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Formal Ontologies Introduction to DLs Formal rmal Foundations oundations of of Ontologies Ontologies and and Reasoning Reasoning Ivan Ivan Varzinczak rzinczak Universit dArtois & CNRS, France http://www.ijv.ovh Ivan

  1. 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 Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 11

  2. Formal Ontologies Introduction to DLs Elements of the language (domain dependent) Atomic concept names • C = def { A 1 , . . . , A n } (Special concepts: ⊤ , ⊥ ) • Intuition: basic classes of a domain of interest • Student, Employee, Parent Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 12

  3. Formal Ontologies Introduction to DLs Elements of the language (domain dependent) Atomic concept names • C = def { A 1 , . . . , A n } (Special concepts: ⊤ , ⊥ ) • Intuition: basic classes of a domain of interest • Student, Employee, Parent Atomic role names • R = def { r 1 , . . . , r m } • Intuition: basic relations between concepts • worksFor, empBy Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 12

  4. Formal Ontologies Introduction to DLs Elements of the language (domain dependent) Atomic concept names • C = def { A 1 , . . . , A n } (Special concepts: ⊤ , ⊥ ) • Intuition: basic classes of a domain of interest • Student, Employee, Parent Atomic role names • R = def { r 1 , . . . , r m } • Intuition: basic relations between concepts • worksFor, empBy Individual names • I = def { a 1 , . . . , a l } • Intuition: names of objects in the domain • john, mary, ibm Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 12

  5. Formal Ontologies Introduction to DLs Elements of the language (domain independent) Boolean constructors ¬ • Concept negation: (class complement) ⊓ • Concept conjunction: (class intersection) ⊔ • Concept disjunction: (class union) Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 13

  6. Formal Ontologies Introduction to DLs Elements of the language (domain independent) Boolean constructors ¬ • Concept negation: (class complement) ⊓ • Concept conjunction: (class intersection) ⊔ • Concept disjunction: (class union) Role restrictions • Existential restriction: ∃ (at least one relationship) ∀ • Value restriction: (all relationships) Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 13

  7. Formal Ontologies Introduction to DLs Elements of the language (domain independent) Boolean constructors ¬ • Concept negation: (class complement) ⊓ • Concept conjunction: (class intersection) ⊔ • Concept disjunction: (class union) Role restrictions • Existential restriction: ∃ (at least one relationship) ∀ • Value restriction: (all relationships) Further constructors: cardinality constraints, inverse roles, . . . (if needed) Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 13

  8. 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 ) ∃ r.C (existential restriction) C ⊓ D (intersection of C and D ) ∀ r.C (value restriction) C ⊔ D (union of C and D ) are all concepts • Nothing else is a concept (at least for now) Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 14

  9. Formal Ontologies Introduction to DLs Exercise Which ones are concepts and which aren’t? • ⊤ ⊓ ⊥ ⊔ ⊤ • C ⊔ ∀ r. ⊓ ¬ D • C ⊔ ¬¬∃ D • ∃ r. ⊤ • ∃ r. ∀ s.C ⊓ D • ∀ r.C ⊓ ¬ D • ∀ r. ( C ⊓ ¬ D ) • ∀∃ r.C Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 15

  10. Formal Ontologies Introduction to DLs Exercise Which ones are concepts and which aren’t? • ⊤ ⊓ ⊥ ⊔ ⊤ � • C ⊔ ∀ r. ⊓ ¬ D • C ⊔ ¬¬∃ D • ∃ r. ⊤ • ∃ r. ∀ s.C ⊓ D • ∀ r.C ⊓ ¬ D • ∀ r. ( C ⊓ ¬ D ) • ∀∃ r.C Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 15

  11. Formal Ontologies Introduction to DLs Exercise Which ones are concepts and which aren’t? • ⊤ ⊓ ⊥ ⊔ ⊤ � • C ⊔ ∀ r. ⊓ ¬ D × • C ⊔ ¬¬∃ D • ∃ r. ⊤ • ∃ r. ∀ s.C ⊓ D • ∀ r.C ⊓ ¬ D • ∀ r. ( C ⊓ ¬ D ) • ∀∃ r.C Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 15

  12. Formal Ontologies Introduction to DLs Exercise Which ones are concepts and which aren’t? • ⊤ ⊓ ⊥ ⊔ ⊤ � • C ⊔ ∀ r. ⊓ ¬ D × • C ⊔ ¬¬∃ D × • ∃ r. ⊤ • ∃ r. ∀ s.C ⊓ D • ∀ r.C ⊓ ¬ D • ∀ r. ( C ⊓ ¬ D ) • ∀∃ r.C Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 15

  13. Formal Ontologies Introduction to DLs Exercise Which ones are concepts and which aren’t? • ⊤ ⊓ ⊥ ⊔ ⊤ � • C ⊔ ∀ r. ⊓ ¬ D × • C ⊔ ¬¬∃ D × • ∃ r. ⊤ � • ∃ r. ∀ s.C ⊓ D • ∀ r.C ⊓ ¬ D • ∀ r. ( C ⊓ ¬ D ) • ∀∃ r.C Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 15

  14. Formal Ontologies Introduction to DLs Exercise Which ones are concepts and which aren’t? • ⊤ ⊓ ⊥ ⊔ ⊤ � • C ⊔ ∀ r. ⊓ ¬ D × • C ⊔ ¬¬∃ D × • ∃ r. ⊤ � • ∃ r. ∀ s.C ⊓ D � • ∀ r.C ⊓ ¬ D • ∀ r. ( C ⊓ ¬ D ) • ∀∃ r.C Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 15

  15. Formal Ontologies Introduction to DLs Exercise Which ones are concepts and which aren’t? • ⊤ ⊓ ⊥ ⊔ ⊤ � • C ⊔ ∀ r. ⊓ ¬ D × • C ⊔ ¬¬∃ D × • ∃ r. ⊤ � • ∃ r. ∀ s.C ⊓ D � • ∀ r.C ⊓ ¬ D � • ∀ r. ( C ⊓ ¬ D ) • ∀∃ r.C Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 15

  16. Formal Ontologies Introduction to DLs Exercise Which ones are concepts and which aren’t? • ⊤ ⊓ ⊥ ⊔ ⊤ � • C ⊔ ∀ r. ⊓ ¬ D × • C ⊔ ¬¬∃ D × • ∃ r. ⊤ � • ∃ r. ∀ s.C ⊓ D � • ∀ r.C ⊓ ¬ D � • ∀ r. ( C ⊓ ¬ D ) � • ∀∃ r.C Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 15

  17. Formal Ontologies Introduction to DLs Exercise Which ones are concepts and which aren’t? • ⊤ ⊓ ⊥ ⊔ ⊤ � • C ⊔ ∀ r. ⊓ ¬ D × • C ⊔ ¬¬∃ D × • ∃ r. ⊤ � • ∃ r. ∀ s.C ⊓ D � • ∀ r.C ⊓ ¬ D � • ∀ r. ( C ⊓ ¬ D ) � • ∀∃ r.C × Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 15

  18. Formal Ontologies Introduction to DLs Building concepts Full negation • Negation of arbitrary concepts • Intuition: the complement of a concept • E.g.: ¬¬ Student ¬ ( Student ⊓ Parent ) Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 16

  19. Formal Ontologies Introduction to DLs Building concepts Full negation • Negation of arbitrary concepts • Intuition: the complement of a concept • E.g.: ¬¬ Student ¬ ( Student ⊓ Parent ) Atomic negation • Some DLs only allow negation of concept names • Good complexity results • E.g.: ¬ Student ¬ Parent Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 16

  20. Formal Ontologies Introduction to DLs Building concepts Concept conjunction • Intuition: the intersection of two concepts • E.g.: Student ⊓ Parent Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 17

  21. Formal Ontologies Introduction to DLs Building concepts Concept conjunction • Intuition: the intersection of two concepts • E.g.: Student ⊓ Parent Concept disjunction • Intuition: the union of two concepts • E.g.: Employee ⊔ Student Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 17

  22. Formal Ontologies Introduction to DLs Building concepts Concept conjunction • Intuition: the intersection of two concepts • E.g.: Student ⊓ Parent Concept disjunction • Intuition: the union of two concepts • E.g.: Employee ⊔ Student So far we have seen the Boolean fragment of our concept language • At least as expressive as classical propositional logic Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 17

  23. Formal Ontologies Introduction to DLs Building concepts Existential restriction • Intuition: there is some link with a concept • E.g.: ∃ pays . Tax Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 18

  24. Formal Ontologies Introduction to DLs Building concepts Existential 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 Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 18

  25. Formal Ontologies Introduction to DLs Building concepts Existential 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 Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 18

  26. Formal Ontologies Introduction to DLs Language Different flavours • ALC : C ::= ⊤ | ⊥ | C | ¬ C | C ⊓ C | C ⊔ C | ∀ r.C | ∃ r.C • ALCQ : C ::= . . . | ≥ nr.C | ≤ nr.C • EL , DL-Lite, SHIQ , SHOQ , SROIQ (basis of OWL 2), . . . Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 19

  27. Formal Ontologies Introduction to DLs Language Different flavours • ALC : C ::= ⊤ | ⊥ | C | ¬ C | C ⊓ C | C ⊔ C | ∀ r.C | ∃ r.C • ALCQ : C ::= . . . | ≥ nr.C | ≤ nr.C • EL , DL-Lite, SHIQ , SHOQ , SROIQ (basis of OWL 2), . . . Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 19

  28. Formal Ontologies Introduction to DLs Language Different flavours • ALC : C ::= ⊤ | ⊥ | C | ¬ C | C ⊓ C | C ⊔ C | ∀ r.C | ∃ r.C • ALCQ : C ::= . . . | ≥ nr.C | ≤ nr.C • EL , DL-Lite, SHIQ , SHOQ , SROIQ (basis of OWL 2), . . . Example ¬ ( Student ⊓ Parent ) Student ⊓ ¬∃ pays . Tax ∃ empBy . Company EmpStud ⊓ ∃ pays . Tax Employee ⊔ Student ⊓ ∃ worksFor . Parent ∀ worksFor . Company Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 19

  29. Formal Ontologies Introduction to DLs Language Different flavours • ALC : C ::= ⊤ | ⊥ | C | ¬ C | C ⊓ C | C ⊔ C | ∀ r.C | ∃ r.C • ALCQ : C ::= . . . | ≥ nr.C | ≤ nr.C • EL , DL-Lite, SHIQ , SHOQ , SROIQ (basis of OWL 2), . . . Example ¬ ( Student ⊓ Parent ) Student ⊓ ¬∃ pays . Tax ∃ empBy . Company EmpStud ⊓ ∃ pays . Tax Employee ⊔ Student ⊓ ∃ worksFor . Parent ∀ worksFor . Company With L ALC we denote the concept language of ALC Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 19

  30. Formal Ontologies Introduction to DLs Semantics Definition (Interpretation) Tuple I = def � ∆ I , · I � , where • ∆ I is a domain (set of objects) • · I is an interpretation function such that A I ⊆ ∆ I r I ⊆ ∆ I × ∆ I a I ∈ ∆ I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 20

  31. Formal Ontologies Introduction to DLs Semantics Definition (Interpretation) Tuple I = def � ∆ I , · I � , where • ∆ I is a domain (set of objects) • · I is an interpretation function such that A I ⊆ ∆ I r I ⊆ ∆ I × ∆ I a I ∈ ∆ I Example Let I = � ∆ I , · I � where: Let C = { A 1 , A 2 , A 3 } , R = { r 1 , r 2 } , I = { a 1 , a 2 , a 3 } . • ∆ I = { x i | 1 ≤ i ≤ 9 } , a I 1 = x 5 , a I 2 = x 1 , a I 3 = x 2 • A I A I A I 1 = { x 1 , x 4 , x 6 , x 7 } , 2 = { x 3 , x 5 , x 9 } , 3 = { x 6 , x 7 , x 8 } • r I r I 1 = { ( x 1 , x 6 ) , ( x 4 , x 8 ) , ( x 2 , x 5 ) } , 2 = { ( x 4 , x 4 ) , ( x 6 , x 4 ) , ( x 5 , x 8 ) , ( x 9 , x 3 ) } Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 20

  32. Formal Ontologies Introduction to DLs Semantics ∆ I I : A I A I 1 2 x 1 ( a 2 ) x 2 ( a 3 ) x 3 r 2 r 1 x 4 r 2 r 1 x 5 ( a 1 ) r 2 r 1 r 2 x 6 x 7 x 8 x 9 A I 3 Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 21

  33. Formal Ontologies Introduction to DLs Semantics Extending DL interpretations ⊤ I = def ∆ I ⊥ I = def ∅ ( ¬ C ) I = def ∆ I \ C I ( C ⊓ D ) I = def C I ∩ D I ( C ⊔ D ) I = def C I ∪ D I ( ∃ r.C ) I = def { x ∈ ∆ I | r I ( x ) ∩ C I � = ∅} ( ∀ r.C ) I = def { x ∈ ∆ I | r I ( x ) ⊆ C I } Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 22

  34. Formal Ontologies Introduction to DLs Semantics Extending DL interpretations ⊤ I = def ∆ I ⊥ I = def ∅ ( ¬ C ) I = def ∆ I \ C I ( C ⊓ D ) I = def C I ∩ D I ( C ⊔ D ) I = def C I ∪ D I ( ∃ r.C ) I = def { x ∈ ∆ I | r I ( x ) ∩ C I � = ∅} ( ∀ r.C ) I = def { x ∈ ∆ I | r I ( x ) ⊆ C I } Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 22

  35. Formal Ontologies Introduction to DLs Semantics Extending DL interpretations ⊤ I = def ∆ I ⊥ I = def ∅ ( ¬ C ) I = def ∆ I \ C I ( C ⊓ D ) I = def C I ∩ D I ( C ⊔ D ) I = def C I ∪ D I ( ∃ r.C ) I = def { x ∈ ∆ I | r I ( x ) ∩ C I � = ∅} ( ∀ r.C ) I = def { x ∈ ∆ I | r I ( x ) ⊆ C I } Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 22

  36. Formal Ontologies Introduction to DLs Semantics Extending DL interpretations ⊤ I = def ∆ I ⊥ I = def ∅ ( ¬ C ) I = def ∆ I \ C I ( C ⊓ D ) I = def C I ∩ D I ( C ⊔ D ) I = def C I ∪ D I ( ∃ r.C ) I = def { x ∈ ∆ I | r I ( x ) ∩ C I � = ∅} ( ∀ r.C ) I = def { x ∈ ∆ I | r I ( x ) ⊆ C I } Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 22

  37. Formal Ontologies Introduction to DLs Semantics Extending DL interpretations ⊤ I = def ∆ I ⊥ I = def ∅ ( ¬ C ) I = def ∆ I \ C I ( C ⊓ D ) I = def C I ∩ D I ( C ⊔ D ) I = def C I ∪ D I ( ∃ r.C ) I = def { x ∈ ∆ I | r I ( x ) ∩ C I � = ∅} ( ∀ r.C ) I = def { x ∈ ∆ I | r I ( x ) ⊆ C I } Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 22

  38. Formal Ontologies Introduction to DLs Semantics Extending DL interpretations ⊤ I = def ∆ I ⊥ I = def ∅ ( ¬ C ) I = def ∆ I \ C I ( C ⊓ D ) I = def C I ∩ D I ( C ⊔ D ) I = def C I ∪ D I ( ∃ r.C ) I = def { x ∈ ∆ I | r I ( x ) ∩ C I � = ∅} ( ∀ r.C ) I = def { x ∈ ∆ I | r I ( x ) ⊆ C I } Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 22

  39. Formal Ontologies Introduction to DLs Semantics Extending DL interpretations ⊤ I = def ∆ I ⊥ I = def ∅ ( ¬ C ) I = def ∆ I \ C I ( C ⊓ D ) I = def C I ∩ D I ( C ⊔ D ) I = def C I ∪ D I ( ∃ r.C ) I = def { x ∈ ∆ I | r I ( x ) ∩ C I � = ∅} ( ∀ r.C ) I = def { x ∈ ∆ I | r I ( x ) ⊆ C I } Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 22

  40. Formal Ontologies Introduction to DLs Semantics Extending DL interpretations ⊤ I = def ∆ I ⊥ I = def ∅ ( ¬ C ) I = def ∆ I \ C I ( C ⊓ D ) I = def C I ∩ D I ( C ⊔ D ) I = def C I ∪ D I ( ∃ r.C ) I = def { x ∈ ∆ I | r I ( x ) ∩ C I � = ∅} ( ∀ r.C ) I = def { x ∈ ∆ I | r I ( x ) ⊆ C I } Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 22

  41. Formal Ontologies Introduction to DLs Semantics Extending DL interpretations ⊤ I = def ∆ I ⊥ I = def ∅ ( ¬ C ) I = def ∆ I \ C I ( C ⊓ D ) I = def C I ∩ D I ( C ⊔ D ) I = def C I ∪ D I ( ∃ r.C ) I = def { x ∈ ∆ I | r I ( x ) ∩ C I � = ∅} ( ∀ r.C ) I = def { x ∈ ∆ I | r I ( x ) ⊆ C I } Definition (Concept Satisfiability) A concept C is satisfiable if there is I = � ∆ I , · I � s.t. C I � = ∅ Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 22

  42. Formal Ontologies Introduction to DLs Semantics ∆ I Individual names a I • At most one element of ∆ I • Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 23

  43. Formal Ontologies Introduction to DLs Semantics ∆ I Individual names a I • At most one element of ∆ I • ∆ I Unique Name Assumption × a I b I • At most one name per object • Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 23

  44. Formal Ontologies Introduction to DLs Semantics ∆ I The ‘top’ concept • Everything is in ⊤ I ⊤ I • Also called Thing Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 24

  45. Formal Ontologies Introduction to DLs Semantics ∆ I The ‘top’ concept • Everything is in ⊤ I ⊤ I • Also called Thing ∆ I The ‘bottom’ concept • ⊥ I is in everything ⊥ I = ∅ • Also called Nothing Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 24

  46. Formal Ontologies Introduction to DLs Semantics ∆ I Arbitrary concept • A class in the domain C I • C I ⊆ ∆ I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 25

  47. Formal Ontologies Introduction to DLs Semantics ∆ I Arbitrary concept • A class in the domain C I • C I ⊆ ∆ I ∆ I Concept negation • The complement of a concept C I • ( ¬ C ) I = ∆ I \ C I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 25

  48. Formal Ontologies Introduction to DLs Semantics ∆ I Concept conjunction • The intersection of two classes C I D I • ( C ⊓ D ) I = C I ∩ D I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 26

  49. Formal Ontologies Introduction to DLs Semantics ∆ I Concept conjunction • The intersection of two classes C I D I • ( C ⊓ D ) I = C I ∩ D I ∆ I Concept disjunction • The union of two classes C I D I • ( C ⊔ D ) I = C I ∪ D I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 26

  50. Formal Ontologies Introduction to DLs Semantics ∆ I r I Existential restriction • At least one value of a class ( ∃ r.C ) I C I • ( ∃ r.C ) I = { x | r I ( x ) ∩ C I � = ∅} Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 27

  51. Formal Ontologies Introduction to DLs Semantics ∆ I r I Existential restriction • At least one value of a class ( ∃ r.C ) I C I • ( ∃ r.C ) I = { x | r I ( x ) ∩ C I � = ∅} ∆ I r I Value restriction • All values of a class ( ∀ r.C ) I C I • ( ∀ r.C ) I = { x | r I ( x ) ⊆ C I } Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 27

  52. Formal Ontologies Introduction to DLs Semantics An interpretation is a complete description of the world I : ∆ I x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 28

  53. Formal Ontologies Introduction to DLs Semantics An interpretation is a complete description of the world I : ∆ I x 0 x 1 x 2 ( mary ) x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 28

  54. Formal Ontologies Introduction to DLs Semantics An interpretation is a complete description of the world I : ∆ I x 0 x 1 x 2 ( mary ) x 3 x 4 x 6 x 5 ( john ) x 7 x 8 x 9 x 10 Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 28

  55. Formal Ontologies Introduction to DLs Semantics An interpretation is a complete description of the world I : ∆ I x 0 x 1 x 2 ( mary ) x 3 x 4 x 5 ( john ) x 6 ( ibm ) x 7 x 8 x 9 x 10 Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 28

  56. Formal Ontologies Introduction to DLs Semantics An interpretation is a complete description of the world I : ∆ I x 0 x 1 x 2 ( mary ) x 3 x 4 x 5 ( john ) x 6 ( ibm ) x 7 x 8 x 9 x 10 Student I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 28

  57. Formal Ontologies Introduction to DLs Semantics An interpretation is a complete description of the world I : ∆ I x 0 x 1 x 2 ( mary ) x 3 Tax I x 4 x 5 ( john ) x 6 ( ibm ) x 7 x 8 x 9 x 10 Student I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 28

  58. Formal Ontologies Introduction to DLs Semantics An interpretation is a complete description of the world I : ∆ I x 0 x 1 x 2 ( mary ) x 3 Tax I Company I x 4 x 5 ( john ) x 6 ( ibm ) x 7 x 8 x 9 x 10 Student I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 28

  59. Formal Ontologies Introduction to DLs Semantics An interpretation is a complete description of the world I : ∆ I Parent I x 0 x 1 x 2 ( mary ) x 3 Tax I Company I x 4 x 5 ( john ) x 6 ( ibm ) x 7 x 8 x 9 x 10 Student I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 28

  60. Formal Ontologies Introduction to DLs Semantics An interpretation is a complete description of the world I : ∆ I Parent I x 0 x 1 x 2 ( mary ) x 3 Tax I Company I x 4 x 5 ( john ) x 6 ( ibm ) x 7 x 8 x 9 x 10 Employee I Student I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 28

  61. Formal Ontologies Introduction to DLs Semantics An interpretation is a complete description of the world I : ∆ I Parent I x 0 x 1 x 2 ( mary ) x 3 EmpStud I Tax I Company I x 4 x 5 ( john ) x 6 ( ibm ) x 7 x 8 x 9 x 10 Employee I Student I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 28

  62. Formal Ontologies Introduction to DLs Semantics An interpretation is a complete description of the world I : ∆ I Parent I x 0 x 1 x 2 ( mary ) x 3 EmpStud I Tax I Company I worksFor x 4 x 5 ( john ) x 6 ( ibm ) worksFor x 7 x 8 x 9 x 10 Employee I Student I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 28

  63. Formal Ontologies Introduction to DLs Semantics An interpretation is a complete description of the world I : ∆ I Parent I x 0 x 1 x 2 ( mary ) x 3 EmpStud I Tax I Company I 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 1) 26 April 2019 28

  64. Formal Ontologies Introduction to DLs Semantics An interpretation is a complete description of the world I : ∆ I Parent I pays x 0 x 1 x 2 ( mary ) x 3 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 1) 26 April 2019 28

  65. Formal Ontologies Introduction to DLs Semantics An interpretation is a complete description of the world I : ∆ I Parent I pays x 0 x 1 x 2 ( mary ) x 3 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 (( EmpStud ⊔ Parent ) ⊓ ∃ pays . ⊤ ) I = { x 1 , x 5 } Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 28

  66. Formal Ontologies Introduction to DLs Exercise Let C = { Company , Employee , EmpStud , Parent , Student , Tax } , R = { empBy , pays , worksFor } I = { ibm , john , mary } 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 1) 26 April 2019 29

  67. Formal Ontologies Introduction to DLs Exercise Let C = { Company , Employee , EmpStud , Parent , Student , Tax } , R = { empBy , pays , worksFor } I = { ibm , john , mary } 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 • ( ¬ Employee ) I =? • ( ¬ EmpStud ⊓ ∀ empBy . Company ) I =? • ( ∃ pays . ⊤ ) I =? • ( ∃ worksFor . ∃ empBy . Parent ) I =? • ( ¬ Parent ⊓ Employee ) I =? • ( Student ⊓ ∀ pays . ⊥ ) I =? Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 29

  68. Formal Ontologies Introduction to DLs Exercise Let C = { Company , Employee , EmpStud , Parent , Student , Tax } , R = { empBy , pays , worksFor } I = { ibm , john , mary } 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 • ( ¬ Employee ) I = { x 0 , x 3 , x 4 , x 6 , x 7 , x 8 , x 10 } • ( ¬ EmpStud ⊓ ∀ empBy . Company ) I =? • ( ∃ pays . ⊤ ) I =? • ( ∃ worksFor . ∃ empBy . Parent ) I =? • ( ¬ Parent ⊓ Employee ) I =? • ( Student ⊓ ∀ pays . ⊥ ) I =? Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 29

  69. Formal Ontologies Introduction to DLs Exercise Let C = { Company , Employee , EmpStud , Parent , Student , Tax } , R = { empBy , pays , worksFor } I = { ibm , john , mary } 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 • ( ¬ Employee ) I = { x 0 , x 3 , x 4 , x 6 , x 7 , x 8 , x 10 } • ( ¬ EmpStud ⊓ ∀ empBy . Company ) I =? • ( ∃ pays . ⊤ ) I = { x 1 , x 5 } • ( ∃ worksFor . ∃ empBy . Parent ) I =? • ( ¬ Parent ⊓ Employee ) I =? • ( Student ⊓ ∀ pays . ⊥ ) I =? Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 29

  70. Formal Ontologies Introduction to DLs Exercise Let C = { Company , Employee , EmpStud , Parent , Student , Tax } , R = { empBy , pays , worksFor } I = { ibm , john , mary } 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 • ( ¬ Employee ) I = { x 0 , x 3 , x 4 , x 6 , x 7 , x 8 , x 10 } • ( ¬ EmpStud ⊓ ∀ empBy . Company ) I =? • ( ∃ pays . ⊤ ) I = { x 1 , x 5 } • ( ∃ worksFor . ∃ empBy . Parent ) I =? • ( ¬ Parent ⊓ Employee ) I = { x 5 , x 9 } • ( Student ⊓ ∀ pays . ⊥ ) I =? Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 29

  71. Formal Ontologies Introduction to DLs Exercise Let C = { Company , Employee , EmpStud , Parent , Student , Tax } , R = { empBy , pays , worksFor } I = { ibm , john , mary } 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 • ( ¬ Employee ) I = { x 0 , x 3 , x 4 , x 6 , x 7 , x 8 , x 10 } • ( ¬ EmpStud ⊓ ∀ empBy . Company ) I = { x 9 } • ( ∃ pays . ⊤ ) I = { x 1 , x 5 } • ( ∃ worksFor . ∃ empBy . Parent ) I =? • ( ¬ Parent ⊓ Employee ) I = { x 5 , x 9 } • ( Student ⊓ ∀ pays . ⊥ ) I =? Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 29

  72. Formal Ontologies Introduction to DLs Exercise Let C = { Company , Employee , EmpStud , Parent , Student , Tax } , R = { empBy , pays , worksFor } I = { ibm , john , mary } 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 • ( ¬ Employee ) I = { x 0 , x 3 , x 4 , x 6 , x 7 , x 8 , x 10 } • ( ¬ EmpStud ⊓ ∀ empBy . Company ) I = { x 9 } • ( ∃ pays . ⊤ ) I = { x 1 , x 5 } • ( ∃ worksFor . ∃ empBy . Parent ) I = ∅ • ( ¬ Parent ⊓ Employee ) I = { x 5 , x 9 } • ( Student ⊓ ∀ pays . ⊥ ) I =? Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 29

  73. Formal Ontologies Introduction to DLs Exercise Let C = { Company , Employee , EmpStud , Parent , Student , Tax } , R = { empBy , pays , worksFor } I = { ibm , john , mary } 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 • ( ¬ Employee ) I = { x 0 , x 3 , x 4 , x 6 , x 7 , x 8 , x 10 } • ( ¬ EmpStud ⊓ ∀ empBy . Company ) I = { x 9 } • ( ∃ pays . ⊤ ) I = { x 1 , x 5 } • ( ∃ worksFor . ∃ empBy . Parent ) I = ∅ • ( ¬ Parent ⊓ Employee ) I = { x 5 , x 9 } • ( Student ⊓ ∀ pays . ⊥ ) I = { x 7 , x 8 } Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 29

  74. Formal Ontologies Introduction to DLs Exercise Let C = { Company , Employee , EmpStud , Parent , Student , Tax } , R = { empBy , pays , worksFor } I = { ibm , john , mary } Find an interpretation I = � ∆ I , · I � such that: • ( Student ⊓ Employee ) I = ∅ , Parent I ⊆ ( Student ⊔ Employee ) I , ( ¬ EmpStud ) I = ∆ I • Student I ⊆ ( ∀ pays . ⊥ ) I , ( ∃ worksFor . ⊤ ) I ⊆ ( ¬ ( Student ⊔ Tax ⊔ Company )) I , Employee I ⊆ ( ∃ empBy . ⊤ ) I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 30

  75. Formal Ontologies Introduction to DLs Exercise Let C = { Company , Employee , EmpStud , Parent , Student , Tax } , R = { empBy , pays , worksFor } I = { ibm , john , mary } Find an interpretation I = � ∆ I , · I � such that: • ( Student ⊓ Employee ) I = ∅ , Parent I ⊆ ( Student ⊔ Employee ) I , ( ¬ EmpStud ) I = ∆ I • Student I ⊆ ( ∀ pays . ⊥ ) I , ( ∃ worksFor . ⊤ ) I ⊆ ( ¬ ( Student ⊔ Tax ⊔ Company )) I , Employee I ⊆ ( ∃ empBy . ⊤ ) I I : ∆ I Parent I x 0 x 1 x 2 ( mary ) x 3 Tax I worksFor empBy Company I x 4 x 5 ( john ) x 6 ( ibm ) worksFor x 7 x 8 x 9 x 10 empBy Student I Employee I Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 30

  76. Formal Ontologies Introduction to DLs Some Properties Lemma For every interpretation I = � ∆ I , · I � , and for every C, D ∈ L ALC • ( ¬¬ C ) I = C I • ( ¬ ( C ⊓ D )) I = ( ¬ C ⊔ ¬ D ) I • ( ¬ ( C ⊔ D )) I = ( ¬ C ⊓ ¬ D ) I • ( ¬∀ r.C ) I = ( ∃ r. ¬ C ) I • ( ¬∃ r.C ) I = ( ∀ r. ¬ C ) I ALC is the smallest propositionally closed DL Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 31

  77. Formal Ontologies Introduction to DLs Some Properties Lemma For every interpretation I = � ∆ I , · I � , and for every C, D ∈ L ALC • ( ¬¬ C ) I = C I • ( ¬ ( C ⊓ D )) I = ( ¬ C ⊔ ¬ D ) I • ( ¬ ( C ⊔ D )) I = ( ¬ C ⊓ ¬ D ) I • ( ¬∀ r.C ) I = ( ∃ r. ¬ C ) I • ( ¬∃ r.C ) I = ( ∀ r. ¬ C ) I ALC is the smallest propositionally closed DL Theorem ALC has the finite model property: if C is satisfiable, then there is I = � ∆ I , · I � such that C I � = ∅ and ∆ I is finite Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 31

  78. Formal Ontologies Introduction to DLs Epilogue Summary • 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 Ivan Varzinczak Formal Foundations of Ontologies and Reasoning (Part 1) 26 April 2019 32

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