Making Statements DL Knowledge Bases Entailment in DLs Formal rmal Foundations oundations of of Ontologies Ontologies and and Reasoning Reasoning Ivan Ivan Varzinczak rzinczak Université d’Artois & CNRS, France http://www.ijv.ovh Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 1
Making Statements DL Knowledge Bases Entailment in DLs Outline of Part 2 Making Statements DL Knowledge Bases Entailment in DLs Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 2
Making Statements DL Knowledge Bases Entailment in DLs Outline of Part 2 Making Statements DL Knowledge Bases Entailment in DLs Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 3
Making Statements DL Knowledge Bases Entailment in DLs Motivation Concept language of ALC ⊤ , ⊥ (constants) A (atomic concept) ¬ C (complement of C ) C ⊓ D (intersection of C and D ) C ⊔ D (union of C and D ) ∃ r.C (existential restriction) ∀ r.C (value restriction) Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 4
Making Statements DL Knowledge Bases Entailment in DLs Motivation Concept language of ALC ⊤ , ⊥ (constants) A (atomic concept) ¬ C (complement of C ) C ⊓ D (intersection of C and D ) C ⊔ D (union of C and D ) ∃ r.C (existential restriction) ∀ r.C (value restriction) Something is missing Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 4
Making Statements DL Knowledge Bases Entailment in DLs Motivation Concept language of ALC ⊤ , ⊥ (constants) A (atomic concept) ¬ C (complement of C ) C ⊓ D (intersection of C and D ) C ⊔ D (union of C and D ) ∃ r.C (existential restriction) ∀ r.C (value restriction) Something is missing • The central notion in logic: C ‘ → ’ D Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 4
Making Statements DL Knowledge Bases Entailment in DLs Motivation Concept language of ALC ⊤ , ⊥ (constants) A (atomic concept) ¬ C (complement of C ) C ⊓ D (intersection of C and D ) C ⊔ D (union of C and D ) ∃ r.C (existential restriction) ∀ r.C (value restriction) Something is missing • The central notion in logic: C ‘ → ’ D • What would C ‘ → ’ D mean here? Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 4
Making Statements DL Knowledge Bases Entailment in DLs Motivation Concept language of ALC ⊤ , ⊥ (constants) A (atomic concept) ¬ C (complement of C ) C ⊓ D (intersection of C and D ) C ⊔ D (union of C and D ) ∃ r.C (existential restriction) ∀ r.C (value restriction) Something is missing • The central notion in logic: C ‘ → ’ D • What would C ‘ → ’ D mean here? (We already have ¬ C ⊔ D ) Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 4
Making Statements DL Knowledge Bases Entailment in DLs Motivation Concept language of ALC ⊤ , ⊥ (constants) A (atomic concept) ¬ C (complement of C ) C ⊓ D (intersection of C and D ) C ⊔ D (union of C and D ) ∃ r.C (existential restriction) ∀ r.C (value restriction) Something is missing • The central notion in logic: C ‘ → ’ D • What would C ‘ → ’ D mean here? (We already have ¬ C ⊔ D ) • DLs have a version of ‘ → ’ that is very special Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 4
Making Statements DL Knowledge Bases Entailment in DLs Statements In many logics meta-language (entailment, etc) object language (formulae) Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 5
Making Statements DL Knowledge Bases Entailment in DLs Statements In many logics In DLs meta-language meta-language (entailment, etc) statements object language (formulae) concept language • Two levels of language • Two notions of ‘entailment’ • Two notions of ‘satisfaction’ Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 5
Making Statements DL Knowledge Bases Entailment in DLs Making statements Subsumption • Concept inclusion • Employed students are students • Employed students are employees Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 6
Making Statements DL Knowledge Bases Entailment in DLs Making statements Subsumption • Concept inclusion • Employed students are students • Employed students are employees Instantiation or assertions • Concept and role membership • John is an employed student (John instantiates employed student) • John works for IBM (John and IBM instantiate works for) Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 6
Making Statements DL Knowledge Bases Entailment in DLs Making statements Subsumption • Concept inclusion • Employed students are students • Employed students are employees Instantiation or assertions • Concept and role membership • John is an employed student (John instantiates employed student) • John works for IBM (John and IBM instantiate works for) Statements talk about concepts, roles and individuals Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 6
Making Statements DL Knowledge Bases Entailment in DLs Making statements Subsumption • Concept inclusion • Employed students are students • Employed students are employees Instantiation or assertions • Concept and role membership • John is an employed student (John instantiates employed student) • John works for IBM (John and IBM instantiate works for) Statements talk about concepts, roles and individuals They are not concepts! They are in the ‘in-between’ language Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 6
Making Statements DL Knowledge Bases Entailment in DLs Subsumption statements C ⊑ D Intuition • D subsumes C (or C is subsumed by D ) • C is more specific than D (or D is more general than C ) • Formalise one aspect of is-a relations Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 7
Making Statements DL Knowledge Bases Entailment in DLs Subsumption statements C ⊑ D Intuition • D subsumes C (or C is subsumed by D ) • C is more specific than D (or D is more general than C ) • Formalise one aspect of is-a relations Example • EmpStud ⊑ Student ⊓ Employee, Employee ⊑ ∃ worksFor . ⊤ • EmpStud ⊑ ∃ pays . Tax, Student ⊓ ¬ Employee ⊑ ¬∃ pays . Tax Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 7
Making Statements DL Knowledge Bases Entailment in DLs Subsumption statements C ⊑ D Intuition • D subsumes C (or C is subsumed by D ) • C is more specific than D (or D is more general than C ) • Formalise one aspect of is-a relations Example • EmpStud ⊑ Student ⊓ Employee, Employee ⊑ ∃ worksFor . ⊤ • EmpStud ⊑ ∃ pays . Tax, Student ⊓ ¬ Employee ⊑ ¬∃ pays . Tax Central notion in DL terminologies (taxonomies) Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 7
Making Statements DL Knowledge Bases Entailment in DLs Subsumption statements C ⊑ D Semantics C I ⊆ D I • I � C ⊑ D ( I satisfies C ⊑ D ) if • First level of ‘entailment’: all C -objects are D -objects Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 8
Making Statements DL Knowledge Bases Entailment in DLs Subsumption statements C ⊑ D Semantics C I ⊆ D I • I � C ⊑ D ( I satisfies C ⊑ D ) if • First level of ‘entailment’: all C -objects are D -objects ∆ I D I C I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 8
Making Statements DL Knowledge Bases Entailment in DLs Subsumption statements C ≡ D Concept equivalence • Just an abbreviation for C ⊑ D and D ⊑ C • I � C ≡ D if I � C ⊑ D and I � D ⊑ C C I = D I • I � C ≡ D if Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 9
Making Statements DL Knowledge Bases Entailment in DLs Subsumption statements C ≡ D Concept equivalence • Just an abbreviation for C ⊑ D and D ⊑ C • I � C ≡ D if I � C ⊑ D and I � D ⊑ C C I = D I • I � C ≡ D if ∆ I C I D I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 9
Making Statements DL Knowledge Bases Entailment in DLs Exercise I : ∆ I Parent I pays x 0 x 1 x 3 x 2 ( mary ) EmpStud I Tax I Company I pays worksFor x 4 x 5 ( john ) x 6 ( ibm ) worksFor x 7 x 8 x 9 x 10 empBy Employee I Student I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 2) 26 April 2019 10
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