Knowledge Representation for the Semantic Web Lecture 3: Description - PowerPoint PPT Presentation

Semantics of Description Logics DL Nomenclature Equivalences Knowledge Representation for the Semantic Web Lecture 3: Description Logics II Daria Stepanova slides based on Reasoning Web 2011 tutorial “ Foundations of Description Logics and OWL ” by S. Rudolph Max Planck Institute for Informatics D5: Databases and Information Systems group WS 2017/18 1 / 34

Semantics of Description Logics DL Nomenclature Equivalences Unit Outline Semantics of Description Logics DL Nomenclature Equivalences 2 / 34

Semantics of Description Logics DL Nomenclature Equivalences Semantics of Description Logics 3 / 34

Semantics of Description Logics DL Nomenclature Equivalences Interpretations Semantics for DLs is defined in a model theoretic way, i.e., based on ”abstract possible worlds”, called interpretations. Def.: An interpretation I = (∆ I , · I ) consists of: ❼ a nonempty set ∆ I , called the interpretation domain (of I ) ❼ an interpretation function · I , which maps ❼ each atomic concept A to a subset A I of ∆ I ❼ each atomic role r to a subset r I of ∆ I × ∆ I . individual names class names N C N I role names N R . . . C . . . . . . a . . . . . . r . . . 4 / 34

Semantics of Description Logics DL Nomenclature Equivalences Interpretations Semantics for DLs is defined in a model theoretic way, i.e., based on ”abstract possible worlds”, called interpretations. Def.: An interpretation I = (∆ I , · I ) consists of: ❼ a nonempty set ∆ I , called the interpretation domain (of I ) ❼ an interpretation function · I , which maps ❼ each atomic concept A to a subset A I of ∆ I ❼ each atomic role r to a subset r I of ∆ I × ∆ I . individual names class names N C N I role names N R . . . C . . . . . . a . . . . . . r . . . ∆ I 4 / 34

Semantics of Description Logics DL Nomenclature Equivalences Interpretations Semantics for DLs is defined in a model theoretic way, i.e., based on ”abstract possible worlds”, called interpretations. Def.: An interpretation I = (∆ I , · I ) consists of: ❼ a nonempty set ∆ I , called the interpretation domain (of I ) ❼ an interpretation function · I , which maps ❼ each atomic concept A to a subset A I of ∆ I ❼ each atomic role r to a subset r I of ∆ I × ∆ I . individual names class names N C N I role names N R . . . C . . . . . . a . . . . . . r . . . · I a I ∆ I 4 / 34

Semantics of Description Logics DL Nomenclature Equivalences Interpretations Semantics for DLs is defined in a model theoretic way, i.e., based on ”abstract possible worlds”, called interpretations. Def.: An interpretation I = (∆ I , · I ) consists of: ❼ a nonempty set ∆ I , called the interpretation domain (of I ) ❼ an interpretation function · I , which maps ❼ each atomic concept A to a subset A I of ∆ I ❼ each atomic role r to a subset r I of ∆ I × ∆ I . individual names class names N C N I role names N R . . . C . . . . . . a . . . . . . r . . . C I · I · I a I ∆ I 4 / 34

Semantics of Description Logics DL Nomenclature Equivalences Interpretations Semantics for DLs is defined in a model theoretic way, i.e., based on ”abstract possible worlds”, called interpretations. Def.: An interpretation I = (∆ I , · I ) consists of: ❼ a nonempty set ∆ I , called the interpretation domain (of I ) ❼ an interpretation function · I , which maps ❼ each atomic concept A to a subset A I of ∆ I ❼ each atomic role r to a subset r I of ∆ I × ∆ I . individual names class names N C N I role names N R . . . C . . . . . . a . . . . . . r . . . C I · I · I · I a I ∆ I r I 4 / 34

Semantics of Description Logics DL Nomenclature Equivalences Interpretations: an Example 5 / 34

Semantics of Description Logics DL Nomenclature Equivalences Interpretation of Individuals Unique Name Assumption (UNA) If c 1 and c 2 are two individuals such that c 1 � = c 2 , then c I 1 � = c I 2 Note: When the UNA holds, equality and distincntness assertions are meaningless. In DLs one can drop UNA. Example: absence of UNA Two fathers ( f 1 , f 2) and two sons ( s 1 , s 2) went to a pizzeria and bought three pizzas for picnic lunch. When they started their lunch, every- one had a whole pizza. How could this happen? 6 / 34

Semantics of Description Logics DL Nomenclature Equivalences Interpretation of Individuals Unique Name Assumption (UNA) If c 1 and c 2 are two individuals such that c 1 � = c 2 , then c I 1 � = c I 2 Note: When the UNA holds, equality and distincntness assertions are meaningless. In DLs one can drop UNA. Standard Name Assumption (SNA) The UNA holds, and moreover individuals are interpreted in the same way in all interpretations. Hence, we can assume that ∆ I contains the set of individuals, and that for each interpretation I , we have that c I = c (then c is called standard name) 6 / 34

Semantics of Description Logics DL Nomenclature Equivalences Interpretation of Concept Expressions Construct Syntax Example Semantics A I ⊆ ∆ I atomic concept A Doctor r I ⊆ ∆ I × ∆ I atomic role r hasChild ∆ I \ A I atomic negation ¬ A ¬ Doctor C I ∩ D I conjunction C ⊓ D Human ⊓ Male unqual. exist. res. 1 { o | ∃ o ′ . ( o, o ′ ) ∈ r I } ∃ r ∃ hasChild { o | ∀ o ′ . ( o, o ′ ) ∈ r I → o ′ ∈ C I } value res. ∀ r.C ∀ hasChild . Male bottom ⊥ ∅ C, D denote arbitrary concepts and r denotes an arbitrary role. The above constructs form the basic language AL 1 Unqualified existential restriction 7 / 34

Semantics of Description Logics DL Nomenclature Equivalences Interpretation of Concept Expressions, cont’d Construct AL Syntax Semantics C I ∪ D I disjunction U C ⊔ D Singer ⊔ Dancer { o | ∃ o ′ . ( o, o ′ ) ∈ r I ∧ o ′ ∈ C I } qual. exist. res. 2 E ∃ R.C ∃ hasChild . Male C ¬ C ¬ ( ∃ hasSibling . Female ) ∆ I \ C I (full) negation { o | # { o ′ | ( o, o ′ ) ∈ r I } ≥ k } num. res. N ( ≥ k r ) ≥ 2 hasSister { o | # { o ′ | ( o, o ′ ) ∈ r I } ≤ k } ( ≤ k r ) ≤ 3 hasBrother { o | # { o ′ | ( o, o ′ ) ∈ r I ∧ o ′ ∈ C I } ≥ k qual. num. res. Q ( ≥ k r.C ) ≥ 2 hasSibling . Female { o | # { o ′ | ( o, o ′ ) ∈ r I ∧ o ′ ∈ C I } ≤ k ( ≥ k r.C ) ≤ 3 hasSibling . Male ∆ I top ⊤ Many different DL constructs and their combinations have been investigated. Combining various constructs we obtain a concrete DL fragment, i.e., language (see slide 26 for further details). 2 Qualified existential restriction 8 / 34

Semantics of Description Logics DL Nomenclature Equivalences Boolean Concept Expressions 9 / 34

Semantics of Description Logics DL Nomenclature Equivalences Boolean Concept Expressions ¬ Politician 9 / 34

Semantics of Description Logics DL Nomenclature Equivalences Boolean Concept Expressions ¬ Politician I 9 / 34

Semantics of Description Logics DL Nomenclature Equivalences Boolean Concept Expressions ¬ Politician I I 9 / 34

Semantics of Description Logics DL Nomenclature Equivalences Boolean Concept Expressions ⊓ Actor Politician ¬ Politician I I 9 / 34

Semantics of Description Logics DL Nomenclature Equivalences Boolean Concept Expressions ⊓ Actor Politician ¬ Politician I I I 9 / 34

Semantics of Description Logics DL Nomenclature Equivalences Boolean Concept Expressions ⊓ Actor Politician ¬ Politician I I I I 9 / 34

Semantics of Description Logics DL Nomenclature Equivalences Boolean Concept Expressions ⊓ Actor Politician ¬ Politician I I I I I 9 / 34

Semantics of Description Logics DL Nomenclature Equivalences Boolean Concept Expressions ⊓ Actor Politician ¬ Politician I I I I I ⊔ Actor Politician 9 / 34

Semantics of Description Logics DL Nomenclature Equivalences Boolean Concept Expressions ⊓ Actor Politician ¬ Politician I I I I I ⊔ Actor Politician I 9 / 34

Semantics of Description Logics DL Nomenclature Equivalences Boolean Concept Expressions ⊓ Actor Politician ¬ Politician I I I I I ⊔ Actor Politician I I 9 / 34

Semantics of Description Logics DL Nomenclature Equivalences Boolean Concept Expressions ⊓ Actor Politician ¬ Politician I I I I I ⊔ Actor Politician I I I 9 / 34

Semantics of Description Logics DL Nomenclature Equivalences Existential Role Restrictions parentOf. ∃ Male 10 / 34

Semantics of Description Logics DL Nomenclature Equivalences Universal Role Restrictions parentOf. ∀ Male 11 / 34

Semantics of Description Logics DL Nomenclature Equivalences Qualified Number Restrictions ≥ 2 parentOf. Male 12 / 34

Semantics of Description Logics DL Nomenclature Equivalences Self-Restrictions ∃ killed . Self 13 / 34

Semantics of Description Logics DL Nomenclature Equivalences Interpretation of Role Expressions Construct Syntax Example Semantics r I ⊆ ∆ I × ∆ I atomic role r hasChild ∆ I × ∆ I \{ ( o, o ′ ) ∈ r I } role negation ¬ r ¬ hasSister hasParent − inverse role r − { ( o, o ′ ) | ( o ′ , o ) ∈ r I } { ( o, o ′ ) | ( o, o ′′ ) ∈ r I , ( o ′′ , o ′ ) ∈ r ′I } r ◦ r ′ transitivity hasChild ◦ hasParent 14 / 34

Semantics of Description Logics DL Nomenclature Equivalences Inverse Role childOf − = parentOf 15 / 34