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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
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
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
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
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
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
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
Introduction to Ontologies Formal Ontologies Introduction to DLs
Reasoning about categories
Tidying up: rudiments of categorisation!
Ivan Varzinczak (CRIL) An Introduction to Description Logics (Part 1) ESSLLI 2018 5
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
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 7
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
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
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
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
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
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
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 10
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 (Part 1) ESSLLI 2018 11
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 (Part 1) ESSLLI 2018 11
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 (Part 1) ESSLLI 2018 12
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 (Part 1) ESSLLI 2018 12
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
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
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
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
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
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
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
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
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
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 16
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 (Part 1) ESSLLI 2018 17
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 (Part 1) ESSLLI 2018 17
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
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
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
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
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
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
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 (Part 1) ESSLLI 2018 20
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 (Part 1) ESSLLI 2018 21
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 (Part 1) ESSLLI 2018 21
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
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
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
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
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
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 (Part 1) ESSLLI 2018 21
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 (Part 1) ESSLLI 2018 21
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 (Part 1) ESSLLI 2018 22
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 (Part 1) ESSLLI 2018 22
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
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
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
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
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
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
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), . . .
Ivan Varzinczak (CRIL) An Introduction to Description Logics (Part 1) ESSLLI 2018 25
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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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), . . .
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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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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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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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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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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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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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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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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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 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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Semantics
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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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 = ∅
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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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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
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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
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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
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SemanticsAn interpretation is a complete description of the world
I : ∆I
x0 x1 x2(mary) x3
x4 x5 x6
x7 x8 x9 x10
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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
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
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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
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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
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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
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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 (Part 1) ESSLLI 2018 34
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
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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
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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
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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 (Part 1) ESSLLI 2018 34
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 (Part 1) ESSLLI 2018 34
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 (Part 1) ESSLLI 2018 35
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=?• (∃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 (Part 1) ESSLLI 2018 35
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=?• (¬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 (Part 1) ESSLLI 2018 35
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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=?
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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=?
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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=?
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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 (Part 1) ESSLLI 2018 35
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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={x7, x8}
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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
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
Introduction to Ontologies Formal Ontologies Introduction to DLs
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 (Part 1) ESSLLI 2018 37
Introduction to Ontologies Formal Ontologies Introduction to DLs
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
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 (Part 1) ESSLLI 2018 38
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 (Part 1) ESSLLI 2018 38
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 (Part 1) ESSLLI 2018 38
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 (Part 1) ESSLLI 2018 39
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 (Part 1) ESSLLI 2018 39
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 (Part 1) ESSLLI 2018 39
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 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