements DL Knowledge Bases Entailment in DLs

Making statements

Subsumption

  • Concept inclusion
  • Employed students are students
  • Employed students are employees

Instantiation or assertions

  • Concept and role membership
  • John is an employed student

(John instantiates employed student)

  • John works for IBM

(John and IBM instantiate works for) Statements talk about concepts, roles and individuals They are not concepts! They are in the ‘in-between’ language

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 6

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SLIDE 16

Making Statements DL Knowledge Bases Entailment in DLs

Subsumption statements

C ⊑ D

Intuition

  • D subsumes C

(or C is subsumed by D)

  • C is more specific than D

(or D is more general than C)

  • Formalise one aspect of is-a relations

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 7

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SLIDE 17

Making Statements DL Knowledge Bases Entailment in DLs

Subsumption statements

C ⊑ D

Intuition

  • D subsumes C

(or C is subsumed by D)

  • C is more specific than D

(or D is more general than C)

  • Formalise one aspect of is-a relations

Example

  • EmpStud ⊑ Student ⊓ Employee,

Employee ⊑ ∃worksFor.⊤

  • EmpStud ⊑ ∃pays.Tax,

Student ⊓ ¬Employee ⊑ ¬∃pays.Tax

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 7

slide-18
SLIDE 18

Making Statements DL Knowledge Bases Entailment in DLs

Subsumption statements

C ⊑ D

Intuition

  • D subsumes C

(or C is subsumed by D)

  • C is more specific than D

(or D is more general than C)

  • Formalise one aspect of is-a relations

Example

  • EmpStud ⊑ Student ⊓ Employee,

Employee ⊑ ∃worksFor.⊤

  • EmpStud ⊑ ∃pays.Tax,

Student ⊓ ¬Employee ⊑ ¬∃pays.Tax Central notion in DL terminologies (taxonomies)

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 7

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SLIDE 19

Making Statements DL Knowledge Bases Entailment in DLs

Subsumption statements

C ⊑ D

Semantics

  • I C ⊑ D

(I satisfies C ⊑ D) if CI ⊆ DI

  • First level of ‘entailment’: all C-objects are D-objects

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 8

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SLIDE 20

Making Statements DL Knowledge Bases Entailment in DLs

Subsumption statements

C ⊑ D

Semantics

  • I C ⊑ D

(I satisfies C ⊑ D) if CI ⊆ DI

  • First level of ‘entailment’: all C-objects are D-objects

∆I CI DI

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 8

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SLIDE 21

Making Statements DL Knowledge Bases Entailment in DLs

Subsumption statements

C ≡ D

Concept equivalence

  • Just an abbreviation for C ⊑ D and D ⊑ C
  • I C ≡ D

if I C ⊑ D and I D ⊑ C

  • I C ≡ D

if CI = DI

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 9

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SLIDE 22

Making Statements DL Knowledge Bases Entailment in DLs

Subsumption statements

C ≡ D

Concept equivalence

  • Just an abbreviation for C ⊑ D and D ⊑ C
  • I C ≡ D

if I C ⊑ D and I D ⊑ C

  • I C ≡ D

if CI = DI

∆I CI DI

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 9

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SLIDE 23

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 10

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SLIDE 24

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I EmpStud ⊑ Student ⊓ Employee ?
  • I ∃worksFor.⊤ ⊑ Employee ?
  • I Employee ⊑ ∃worksFor.⊤ ?
  • I EmpStud ⊑ ∀pays.Tax ?
  • I Parent ⊓ ¬Employee ⊑ Tax ⊔ ¬Student ?
  • I EmpStud ⊓ Parent ⊑ ∃pays.Tax ?
  • I ∃empBy.⊤ ⊑ ∃worksFor.Company ?
  • I ∀pays.Tax ⊑ ∃empBy.Company ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 10

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SLIDE 25

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I EmpStud ⊑ Student ⊓ Employee Yep!
  • I ∃worksFor.⊤ ⊑ Employee ?
  • I Employee ⊑ ∃worksFor.⊤ ?
  • I EmpStud ⊑ ∀pays.Tax ?
  • I Parent ⊓ ¬Employee ⊑ Tax ⊔ ¬Student ?
  • I EmpStud ⊓ Parent ⊑ ∃pays.Tax ?
  • I ∃empBy.⊤ ⊑ ∃worksFor.Company ?
  • I ∀pays.Tax ⊑ ∃empBy.Company ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 10

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SLIDE 26

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I EmpStud ⊑ Student ⊓ Employee Yep!
  • I ∃worksFor.⊤ ⊑ Employee Yep!
  • I Employee ⊑ ∃worksFor.⊤ ?
  • I EmpStud ⊑ ∀pays.Tax ?
  • I Parent ⊓ ¬Employee ⊑ Tax ⊔ ¬Student ?
  • I EmpStud ⊓ Parent ⊑ ∃pays.Tax ?
  • I ∃empBy.⊤ ⊑ ∃worksFor.Company ?
  • I ∀pays.Tax ⊑ ∃empBy.Company ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 10

slide-27
SLIDE 27

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I EmpStud ⊑ Student ⊓ Employee Yep!
  • I ∃worksFor.⊤ ⊑ Employee Yep!
  • I Employee ⊑ ∃worksFor.⊤ No!
  • I EmpStud ⊑ ∀pays.Tax ?
  • I Parent ⊓ ¬Employee ⊑ Tax ⊔ ¬Student ?
  • I EmpStud ⊓ Parent ⊑ ∃pays.Tax ?
  • I ∃empBy.⊤ ⊑ ∃worksFor.Company ?
  • I ∀pays.Tax ⊑ ∃empBy.Company ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 10

slide-28
SLIDE 28

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I EmpStud ⊑ Student ⊓ Employee Yep!
  • I ∃worksFor.⊤ ⊑ Employee Yep!
  • I Employee ⊑ ∃worksFor.⊤ No!
  • I EmpStud ⊑ ∀pays.Tax No!
  • I Parent ⊓ ¬Employee ⊑ Tax ⊔ ¬Student ?
  • I EmpStud ⊓ Parent ⊑ ∃pays.Tax ?
  • I ∃empBy.⊤ ⊑ ∃worksFor.Company ?
  • I ∀pays.Tax ⊑ ∃empBy.Company ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 10

slide-29
SLIDE 29

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I EmpStud ⊑ Student ⊓ Employee Yep!
  • I ∃worksFor.⊤ ⊑ Employee Yep!
  • I Employee ⊑ ∃worksFor.⊤ No!
  • I EmpStud ⊑ ∀pays.Tax No!
  • I Parent ⊓ ¬Employee ⊑ Tax ⊔ ¬Student Yep!
  • I EmpStud ⊓ Parent ⊑ ∃pays.Tax ?
  • I ∃empBy.⊤ ⊑ ∃worksFor.Company ?
  • I ∀pays.Tax ⊑ ∃empBy.Company ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 10

slide-30
SLIDE 30

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I EmpStud ⊑ Student ⊓ Employee Yep!
  • I ∃worksFor.⊤ ⊑ Employee Yep!
  • I Employee ⊑ ∃worksFor.⊤ No!
  • I EmpStud ⊑ ∀pays.Tax No!
  • I Parent ⊓ ¬Employee ⊑ Tax ⊔ ¬Student Yep!
  • I EmpStud ⊓ Parent ⊑ ∃pays.Tax No!
  • I ∃empBy.⊤ ⊑ ∃worksFor.Company ?
  • I ∀pays.Tax ⊑ ∃empBy.Company ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 10

slide-31
SLIDE 31

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I EmpStud ⊑ Student ⊓ Employee Yep!
  • I ∃worksFor.⊤ ⊑ Employee Yep!
  • I Employee ⊑ ∃worksFor.⊤ No!
  • I EmpStud ⊑ ∀pays.Tax No!
  • I Parent ⊓ ¬Employee ⊑ Tax ⊔ ¬Student Yep!
  • I EmpStud ⊓ Parent ⊑ ∃pays.Tax No!
  • I ∃empBy.⊤ ⊑ ∃worksFor.Company Yep!
  • I ∀pays.Tax ⊑ ∃empBy.Company ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 10

slide-32
SLIDE 32

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I EmpStud ⊑ Student ⊓ Employee Yep!
  • I ∃worksFor.⊤ ⊑ Employee Yep!
  • I Employee ⊑ ∃worksFor.⊤ No!
  • I EmpStud ⊑ ∀pays.Tax No!
  • I Parent ⊓ ¬Employee ⊑ Tax ⊔ ¬Student Yep!
  • I EmpStud ⊓ Parent ⊑ ∃pays.Tax No!
  • I ∃empBy.⊤ ⊑ ∃worksFor.Company Yep!
  • I ∀pays.Tax ⊑ ∃empBy.Company No!

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 10

slide-33
SLIDE 33

Making Statements DL Knowledge Bases Entailment in DLs

Assertions

a : C (a, b) : r

Intuition

  • a is an instance of C
  • a and b are related via r

(or (a, b) is an instance of r)

  • Formalise another aspect of is-a relations

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 11

slide-34
SLIDE 34

Making Statements DL Knowledge Bases Entailment in DLs

Assertions

a : C (a, b) : r

Intuition

  • a is an instance of C
  • a and b are related via r

(or (a, b) is an instance of r)

  • Formalise another aspect of is-a relations

Example

  • john : EmpStud,

mary : Parent ⊓ ¬∃worksFor.⊤ ⊓ ¬∃pays.Tax

  • (john, ibm) : worksFor

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 11

slide-35
SLIDE 35

Making Statements DL Knowledge Bases Entailment in DLs

Assertions

a : C (a, b) : r

Intuition

  • a is an instance of C
  • a and b are related via r

(or (a, b) is an instance of r)

  • Formalise another aspect of is-a relations

Example

  • john : EmpStud,

mary : Parent ⊓ ¬∃worksFor.⊤ ⊓ ¬∃pays.Tax

  • (john, ibm) : worksFor

(in some DLs: (john, ibm) : ¬empBy)

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 11

slide-36
SLIDE 36

Making Statements DL Knowledge Bases Entailment in DLs

Assertions

a : C (a, b) : r

Intuition

  • a is an instance of C
  • a and b are related via r

(or (a, b) is an instance of r)

  • Formalise another aspect of is-a relations

Example

  • john : EmpStud,

mary : Parent ⊓ ¬∃worksFor.⊤ ⊓ ¬∃pays.Tax

  • (john, ibm) : worksFor

(in some DLs: (john, ibm) : ¬empBy) Central notion in DL ‘databases’

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 11

slide-37
SLIDE 37

Making Statements DL Knowledge Bases Entailment in DLs

Assertions

a : C (a, b) : r

Semantics

  • I a : C

(I satisfies a : C) if aI ∈ CI

  • I (a, b) : r

(I satisfies (a, b) : r) if (aI, bI) ∈ rI

  • First level of ‘satisfaction’: a is a ‘model’ of C,

(a, b) is a ‘model’ of r

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 12

slide-38
SLIDE 38

Making Statements DL Knowledge Bases Entailment in DLs

Assertions

a : C (a, b) : r

Semantics

  • I a : C

(I satisfies a : C) if aI ∈ CI

  • I (a, b) : r

(I satisfies (a, b) : r) if (aI, bI) ∈ rI

  • First level of ‘satisfaction’: a is a ‘model’ of C,

(a, b) is a ‘model’ of r

∆I aI CI ∆I aI bI rI

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 12

slide-39
SLIDE 39

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 13

slide-40
SLIDE 40

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I john : Employee ⊓ ∃pays.Tax ?
  • I mary : ∀pays.Tax ?
  • I (mary, ibm) : empBy ?
  • I (ibm, john) : worksFor ?
  • I mary : Parent ⊓ ¬∃worksFor.⊤ ⊓ ¬∃pays.Tax ?
  • I mary : Employee ⊓ ∃empBy.⊤ ?
  • I john : ∀empBy.Company ?
  • I john : ∃worksFor.∀pays.⊥ ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 13

slide-41
SLIDE 41

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I john : Employee ⊓ ∃pays.Tax Yep!
  • I mary : ∀pays.Tax ?
  • I (mary, ibm) : empBy ?
  • I (ibm, john) : worksFor ?
  • I mary : Parent ⊓ ¬∃worksFor.⊤ ⊓ ¬∃pays.Tax ?
  • I mary : Employee ⊓ ∃empBy.⊤ ?
  • I john : ∀empBy.Company ?
  • I john : ∃worksFor.∀pays.⊥ ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 13

slide-42
SLIDE 42

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I john : Employee ⊓ ∃pays.Tax Yep!
  • I mary : ∀pays.Tax Yep!
  • I (mary, ibm) : empBy ?
  • I (ibm, john) : worksFor ?
  • I mary : Parent ⊓ ¬∃worksFor.⊤ ⊓ ¬∃pays.Tax ?
  • I mary : Employee ⊓ ∃empBy.⊤ ?
  • I john : ∀empBy.Company ?
  • I john : ∃worksFor.∀pays.⊥ ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 13

slide-43
SLIDE 43

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I john : Employee ⊓ ∃pays.Tax Yep!
  • I mary : ∀pays.Tax Yep!
  • I (mary, ibm) : empBy No!
  • I (ibm, john) : worksFor ?
  • I mary : Parent ⊓ ¬∃worksFor.⊤ ⊓ ¬∃pays.Tax ?
  • I mary : Employee ⊓ ∃empBy.⊤ ?
  • I john : ∀empBy.Company ?
  • I john : ∃worksFor.∀pays.⊥ ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 13

slide-44
SLIDE 44

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I john : Employee ⊓ ∃pays.Tax Yep!
  • I mary : ∀pays.Tax Yep!
  • I (mary, ibm) : empBy No!
  • I (ibm, john) : worksFor No!
  • I mary : Parent ⊓ ¬∃worksFor.⊤ ⊓ ¬∃pays.Tax ?
  • I mary : Employee ⊓ ∃empBy.⊤ ?
  • I john : ∀empBy.Company ?
  • I john : ∃worksFor.∀pays.⊥ ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 13

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Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I john : Employee ⊓ ∃pays.Tax Yep!
  • I mary : ∀pays.Tax Yep!
  • I (mary, ibm) : empBy No!
  • I (ibm, john) : worksFor No!
  • I mary : Parent ⊓ ¬∃worksFor.⊤ ⊓ ¬∃pays.Tax Yep!
  • I mary : Employee ⊓ ∃empBy.⊤ ?
  • I john : ∀empBy.Company ?
  • I john : ∃worksFor.∀pays.⊥ ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 13

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Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I john : Employee ⊓ ∃pays.Tax Yep!
  • I mary : ∀pays.Tax Yep!
  • I (mary, ibm) : empBy No!
  • I (ibm, john) : worksFor No!
  • I mary : Parent ⊓ ¬∃worksFor.⊤ ⊓ ¬∃pays.Tax Yep!
  • I mary : Employee ⊓ ∃empBy.⊤ No!
  • I john : ∀empBy.Company ?
  • I john : ∃worksFor.∀pays.⊥ ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 13

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Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I john : Employee ⊓ ∃pays.Tax Yep!
  • I mary : ∀pays.Tax Yep!
  • I (mary, ibm) : empBy No!
  • I (ibm, john) : worksFor No!
  • I mary : Parent ⊓ ¬∃worksFor.⊤ ⊓ ¬∃pays.Tax Yep!
  • I mary : Employee ⊓ ∃empBy.⊤ No!
  • I john : ∀empBy.Company Yep!
  • I john : ∃worksFor.∀pays.⊥ ?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 13

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Making Statements DL Knowledge Bases Entailment in DLs

Exercise

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy

  • I john : Employee ⊓ ∃pays.Tax Yep!
  • I mary : ∀pays.Tax Yep!
  • I (mary, ibm) : empBy No!
  • I (ibm, john) : worksFor No!
  • I mary : Parent ⊓ ¬∃worksFor.⊤ ⊓ ¬∃pays.Tax Yep!
  • I mary : Employee ⊓ ∃empBy.⊤ No!
  • I john : ∀empBy.Company Yep!
  • I john : ∃worksFor.∀pays.⊥ Yep!

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 13

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Making Statements DL Knowledge Bases Entailment in DLs

Subsumptions and assertions

Validity

  • Let α denote a statement
  • |

= α (α is valid) if I α for every I

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Making Statements DL Knowledge Bases Entailment in DLs

Subsumptions and assertions

Validity

  • Let α denote a statement
  • |

= α (α is valid) if I α for every I

Example

  • |

= ¬(C ⊓ D) ≡ (¬C ⊔ ¬D)

  • |

= ∀r.(C ⊓ D) ⊑ ∀r.C

  • |

= ∃r.⊤ ⊑ ∃r.C

  • |

= a : C ⊔ ¬C

  • |

= ¬(C ⊔ D) ≡ (¬C ⊓ ¬D)

  • |

= ∀r.C ⊑ ∀r.(C ⊓ D)

  • |

= ∃r.C ⊑ ∃r.⊤

  • |

= (a, b) : r

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 14

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Making Statements DL Knowledge Bases Entailment in DLs

Subsumptions and assertions

Validity

  • Let α denote a statement
  • |

= α (α is valid) if I α for every I

Example

  • |

= ¬(C ⊓ D) ≡ (¬C ⊔ ¬D)

  • |

= ∀r.(C ⊓ D) ⊑ ∀r.C

  • |

= ∃r.⊤ ⊑ ∃r.C

  • |

= a : C ⊔ ¬C

  • |

= ¬(C ⊔ D) ≡ (¬C ⊓ ¬D)

  • |

= ∀r.C ⊑ ∀r.(C ⊓ D)

  • |

= ∃r.C ⊑ ∃r.⊤

  • |

= (a, b) : r Watch out: Statements can be valid; concepts cannot!

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 14

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Making Statements DL Knowledge Bases Entailment in DLs

Outline of Part 2

Making Statements DL Knowledge Bases Entailment in DLs

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 15

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Making Statements DL Knowledge Bases Entailment in DLs

TBoxes and ABoxes

Intensional knowledge

  • Set of subsumption statements
  • Intuition: provide definitions of concepts (a terminology)
  • Called the TBox (terminological box). Notation: T

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 16

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Making Statements DL Knowledge Bases Entailment in DLs

TBoxes and ABoxes

Intensional knowledge

  • Set of subsumption statements
  • Intuition: provide definitions of concepts (a terminology)
  • Called the TBox (terminological box). Notation: T

Extensional knowledge

  • Concept and role assertions
  • Intuition: provide an instantiation of concepts and roles (a ‘database’)
  • Called the ABox (assertion box). Notation: A

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 16

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Making Statements DL Knowledge Bases Entailment in DLs

TBoxes and ABoxes

Intensional knowledge

  • Set of subsumption statements
  • Intuition: provide definitions of concepts (a terminology)
  • Called the TBox (terminological box). Notation: T

Extensional knowledge

  • Concept and role assertions
  • Intuition: provide an instantiation of concepts and roles (a ‘database’)
  • Called the ABox (assertion box). Notation: A

Definition (Knowledge base)

A DL knowledge base (a.k.a. ontology) is a tuple KB =def T , A

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 16

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Making Statements DL Knowledge Bases Entailment in DLs

Knowledge bases

Example (The student ontology in DL)

T =

                  

EmpStud ≡ Student ⊓ Employee, Student ⊓ ¬Employee ⊑ ¬∃pays.Tax, EmpStud ⊓ ¬Parent ⊑ ∃pays.Tax, EmpStud ⊓ Parent ⊑ ¬∃pays.Tax, ∃worksFor.Company ⊑ Employee

                  

A =

            

ibm : Company, mary : Parent, john : EmpStud, (john, ibm) : worksFor

            

classes relations individuals

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 17

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Making Statements DL Knowledge Bases Entailment in DLs

Knowledge bases

Semantics

  • I T

if I C ⊑ D for every C ⊑ D ∈ T

  • I A

if:

  • I a : C

for every a : C ∈ A, and

  • I (a, b) : r

for every (a, b) : r ∈ A

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 18

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Making Statements DL Knowledge Bases Entailment in DLs

Knowledge bases

Semantics

  • I T

if I C ⊑ D for every C ⊑ D ∈ T

  • I A

if:

  • I a : C

for every a : C ∈ A, and

  • I (a, b) : r

for every (a, b) : r ∈ A

Moreover

  • If I T , we say I is a model of T
  • If I A, we say I is a model of A
  • If I T ∪ A, then I is a model of KB = T , A
  • KB is satisfiable if it has a model

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 18

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Making Statements DL Knowledge Bases Entailment in DLs

Exercise

  • Let KB = T , A, where:

T =

                  

EmpStud ≡ Student ⊓ Employee, Student ⊓ ¬Employee ⊑ ¬∃pays.Tax, EmpStud ⊓ ¬Parent ⊑ ∃pays.Tax, EmpStud ⊓ Parent ⊑ ¬∃pays.Tax, ∃worksFor.Company ⊑ Employee

                  

A =

            

ibm : Company, mary : Parent, john : EmpStud, (john, ibm) : worksFor

            

  • Is the interpretation I below a model of KB ?

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 19

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SLIDE 60

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

  • Let KB = T , A, where:

T =

                  

EmpStud ≡ Student ⊓ Employee, Student ⊓ ¬Employee ⊑ ¬∃pays.Tax, EmpStud ⊓ ¬Parent ⊑ ∃pays.Tax, EmpStud ⊓ Parent ⊑ ¬∃pays.Tax, ∃worksFor.Company ⊑ Employee

                  

A =

            

ibm : Company, mary : Parent, john : EmpStud, (john, ibm) : worksFor

            

  • Is the interpretation I below a model of KB ? Yep!

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 19

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Making Statements DL Knowledge Bases Entailment in DLs

Exercise

  • Let KB = T , A, where:

T =

                  

EmpStud ≡ Student ⊓ Employee, Student ⊓ ¬Employee ⊑ ¬∃pays.Tax, EmpStud ⊓ ¬Parent ⊑ ∃pays.Tax, EmpStud ⊓ Parent ⊑ ¬∃pays.Tax, ∃worksFor.Company ⊑ Employee

                  

A =

            

ibm : Company, mary : Parent, john : EmpStud, (john, ibm) : worksFor

            

  • Find a counter-model for this knowledge base

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 20

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SLIDE 62

Making Statements DL Knowledge Bases Entailment in DLs

Exercise

  • Let KB = T , A, where:

T =

                  

EmpStud ≡ Student ⊓ Employee, Student ⊓ ¬Employee ⊑ ¬∃pays.Tax, EmpStud ⊓ ¬Parent ⊑ ∃pays.Tax, EmpStud ⊓ Parent ⊑ ¬∃pays.Tax, ∃worksFor.Company ⊑ Employee

                  

A =

            

ibm : Company, mary : Parent, john : EmpStud, (john, ibm) : worksFor

            

  • Find a counter-model for this knowledge base

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1 x2(mary) x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 20

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SLIDE 63

Making Statements DL Knowledge Bases Entailment in DLs

Outline of Part 2

Making Statements DL Knowledge Bases Entailment in DLs

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 21

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Making Statements DL Knowledge Bases Entailment in DLs

What does follow from a KB?

Example

  • Is an employed parent an employee who is also a student?
  • Is being an employee the same as being employed by someone?
  • Is Mary an employed parent?
  • Is being an employee more general than being a employed student?
  • Is there anybody who is a student and a parent at the same time?
  • Is my knowledge base consistent?
  • Tell me, briefly, what John is.
  • Who are the employed students who work for companies?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 22

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Making Statements DL Knowledge Bases Entailment in DLs

What does follow from a KB?

Entailment from KBs

  • Defined on the level of statements (not concepts)
  • Remember:

In many logics

meta-language (entailment, etc)

  • bject language

(formulae)

In DLs

meta-language statements concept language

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 23

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Making Statements DL Knowledge Bases Entailment in DLs

What does follow from a KB?

Entailment from KBs

  • Given a TBox T , what other subsumptions follow?
  • Given an ABox A, what other assertions follow?
  • Given a knowledge base KB = T , A, what statements follow from it?

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 24

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Making Statements DL Knowledge Bases Entailment in DLs

What does follow from a KB?

Entailment from KBs

  • Given a TBox T , what other subsumptions follow?
  • Given an ABox A, what other assertions follow?
  • Given a knowledge base KB = T , A, what statements follow from it?

Obvious definition of entailment

  • T |

= α if I α for every I s.t. I T

  • A |

= α if I α for every I s.t. I A

  • KB |

= α if I α for every I s.t. I KB

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 24

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Making Statements DL Knowledge Bases Entailment in DLs

What does follow from a KB?

Example

T =

                  

EmpStud ≡ Student ⊓ Employee, Student ⊓ ¬Employee ⊑ ¬∃pays.Tax, EmpStud ⊓ ¬Parent ⊑ ∃pays.Tax, EmpStud ⊓ Parent ⊑ ¬∃pays.Tax, ∃worksFor.Company ⊑ Employee

                  

A =

            

ibm : Company, mary : Parent, john : EmpStud, (john, ibm) : worksFor

            

  • KB |

= Student ⊓ ∃worksFor.Company ⊓ ¬Parent ⊑ EmpStud ⊓ ∃pays.Tax

  • KB |

= john : Student ⊓ ∃worksFor.Company

  • KB |

= mary : ¬∃pays.Tax

  • KB |

= Employee ⊑ ∃empBy.Company

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 25

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Making Statements DL Knowledge Bases Entailment in DLs

What does follow from a KB?

Example

  • KB |

= mary : ¬∃pays.Tax

  • KB |

= Employee ⊑ ∃empBy.Company

I : ∆I TaxI ParentI StudentI EmployeeI CompanyI EmpStudI x0 x1(mary) x2 x3 x4 x5(john) x6(ibm) x7 x8 x9 x10

pays pays worksFor worksFor empBy Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 26

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Making Statements DL Knowledge Bases Entailment in DLs

Open- v. closed-world assumption

Closed-world assumption (CWA)

  • KB contains all information
  • Non-derivable statements are assumed to be false

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 27

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Making Statements DL Knowledge Bases Entailment in DLs

Open- v. closed-world assumption

Closed-world assumption (CWA)

  • KB contains all information
  • Non-derivable statements are assumed to be false

Open-world assumption (OWA)

  • KB may be incomplete
  • Truth of non-derivable statements is just unknown

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 27

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Making Statements DL Knowledge Bases Entailment in DLs

Open- v. closed-world assumption

Closed-world assumption (CWA)

  • KB contains all information
  • Non-derivable statements are assumed to be false

Open-world assumption (OWA)

  • KB may be incomplete
  • Truth of non-derivable statements is just unknown

Example

{(john, ibm) : worksFor, ibm : Company)} | = john : ∀worksFor.Company ?

  • In Prolog: “Yep!”
  • In DL-based systems: “Uh, I don’t know . . . ”

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 27

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Making Statements DL Knowledge Bases Entailment in DLs

Epilogue

Summary

  • Intensional and extensional knowledge
  • Specifying DL knowledge bases
  • TBox: categories
  • ABox: partial view of the world
  • What follows from a DL ontology

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Making Statements DL Knowledge Bases Entailment in DLs

Epilogue

Summary

  • Intensional and extensional knowledge
  • Specifying DL knowledge bases
  • TBox: categories
  • ABox: partial view of the world
  • What follows from a DL ontology

What next?

  • Reasoning with DL ontologies

Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 28