An Introduction to Description Logicsesslli2018.folli.info/wp-content/uploads/ESSLLI-Part2.pdfIvan...

Making Statements DL Knowledge Bases Entailment in DLs

An Introduction to Description LogicsPart 2: Formal Ontologies in ALC

Ivan Varzinczak

CRIL, Univ. Artois & CNRSLens, France

http://www.ijv.ovh

ESSLLI 2018

Ivan Varzinczak (CRIL) An Introduction to Description Logics (Part 2) ESSLLI 2018 1

http://www.ijv.ovh


Recap: Main ingredients in formal ontologiesA common vocabulary and a shared understanding

Classes or concepts• Describe concrete or abstract entities within the domain of interest

• E.g.: Employed student, Parent

Relations or roles• Describe relationships between concepts or attributes of a concept

• E.g.: work for someone, being employed by someone

Instances of classes and relations• Name objects of the domain and denote representatives of a concept

• E.g.: John, John is an employed student, John works for IBM



Recap: Why Description Logics?Expressivity• Concepts X

• Relations X

• Instances X

DLs have all one needs to formalise ontologies!

Available tools

Computational properties• Amenability to implementation X

• Decidability X

• Good trade-off between expressivity and complexity X

Most DL-based systems satisfy all of these!

FaCT++

Pellet

HermiT

CEL

· · ·



Recap: Concept languageAtomic concept names• C =def {A1, . . . , An} (Special concepts: >, ⊥)• Intuition: basic classes of a domain of interest• Student, Employee, Parent

Atomic role names• R =def {r1, . . . , rm}• Intuition: basic relations between concepts• worksFor, empBy

Individual names• I =def {a1, . . . , al}• Intuition: names of objects in the domain• john, mary, ibm



Recap: Concept languageBoolean constructors• Concept negation: ¬ (class complement)• Concept conjunction: u (class intersection)• Concept disjunction: t (class union)Role restrictions

• Existential restriction: ∃ (at least one relationship)• Value restriction: ∀ (all relationships)Further constructors: cardinality constraints, inverse roles, . . . (if needed)



Recap: Concept language

ALC complex concepts: LALC

>, ⊥ (constants)

A (atomic concept)

¬C (complement)

C uD (conjunction)

C tD (disjunction)

∃r.C (existential restriction)

∀r.C (value restriction)

PlusIndividual names a

Semantics: I = 〈∆I , ·I〉

>I = ∆I , ⊥I = ∅

AI ⊆ ∆I

∆I \ CI

CI ∩DI

CI ∪DI

{x ∈ ∆I | rI(x) ∩ CI 6= ∅}

{x ∈ ∆I | rI(x) ⊆ CI}

AndaI ∈ ∆I



Outline

Making Statements

DL Knowledge Bases

Entailment in DLs



MotivationConcept language of ALC

>, ⊥ (constants)A (atomic concept)¬C (complement of C)C uD (intersection of C and D)C tD (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?• DLs have a version of ‘→’ that is very special





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

• What would C ‘→’ D mean here?• DLs have a version of ‘→’ that is very special





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

• DLs have a version of ‘→’ that is very special





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

• DLs have a version of ‘→’ that is very special





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



Statements

In many logics

meta-language(entailment, etc)

object language(formulae)

In DLs

meta-language

statements

concept language

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



Making statementsSubsumption• 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 individualsThey are not concepts! They are in the ‘in-between’ language









Statements talk about concepts, roles and individuals

They are not concepts! They are in the ‘in-between’ language









Statements talk about concepts, roles and individualsThey are not concepts! They are in the ‘in-between’ language



Subsumption statements

C v DIntuition• 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 v Student u Employee, Employee v ∃worksFor.>

• EmpStud v ∃pays.Tax, Student u ¬Employee v ¬∃pays.Tax

Central notion in DL terminologies (taxonomies)




C v DSemantics• I C v D (I satisfies C v D) if CI ⊆ DI

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

∆I

CI

DI




C ≡ DConcept equivalence• Just an abbreviation for C v D and D v C• I C ≡ D if I C v D and I D v C• I C ≡ D if CI = DI

∆I

CI DI



Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy

• I EmpStud v Student u Employee• I ∃worksFor.> v Employee• I Employee v ∃worksFor.>• I EmpStud v ∀pays.Tax

• I Parent u ¬Employee v Tax t ¬Student• I EmpStud u Parent v ∃pays.Tax• I ∃empBy.> v ∃worksFor.Company• I ∀pays.Tax v ∃empBy.Company



Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy

• I EmpStud v Student u Employee ?• I ∃worksFor.> v Employee ?• I Employee v ∃worksFor.> ?• I EmpStud v ∀pays.Tax ?

• I Parent u ¬Employee v Tax t ¬Student ?• I EmpStud u Parent v ∃pays.Tax ?• I ∃empBy.> v ∃worksFor.Company ?• I ∀pays.Tax v ∃empBy.Company ?



Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy

• I EmpStud v Student u Employee Yep!• I ∃worksFor.> v Employee ?• I Employee v ∃worksFor.> ?• I EmpStud v ∀pays.Tax ?




Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy

• I EmpStud v Student u Employee Yep!• I ∃worksFor.> v Employee Yep!• I Employee v ∃worksFor.> ?• I EmpStud v ∀pays.Tax ?




Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy

• I EmpStud v Student u Employee Yep!• I ∃worksFor.> v Employee Yep!• I Employee v ∃worksFor.> No!• I EmpStud v ∀pays.Tax ?




Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy

• I EmpStud v Student u Employee Yep!• I ∃worksFor.> v Employee Yep!• I Employee v ∃worksFor.> No!• I EmpStud v ∀pays.Tax No!




Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy


• I Parent u ¬Employee v Tax t ¬Student Yep!• I EmpStud u Parent v ∃pays.Tax ?• I ∃empBy.> v ∃worksFor.Company ?• I ∀pays.Tax v ∃empBy.Company ?



Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy


• I Parent u ¬Employee v Tax t ¬Student Yep!• I EmpStud u Parent v ∃pays.Tax No!• I ∃empBy.> v ∃worksFor.Company ?• I ∀pays.Tax v ∃empBy.Company ?



Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy


• I Parent u ¬Employee v Tax t ¬Student Yep!• I EmpStud u Parent v ∃pays.Tax No!• I ∃empBy.> v ∃worksFor.Company Yep!• I ∀pays.Tax v ∃empBy.Company ?



Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy


• I Parent u ¬Employee v Tax t ¬Student Yep!• I EmpStud u Parent v ∃pays.Tax No!• I ∃empBy.> v ∃worksFor.Company Yep!• I ∀pays.Tax v ∃empBy.Company No!



Assertions

a : C (a, b) : rIntuition• 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 u ¬∃worksFor.> u ¬∃pays.Tax

• (john, ibm) : worksFor (in some DLs: (john, ibm) : ¬empBy)

Central notion in DL ‘databases’



Assertions





• (john, ibm) : worksFor

(in some DLs: (john, ibm) : ¬empBy)




Assertions





• (john, ibm) : worksFor (in some DLs: (john, ibm) : ¬empBy)




Assertions

a : C (a, b) : rSemantics• 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



Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy

• I john : Employee u ∃pays.Tax• I mary : ∀pays.Tax• I (mary, ibm) : empBy• I (ibm, john) : worksFor

• I mary : Parent u ¬∃worksFor.> u ¬∃pays.Tax• I mary : Employee u ∃empBy.>• I john : ∀empBy.Company• I john : ∃worksFor.∀pays.⊥



Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy

• I john : Employee u ∃pays.Tax ?• I mary : ∀pays.Tax ?• I (mary, ibm) : empBy ?• I (ibm, john) : worksFor ?

• I mary : Parent u ¬∃worksFor.> u ¬∃pays.Tax ?• I mary : Employee u ∃empBy.> ?• I john : ∀empBy.Company ?• I john : ∃worksFor.∀pays.⊥ ?



Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy

• I john : Employee u ∃pays.Tax Yep!• I mary : ∀pays.Tax ?• I (mary, ibm) : empBy ?• I (ibm, john) : worksFor ?




Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy

• I john : Employee u ∃pays.Tax Yep!• I mary : ∀pays.Tax Yep!• I (mary, ibm) : empBy ?• I (ibm, john) : worksFor ?




Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy

• I john : Employee u ∃pays.Tax Yep!• I mary : ∀pays.Tax Yep!• I (mary, ibm) : empBy No!• I (ibm, john) : worksFor ?




Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy

• I john : Employee u ∃pays.Tax Yep!• I mary : ∀pays.Tax Yep!• I (mary, ibm) : empBy No!• I (ibm, john) : worksFor No!




Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy


• I mary : Parent u ¬∃worksFor.> u ¬∃pays.Tax Yep!• I mary : Employee u ∃empBy.> ?• I john : ∀empBy.Company ?• I john : ∃worksFor.∀pays.⊥ ?



Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy


• I mary : Parent u ¬∃worksFor.> u ¬∃pays.Tax Yep!• I mary : Employee u ∃empBy.> No!• I john : ∀empBy.Company ?• I john : ∃worksFor.∀pays.⊥ ?



Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy


• I mary : Parent u ¬∃worksFor.> u ¬∃pays.Tax Yep!• I mary : Employee u ∃empBy.> No!• I john : ∀empBy.Company Yep!• I john : ∃worksFor.∀pays.⊥ ?



Exercise

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy


• I mary : Parent u ¬∃worksFor.> u ¬∃pays.Tax Yep!• I mary : Employee u ∃empBy.> No!• I john : ∀empBy.Company Yep!• I john : ∃worksFor.∀pays.⊥ Yep!



Outline

Making Statements

DL Knowledge Bases

Entailment in DLs



TBoxes and ABoxesIntensional 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〉



Knowledge bases

Example (The student ontology in DL)

T =

EmpStud ≡ Student u Employee,Student u ¬Employee v ¬∃pays.Tax,

EmpStud u ¬Parent v ∃pays.Tax,EmpStud u Parent v ¬∃pays.Tax,∃worksFor.Company v Employee

A =

ibm : Company,mary : Parent,

john : EmpStud,(john, ibm) : worksFor

classesrelations

individuals



Knowledge basesSemantics• I T if I C v D for every C v 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



Exercise• Let KB = 〈T ,A〉, where:

T =



A =



• Is the interpretation I below a model of KB ?

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyIEm

pStu

dI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy




T =



A =



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

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyIEm

pStu

dI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy




T =



A =



• Find a counter-model for this knowledge base

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy




T =



A =



• Find a counter-model for this knowledge base

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyIEm

pStu

dI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy



Translating KBs. . .. . . in FOLExtend τ to a mapping from KB = 〈T ,A〉 to conjunctions of FOL formulae

τ(C v D) = ∀x.[τ(C, x)→ τ(D,x)]τ(C ≡ D) = ∀x.[τ(C, x)↔ τ(D,x)]

τ(a : C) = τ(C, ca)τ((a, b) : r) = pr(ca, cb)

Then• τ(KB) = (

∧α∈T τ(α)) ∧ (

∧α∈A τ(α))






Then• τ(KB) = (

∧α∈T τ(α)) ∧ (

∧α∈A τ(α)) ∧ (

∧distinct a,b occurring in A ca 6= cb)






Then• τ(KB) = (

∧α∈T τ(α)) ∧ (

∧α∈A τ(α)) ∧ (

∧distinct a,b occurring in A ca 6= cb)

TheoremKB is satisfiable iff τ(KB) is FOL-satisfiable



Translating KBs. . .. . . in modal logicWe need a universal modality and nominals

ThenWe extend η to a mapping from statements to formulae of hybrid logic

η(C v D) = 2(η(C)→ η(D))η(C ≡ D) = 2(η(C)↔ η(D))

η(a : C) = @naη(C)η((a, b) : r) = @na〈ir〉nb

TheoremKB is satisfiable iff η(KB) is modally satisfiable



Outline

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?



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)

object language(formulae)

In DLs

meta-language

statements

concept language



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




Example

T =



A =



• KB |= Student u ∃worksFor.Company u ¬Parent v EmpStud u ∃pays.Tax

• KB |= john : Student u ∃worksFor.Company

• KB 6|= mary : ¬∃pays.Tax

• KB 6|= Employee v ∃empBy.Company




Example• KB 6|= mary : ¬∃pays.Tax• KB 6|= Employee v ∃empBy.Company

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyIEm

pStu

dI

x0 x1(mary) x2 x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

pays

pays worksFor

worksForempBy



Open- v. closed-world assumptionClosed-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 . . . ”



DLs are (very) classical

Definition (Consequence operator)Given KB = 〈T ,A〉, the set of all consequences of KB is

Cn(KB) =def {α | KB |= α}

TheoremCn(·) satisfies the following properties:

• KB ⊆ Cn(KB) (Inclusion)

• Cn(KB) = Cn(Cn(KB)) (Idempotency)

• If KB1 ⊆ KB2, then Cn(KB1) ⊆ Cn(KB2) (Monotonicity)

Hence, Cn(·) is a Tarskian consequence operator



EpilogueSummary• 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



EpilogueSummary• 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


Making StatementsDL Knowledge BasesEntailment in DLs

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