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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
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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)
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 3
http://arxiv.org/pdf/1201.4089v3.pdfhttp://protege.stanford.eduhttp://dl.kr.orghttps://tinyurl.com/ESSLLI2018DL
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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)
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 3
http://arxiv.org/pdf/1201.4089v3.pdfhttp://protege.stanford.eduhttp://dl.kr.orghttps://tinyurl.com/ESSLLI2018DL
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
Reasoning about categories
Tidying up: rudiments of categorisation!
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
Outline
Introduction to Ontologies
Formal Ontologies
Introduction to DLs
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 6
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
Outline
Introduction to Ontologies
Formal Ontologies
Introduction to DLs
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies 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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies 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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 8
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 9
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
classes relations individuals
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 9
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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 and IBM are in works for
classes relations individualsspecialisation and
instantiation
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 9
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
Outline
Introduction to Ontologies
Formal Ontologies
Introduction to DLs
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies 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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies 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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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)
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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)
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 13
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 13
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 14
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 14
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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!
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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!
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 15
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
· · ·
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 15
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
Outline
Introduction to Ontologies
Formal Ontologies
Introduction to DLs
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies 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. . .
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies 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. . .
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 18
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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)
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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)
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 19
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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)
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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)
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
ExerciseWhich ones are concepts and which aren’t?• > u⊥ t>
X
• C t ∀r. u ¬D ו C t ¬¬∃D
• ∃r.>
• ∃r.∀s.C uD
• ∀r.C u ¬D
• ∀r.(C u ¬D)
• ∀∃r.C
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 21
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
ExerciseWhich ones are concepts and which aren’t?• > u⊥ t>
X
• C t ∀r. u ¬D ו C t ¬¬∃D ו ∃r.>
• ∃r.∀s.C uD
• ∀r.C u ¬D
• ∀r.(C u ¬D)
• ∀∃r.C
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 21
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
ExerciseWhich ones are concepts and which aren’t?• > u⊥ t>
X
• C t ∀r. u ¬D ו C t ¬¬∃D ו ∃r.> X
• ∃r.∀s.C uD
• ∀r.C u ¬D
• ∀r.(C u ¬D)
• ∀∃r.C
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 21
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
ExerciseWhich ones are concepts and which aren’t?• > u⊥ t>
X
• C t ∀r. u ¬D ו C t ¬¬∃D ו ∃r.> X
• ∃r.∀s.C uD X
• ∀r.C u ¬D
• ∀r.(C u ¬D)
• ∀∃r.C
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 21
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
ExerciseWhich ones are concepts and which aren’t?• > u⊥ t>
X
• C t ∀r. u ¬D ו C t ¬¬∃D ו ∃r.> X
• ∃r.∀s.C uD X
• ∀r.C u ¬D X
• ∀r.(C u ¬D)
• ∀∃r.C
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 21
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
ExerciseWhich ones are concepts and which aren’t?• > u⊥ t>
X
• C t ∀r. u ¬D ו C t ¬¬∃D ו ∃r.> X
• ∃r.∀s.C uD X
• ∀r.C u ¬D X
• ∀r.(C u ¬D) X
• ∀∃r.C
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
ExerciseWhich ones are concepts and which aren’t?• > u⊥ t>
X
• C t ∀r. u ¬D ו C t ¬¬∃D ו ∃r.> X
• ∃r.∀s.C uD X
• ∀r.C u ¬D X
• ∀r.(C u ¬D) X
• ∀∃r.C×
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 23
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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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)
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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)
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 24
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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)
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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), . . .
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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), . . .
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Introduction to Ontologies Formal Ontologies Introduction to
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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), . . .
Example
¬(Student u Parent)
∃empBy.Company
Employee t Student u ∃worksFor.Parent
Student u ¬∃pays.Tax
EmpStud u ∃pays.Tax
∀worksFor.Company
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Introduction to Ontologies Formal Ontologies Introduction to
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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), . . .
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
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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)}
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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)}
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I : ∆I
AI1 AI2
AI3
x1(a2) x2(a3) x3
x4 x5(a1)
x6 x7 x8 x9
r1
r2 r1 r2
r1
r2
r2
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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}
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Introduction to Ontologies Formal Ontologies Introduction to
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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}
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Introduction to Ontologies Formal Ontologies Introduction to
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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}
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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}
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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}
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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}
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 28
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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}
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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}
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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= ∅
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Individual names• At most one element of ∆I
∆I
aI•
Unique Name Assumption• At most one name per object
∆I
aI bIו
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Individual names• At most one element of ∆I
∆I
aI•
Unique Name Assumption• At most one name per object
∆I
aI bIו
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The ‘top’ concept• Everything is in >I
• Also called Thing
∆I
>I
The ‘bottom’ concept• ⊥I is in everything
• Also called Nothing
∆I
⊥I = ∅
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The ‘top’ concept• Everything is in >I
• Also called Thing
∆I
>I
The ‘bottom’ concept• ⊥I is in everything
• Also called Nothing
∆I
⊥I = ∅
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Arbitrary concept• A class in the domain
• CI ⊆ ∆I
∆I
CI
Concept negation• The complement of a concept
• (¬C)I = ∆I \ CI
∆I
CI
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Arbitrary concept• A class in the domain
• CI ⊆ ∆I
∆I
CI
Concept negation• The complement of a concept
• (¬C)I = ∆I \ CI
∆I
CI
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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
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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
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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
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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
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
SemanticsAn interpretation is a complete description of the
world
I : ∆I
x0 x1 x2 x3
x4 x5 x6
x7 x8 x9 x10
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
SemanticsAn interpretation is a complete description of the
world
I : ∆I
x0 x1 x2(mary) x3
x4 x5 x6
x7 x8 x9 x10
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
SemanticsAn interpretation is a complete description of the
world
I : ∆I
x0 x1 x2(mary) x3
x4 x5(john) x6
x7 x8 x9 x10
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 34
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
SemanticsAn interpretation is a complete description of the
world
I : ∆I
x0 x1 x2(mary) x3
x4 x5(john) x6(ibm)
x7 x8 x9 x10
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
SemanticsAn interpretation is a complete description of the
world
I : ∆I
StudentI
x0 x1 x2(mary) x3
x4 x5(john) x6(ibm)
x7 x8 x9 x10
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
SemanticsAn interpretation is a complete description of the
world
I : ∆I
TaxI
StudentI
x0 x1 x2(mary) x3
x4 x5(john) x6(ibm)
x7 x8 x9 x10
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
SemanticsAn interpretation is a complete description of the
world
I : ∆I
TaxI
StudentI
CompanyI
x0 x1 x2(mary) x3
x4 x5(john) x6(ibm)
x7 x8 x9 x10
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
SemanticsAn interpretation is a complete description of the
world
I : ∆I
TaxI
ParentI
StudentI
CompanyI
x0 x1 x2(mary) x3
x4 x5(john) x6(ibm)
x7 x8 x9 x10
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
SemanticsAn interpretation is a complete description of the
world
I : ∆I
TaxI
ParentI
StudentI EmployeeI
CompanyI
x0 x1 x2(mary) x3
x4 x5(john) x6(ibm)
x7 x8 x9 x10
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
SemanticsAn interpretation is a complete description of the
world
I : ∆I
TaxI
ParentI
StudentI EmployeeI
CompanyI
EmpS
tudI
x0 x1 x2(mary) x3
x4 x5(john) x6(ibm)
x7 x8 x9 x10
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
SemanticsAn interpretation is a complete description of the
world
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
SemanticsAn interpretation is a complete description of the
world
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
SemanticsAn interpretation is a complete description of the
world
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
SemanticsAn interpretation is a complete description of the
world
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}
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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=
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
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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=?
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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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={x0, x3, x4, x6, x7, x8, x10}• (∃pays.>)I=?•
(¬Parent u Employee)I=?
• (¬EmpStud u ∀empBy.Company)I=?• (∃worksFor.∃empBy.Parent)I=?•
(Student u ∀pays.⊥)I=?
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
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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={x0, x3, x4, x6, x7, x8, x10}• (∃pays.>)I={x1,
x5}• (¬Parent u Employee)I=?
• (¬EmpStud u ∀empBy.Company)I=?• (∃worksFor.∃empBy.Parent)I=?•
(Student u ∀pays.⊥)I=?
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
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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={x0, x3, x4, x6, x7, x8, x10}• (∃pays.>)I={x1,
x5}• (¬Parent u Employee)I={x5, x9}
• (¬EmpStud u ∀empBy.Company)I=?• (∃worksFor.∃empBy.Parent)I=?•
(Student u ∀pays.⊥)I=?
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
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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={x0, x3, x4, x6, x7, x8, x10}• (∃pays.>)I={x1,
x5}• (¬Parent u Employee)I={x5, x9}
• (¬EmpStud u ∀empBy.Company)I={x9}•
(∃worksFor.∃empBy.Parent)I=?• (Student u ∀pays.⊥)I=?
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
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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={x0, x3, x4, x6, x7, x8, x10}• (∃pays.>)I={x1,
x5}• (¬Parent u Employee)I={x5, x9}
• (¬EmpStud u ∀empBy.Company)I={x9}• (∃worksFor.∃empBy.Parent)I=
∅• (Student u ∀pays.⊥)I=?
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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={x0, x3, x4, x6, x7, x8, x10}• (∃pays.>)I={x1,
x5}• (¬Parent u Employee)I={x5, x9}
• (¬EmpStud u ∀empBy.Company)I={x9}• (∃worksFor.∃empBy.Parent)I=
∅• (Student u ∀pays.⊥)I={x7, x8}
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
ExerciseLet C = {Company,Employee,EmpStud,Parent,Student,Tax}, R
= {empBy, pays,worksFor} I = {ibm, john,mary}
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 36
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
ExerciseLet C = {Company,Employee,EmpStud,Parent,Student,Tax}, R
= {empBy, pays,worksFor} I = {ibm, john,mary}
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
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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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
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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
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
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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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
DLs
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
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Introduction to Ontologies Formal Ontologies Introduction to
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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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 40
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Introduction to Ontologies Formal Ontologies Introduction to
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
Ivan Varzinczak (CRIL) An Introduction to Description Logics
(Part 1) ESSLLI 2018 40
Introduction to OntologiesFormal OntologiesIntroduction to
DLs