An Introduction to Description Logics -...

Introduction to Ontologies Formal Ontologies Introduction to DLs

An Introduction to Description LogicsPart 1: Introduction to Ontologies and DLs

Ivan Varzinczak

CRIL, Univ. Artois & CNRSLens, France

http://www.ijv.ovh

ESSLLI 2018

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

http://www.ijv.ovh


Overview of the courseMain parts1. Introduction to ontologies and description logics

2. The description logic ALC

3. Introduction to modelling and reasoning with ALC

4. Reasoning with ontologies

5. More and less expressive DLs

6. Formal ontologies in OWL and Protégé

There will be• Examples

• Exercises

• A lot of interaction (I hope)



Overview of the courseMain parts1. Introduction to ontologies and description logics

2. The description logic ALC

3. Introduction to modelling and reasoning with ALC

4. Reasoning with ontologies

5. More and less expressive DLs

6. Formal ontologies in OWL and Protégé

There will be• Examples

• Exercises

• A lot of interaction (I hope)Ivan Varzinczak (CRIL) An Introduction to Description Logics (Part 1) ESSLLI 2018 2


Overview of the courseBibliography• F. Baader, D. Calvanese, D. McGuinness, D. Nardi, and P. Patel-Schneider

(eds.): The Description Logic Handbook: Theory, Implementation andApplications. Cambridge University Press, 2nd edition, 2007.

• F. Baader, I. Horrocks, C. Lutz, and U. Sattler. An Introduction to DescriptionLogic. Cambridge University Press, 2017.

• M. Krötzsch, F. Simančík, and I. Horrocks. Description Logic Primer.http://arxiv.org/pdf/1201.4089v3.pdf

• The Protégé Ontology Editor. http://protege.stanford.edu

• The Description Logic workshop series. http://dl.kr.org

Course website• https://tinyurl.com/ESSLLI2018DL (for slides and more exercises)


http://arxiv.org/pdf/1201.4089v3.pdfhttp://protege.stanford.eduhttp://dl.kr.orghttps://tinyurl.com/ESSLLI2018DL


Knowledge we grasp and reason about

• Basic coarse facts

• Quantification

• Actions and causation

• Time

• Knowledge and belief

• Rules and obligations

• Interacting combinations thereof

Propositional Logic

First-Order Logic

Dynamic Logic

Temporal Logic

Epistemic Logic

Deontic Logic

...



Knowledge we grasp and reason about

• Basic coarse facts

• Quantification

• Actions and causation

• Time

• Knowledge and belief

• Rules and obligations

• Interacting combinations thereof

Propositional Logic

First-Order Logic

Dynamic Logic

Temporal Logic

Epistemic Logic

Deontic Logic

...Ivan Varzinczak (CRIL) An Introduction to Description Logics (Part 1) ESSLLI 2018 4


Reasoning about categories

Tidying up: rudiments of categorisation!



Outline

Introduction to Ontologies

Formal Ontologies

Introduction to DLs



Outline


Formal Ontologies

Introduction to DLs



Ontology v. ontologiesIn philosophy• Oντoλoγια

• The nature of being and reality

• Concerned with the existence of objects

• Related to metaphysics

In science• Objects and phenomena of study

• Relationships between them

• Rigorous description of some aspect of the world

• Explicit specification of a shared conceptualisation



OntologiesExplicit specification of a shared conceptualisation

Example (The student ontology)• Employed students are students and employees

• Students are not taxpayers (do not pay taxes)

• Employed students are taxpayers (pay taxes)

• Employed students who are parents are not taxpayers (do not pay taxes)

• To work for is to be employed by

• John is an employed student, John works for IBM









• John is an employed student, John works for IBM

classes relations individuals









• John is an employed student, John and IBM are in works for

classes relations individualsspecialisation and instantiation



Outline


Formal Ontologies

Introduction to DLs



Formal ontologiesIn computer and information sciences• Ontology: a model of the world

• Classes within a domain

• Relationships between classes

• Instantiations of classes

A formal, explicit specification of a shared conceptualisation• Conceptualisation: Abstract model of some domain

• Formal: Machine processable and unambiguous

• Explicit: Tangible

• Shared: Common ground for interoperability



Formal ontologies are everywhereIn medicine• Classification of medical terms: diseases, body parts, drugs

• Effective recording of clinical data: improve patient care

• Ex.: SNOMED-CT (320,000 terms, 1.4 million relations on terms)

In life sciences• Taxonomies: classification of bio-diversity

• Improve environmental and agricultural policies

• Ex.: Gene ontology (25,000 terms)



Formal ontologies are everywhereIn semantic technologies• Shared web resources

• Interoperability of smart applications, enhanced NLP

• Ex.: the (future) Semantic Web

Also in decision-making for organisations• Coherent and unified conceptual view of data sources

• Help in finding redundancy and incoherence in conceptual models

• Ex.: Enhancing modelling languages with ontology-based reasoning



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



Why Description Logics?Expressivity• Concepts X

• Relations X

• Instances X

DLs have all one needs to formalise ontologies!

Computational properties• Amenability to implementation X

• Decidability X

• Good trade-off between expressivity and complexity X

Most DL-based systems satisfy all of these!



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

· · ·



Outline


Formal Ontologies

Introduction to DLs



First of all, what are DLs?Decidability• Some logics can be made decidable by sacrificing expressive power

• DLs are less expressive than full first-order logic

• DLs are decidable, but what complexity is “OK”?

Technically• DLs are a family of fragments of first-order logic

• Only two variable names

• For the cognoscenti: correspond to guarded fragments of FOL

• But much, much simpler than FOL. . .



Elements of the language (domain dependent)Atomic 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



Elements of the language (domain independent)Boolean 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)



Elements of the language (domain independent)Boolean 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)



Elements of the language (domain independent)Boolean 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)



Building concepts

Definition (Complex concepts)• > and ⊥ are concepts• Every concept name A ∈ C is a concept• If C and D are concepts and r ∈ R, then

¬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)

are all concepts• Nothing else is a concept (at least for now)



ExerciseWhich ones are concepts and which aren’t?• > u⊥ t>

• C t ∀r. u ¬D

• C t ¬¬∃D

• ∃r.>

• ∃r.∀s.C uD

• ∀r.C u ¬D

• ∀r.(C u ¬D)

• ∀∃r.C



ExerciseWhich ones are concepts and which aren’t?• > u⊥ t> X

• C t ∀r. u ¬D

• C t ¬¬∃D

• ∃r.>

• ∃r.∀s.C uD

• ∀r.C u ¬D

• ∀r.(C u ¬D)

• ∀∃r.C




• C t ∀r. u ¬D ×• C t ¬¬∃D

• ∃r.>

• ∃r.∀s.C uD

• ∀r.C u ¬D

• ∀r.(C u ¬D)

• ∀∃r.C




• C t ∀r. u ¬D ×• C t ¬¬∃D ×• ∃r.>

• ∃r.∀s.C uD

• ∀r.C u ¬D

• ∀r.(C u ¬D)

• ∀∃r.C




• C t ∀r. u ¬D ×• C t ¬¬∃D ×• ∃r.> X

• ∃r.∀s.C uD

• ∀r.C u ¬D

• ∀r.(C u ¬D)

• ∀∃r.C




• C t ∀r. u ¬D ×• C t ¬¬∃D ×• ∃r.> X

• ∃r.∀s.C uD X

• ∀r.C u ¬D

• ∀r.(C u ¬D)

• ∀∃r.C




• C t ∀r. u ¬D ×• C t ¬¬∃D ×• ∃r.> X


• ∀r.C u ¬D X

• ∀r.(C u ¬D)

• ∀∃r.C




• C t ∀r. u ¬D ×• C t ¬¬∃D ×• ∃r.> X


• ∀r.C u ¬D X

• ∀r.(C u ¬D) X

• ∀∃r.C




• C t ∀r. u ¬D ×• C t ¬¬∃D ×• ∃r.> X


• ∀r.C u ¬D X

• ∀r.(C u ¬D) X

• ∀∃r.C×



Building conceptsFull negation• Negation of arbitrary concepts

• Intuition: the complement of a concept

• E.g.: ¬¬Student ¬(Student u Parent)

Atomic negation• Some DLs only allow negation of concept names

• Good complexity results

• E.g.: ¬Student ¬Parent



Building conceptsConcept conjunction• Intuition: the intersection of two concepts

• E.g.: Student u Parent

Concept disjunction• Intuition: the union of two concepts

• E.g.: Employee t Student

So far we have seen the Boolean fragment of our concept language

• At least as expressive as classical propositional logic



Building conceptsExistential restriction• Intuition: there is some link with a concept

• E.g.: ∃pays.Tax

Value restriction• Intuition: all links with a concept

• E.g.: ∀empBy.Company

So far we have got ALC (Attributive Language with Complement)

• Prototypical concept description language (there are others)



LanguageDifferent flavours• ALC: C ::= > | ⊥ | C | ¬C | C u C | C t C | ∀r.C | ∃r.C

• ALCQ: C ::= . . . | ≥ nr.C | ≤ nr.C

• EL, DL-Lite, SHIQ, SHOQ, SROIQ (basis of OWL 2), . . .




• ALCQ: C ::= . . . | ≥ nr.C | ≤ nr.C


Example

¬(Student u Parent)

∃empBy.Company

Employee t Student u ∃worksFor.Parent

Student u ¬∃pays.Tax

EmpStud u ∃pays.Tax

∀worksFor.Company




• ALCQ: C ::= . . . | ≥ nr.C | ≤ nr.C


Example

¬(Student u Parent)

∃empBy.Company

Employee t Student u ∃worksFor.Parent

Student u ¬∃pays.Tax

EmpStud u ∃pays.Tax

∀worksFor.Company

With LALC we denote the concept language of ALC



Semantics

Definition (Interpretation)Tuple I =def 〈∆I , ·I〉, where

• ∆I is a domain (set of objects)

• ·I is an interpretation function such that

AI ⊆ ∆I rI ⊆ ∆I ×∆I aI ∈ ∆I

ExampleLet C = {A1, A2, A3}, R = {r1, r2}, I = {a1, a2, a3}. Let I = 〈∆I , ·I〉 where:

• ∆I = {xi | 1 ≤ i ≤ 9}, aI1 = x5, aI2 = x1, aI3 = x2• AI1 = {x1, x4, x6, x7}, AI2 = {x3, x5, x9}, AI3 = {x6, x7, x8}

• rI1 = {(x1, x6), (x4, x8), (x2, x5)}, rI2 = {(x4, x4), (x6, x4), (x5, x8), (x9, x3)}



Semantics

I : ∆I

AI1 AI2

AI3

x1(a2) x2(a3) x3

x4 x5(a1)

x6 x7 x8 x9

r1

r2 r1 r2

r1

r2

r2



SemanticsExtending DL interpretations

>I =def ∆I ⊥I =def ∅ (¬C)I =def ∆I \ CI

(C uD)I =def CI ∩DI (C tD)I =def CI ∪DI

(∃r.C)I =def {x ∈ ∆I | rI(x) ∩ CI 6= ∅}

(∀r.C)I =def {x ∈ ∆I | rI(x) ⊆ CI}



SemanticsExtending DL interpretations

>I =def ∆I ⊥I =def ∅ (¬C)I =def ∆I \ CI

(C uD)I =def CI ∩DI (C tD)I =def CI ∪DI

(∃r.C)I =def {x ∈ ∆I | rI(x) ∩ CI 6= ∅}

(∀r.C)I =def {x ∈ ∆I | rI(x) ⊆ CI}

Definition (Concept Satisfiability)A concept C is satisfiable if there is I = 〈∆I , ·I〉 s.t. CI 6= ∅



Semantics

Individual names• At most one element of ∆I

∆I

aI•

Unique Name Assumption• At most one name per object

∆I

aI bI×•



Semantics

The ‘top’ concept• Everything is in >I

• Also called Thing

∆I

>I

The ‘bottom’ concept• ⊥I is in everything

• Also called Nothing

∆I

⊥I = ∅



Semantics

Arbitrary concept• A class in the domain

• CI ⊆ ∆I

∆I

CI

Concept negation• The complement of a concept

• (¬C)I = ∆I \ CI

∆I

CI



Semantics

Concept conjunction• The intersection of two classes

• (C uD)I = CI ∩DI

∆I

CI DI

Concept disjunction• The union of two classes

• (C tD)I = CI ∪DI

∆I

CI DI



Semantics

Existential restriction• At least one value of a class

• (∃r.C)I = {x | rI(x) ∩ CI 6= ∅}

∆I

(∃r.C)I CI

rI

Value restriction• All values of a class

• (∀r.C)I = {x | rI(x) ⊆ CI}

∆I

(∀r.C)I CI

rI



SemanticsAn interpretation is a complete description of the world

I : ∆I

x0 x1 x2 x3

x4 x5 x6

x7 x8 x9 x10




I : ∆I

x0 x1 x2(mary) x3

x4 x5 x6

x7 x8 x9 x10




I : ∆I

x0 x1 x2(mary) x3

x4 x5(john) x6

x7 x8 x9 x10




I : ∆I

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10




I : ∆I

StudentI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10




I : ∆I

TaxI

StudentI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10




I : ∆I

TaxI

StudentI

CompanyI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10




I : ∆I

TaxI

ParentI

StudentI

CompanyI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10




I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10




I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10




I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

worksFor

worksFor




I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

EmpS

tudI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

worksFor

worksForempBy




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 : ∆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

((EmpStud t Parent) u ∃pays.>)I = {x1, x5}



ExerciseLet C = {Company,Employee,EmpStud,Parent,Student,Tax}, R = {empBy, pays,worksFor} I = {ibm, john,mary}

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

• (¬Employee)I=• (∃pays.>)I=• (¬Parent u Employee)I=

• (¬EmpStud u ∀empBy.Company)I=• (∃worksFor.∃empBy.Parent)I=• (Student u ∀pays.⊥)I=




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

• (¬Employee)I=?• (∃pays.>)I=?• (¬Parent u Employee)I=?

• (¬EmpStud u ∀empBy.Company)I=?• (∃worksFor.∃empBy.Parent)I=?• (Student u ∀pays.⊥)I=?




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

• (¬Employee)I={x0, x3, x4, x6, x7, x8, x10}• (∃pays.>)I=?• (¬Parent u Employee)I=?





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

• (¬Employee)I={x0, x3, x4, x6, x7, x8, x10}• (∃pays.>)I={x1, x5}• (¬Parent u Employee)I=?





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

• (¬Employee)I={x0, x3, x4, x6, x7, x8, x10}• (∃pays.>)I={x1, x5}• (¬Parent u Employee)I={x5, x9}





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


• (¬EmpStud u ∀empBy.Company)I={x9}• (∃worksFor.∃empBy.Parent)I=?• (Student u ∀pays.⊥)I=?




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


• (¬EmpStud u ∀empBy.Company)I={x9}• (∃worksFor.∃empBy.Parent)I= ∅• (Student u ∀pays.⊥)I=?




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


• (¬EmpStud u ∀empBy.Company)I={x9}• (∃worksFor.∃empBy.Parent)I= ∅• (Student u ∀pays.⊥)I={x7, x8}




Find an interpretation I = 〈∆I , ·I〉 such that:• (Student u Employee)I = ∅, ParentI ⊆ (Student t Employee)I , (¬EmpStud)I = ∆I• StudentI ⊆ (∀pays.⊥)I , (∃worksFor.>)I ⊆ (¬(Student t Tax t Company))I , EmployeeI ⊆ (∃empBy.>)I

I : ∆I

TaxI

ParentI

StudentI EmployeeI

CompanyI

x0 x1 x2(mary) x3

x4 x5(john) x6(ibm)

x7 x8 x9 x10

worksForempBy

worksForempBy



Some Properties

LemmaFor every interpretation I = 〈∆I , ·I〉, and for every C,D ∈ LALC• (¬¬C)I = CI

• (¬(C uD))I = (¬C t ¬D)I

• (¬(C tD))I = (¬C u ¬D)I

• (¬∀r.C)I = (∃r.¬C)I

• (¬∃r.C)I = (∀r.¬C)I

ALC is the smallest propositionally closed DL

TheoremALC has the finite model property: if C is satisfiable, then there isI = 〈∆I , ·I〉 such that CI 6= ∅ and ∆I is finite



Some Properties

LemmaFor every interpretation I = 〈∆I , ·I〉, and for every C,D ∈ LALC• (¬¬C)I = CI

• (¬(C uD))I = (¬C t ¬D)I

• (¬(C tD))I = (¬C u ¬D)I

• (¬∀r.C)I = (∃r.¬C)I

• (¬∃r.C)I = (∀r.¬C)I

ALC is the smallest propositionally closed DL

TheoremALC has the finite model property: if C is satisfiable, then there isI = 〈∆I , ·I〉 such that CI 6= ∅ and ∆I is finiteIvan Varzinczak (CRIL) An Introduction to Description Logics (Part 1) ESSLLI 2018 37


Translating ALC-concepts into FOL-formulaeTranslation of non-logical symbols• Individual name a ∈ I 7→ constant ca• Concept name A ∈ C 7→ unary predicate pA• Role name r ∈ R 7→ binary predicate pr

Translation function τ : LALC × V ar −→ LF OL• τ(A, x) = pA(x)• τ(>, x) = >• τ(⊥, x) = ⊥• τ(¬C, x) = ¬τ(C, x)

• τ(C uD,x) = τ(C, x) ∧ τ(D,x)• τ(C tD,x) = τ(C, x) ∨ τ(D,x)• τ(∀r.C, x) = ∀y[pr(x, y)→ τ(C, y)] (y new)• τ(∃r.C, x) = ∃y[pr(x, y) ∧ τ(C, y)] (y new)

TheoremA concept C ∈ LALC is satisfiable iff τ(C, x) is satisfiable in FOL



Translating ALC-concepts to modal formulaeTranslation of non-logical symbols• Concept name A ∈ C 7→ proposition pA• Role name r ∈ R 7→ modality ir

Translation function η : LALC −→ LML• η(A) = pA• η(>) = >• η(⊥) = ⊥• η(¬C) = ¬η(C)

• η(C uD) = η(C) ∧ η(D)• η(C tD) = η(C) ∨ η(D)• η(∀r.C) = [ir]η(C)• η(∃r.C) = 〈ir〉η(C)

TheoremA concept C ∈ LALC is satisfiable iff η(C) is satisfiable



EpilogueSummary• What we mean by ontology

• Formal ontologies and their main ingredients

• Basic description logics

• The concept language and its semantics

• How DLs relate to other formalisms

What next?• A fundamental notion in DLs

• Formalising ontologies with DLs



EpilogueSummary• What we mean by ontology

• Formal ontologies and their main ingredients

• Basic description logics

• The concept language and its semantics

• How DLs relate to other formalisms

What next?• A fundamental notion in DLs

• Formalising ontologies with DLs


Introduction to OntologiesFormal OntologiesIntroduction to DLs

An Introduction to Description Logics -...

Documents

Proof theory for hybrid(ised) logics I - COnnecting REpositories · 2019. 10. 24. · logics [7], quantum logics [13], hidden and observational logics [11, 9, 43], algebrocaic logics

Problems of syntax-semantics interface ESSLLI 02 Trento

Knowledge Representation (Overview)mjs/teaching/KnowledgeRep491/... · Knowledge Representation and Reasoning Paraconsistent logics Nonmonotonic logics, Default Logics, Conditional

Blocs logics

Description Logics Structural Description Logics

Logics S.R.L

Script logics

Modal Logics, Description Logics and Arithmetic Reasoning · Modal Logics, Description Logics and Arithmetic Reasoning Hans J¨urgen Ohlbach Dept. of Computer Science, King’s College

Paraconsistent Logics

Proceedings of the ESSLLI Workshop on Distributional ...events.illc.uva.nl/ESSLLI2008/Materials/BaroniEvertLenci/ESSLLI... · ESSLLI Workshop on Distributional Lexical Semantics Bridging

Logics (Mid)

Computational Semantics coli.uni-sb.de/cl/projects/milca/esslli

Timed Logics

Seperator Logics

An Introduction to Description Logicsesslli2018.folli.info/wp-content/uploads/ESSLLI-Part2.pdfIvan Varzinczak (CRIL) An Introduction to Description Logics (Part 2) ESSLLI 2018 15 Making

ESSLLI 2013: Computational Morphology - …hana/teaching/2013-esslli/2013...Phonology/Phonetics in 5 slides What is morphology Words, Morphemes, etc. Morphological processes Typology

VEE Logics

Visual Analytics for Linguistics - Day 4 ESSLLI - structured data

ESSLLI 2018 course =1=XLogics for Epistemic and Strategic ... · Lecture 3: Logics for strategic reasoning with complete information Valentin Goranko Stockholm University ESSLLI 2018

Non Standard Logics & Modal Logics