Semantic Modeling with Frames Rainer Osswald & Wiebke Petersen Department of Linguistics and Information Science Heinrich-Heine-Universit¨ at D¨ usseldorf ESSLLI 2018 Introductory Course Sofia University 06. 08. – 10. 08. 2018 SFB 991
Part 5 Nominal frames and Shifs Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 1
frames as atribute-value structures crown crown Given a set of types T and a set of atributes A . A basic frame over ( T , A ) is a tuple trunk bark � Q , ¯ q , arg , θ , δ � with tree trunk bark Q is a finite set of nodes. tree -frame q ∈ Q is the central node ¯ arg ⊆ Q argument nodes. tree crown: crown ind ⊆ Q individual nodes. � � trunk θ : Q → T is the typing function trunk: bark : bark δ : Q n × A → Q is the partial transition function. such that the underlying graph ( Q , E ) with edge set E = {{ q i , q }|∃ a ∈ A , q i ∈ { q 1 , . . . , q n } : δ ( q 1 , . . . q n , a ) = q } is connected. L Carpenter 1992 L Petersen 2007 Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 2
frames as atribute-value structures crown crown Given a set of types T and a set of atributes A . A basic frame over ( T , A ) is a tuple trunk bark � Q , ¯ q , arg , θ , δ � with tree trunk bark Q is a finite set of nodes. tree -frame q ∈ Q is the central node ¯ arg ⊆ Q argument nodes. trunk bark tree trunk bark ind ⊆ Q individual nodes. trunk (of) -frame θ : Q → T is the typing function δ : Q n × A → Q is the partial transition function. class class class such that the underlying graph ( Q , E ) with edge set E = {{ q i , q }|∃ a ∈ A , q i ∈ { q 1 , . . . , q n } : person person δ ( q 1 , . . . q n , a ) = q } is connected. classmate (of) -frame L Carpenter 1992 L Petersen 2007 Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 2
Remarks The definition and the notation differs slightly from the one we used the last days frames are connected , directed graphs with labeled nodes (by types) and arcs (by atributes) argument nodes are represented as rectangular nodes individual nodes are marked by small pointing arrows the central node is marked by a double line the restriction that every node can be reached via directed arcs from an argument or the central node is dropped the types are writen inside the nodes m o v e r man locomotion o r a c t p a manner t h path walking Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 3
Definition (Subsumption) A frame F 1 = � Q 1 , ¯ q 1 , θ 1 , δ 1 � subsumes a frame F 2 = � Q 2 , ¯ q 2 , θ 2 , δ 2 � ( F ⊑ F ′ ) iff there is a total function h : Q 1 → Q 2 with h ( ¯ q 1 ) = ¯ q 2 , ∀ q ∈ Q 1 : θ 1 ( q ) ⊑ θ 2 ( h ( q )) , if δ 1 ( f , q ) is defined, then h ( δ 1 ( f , q )) = δ 2 ( f , h ( q )) . Definition (Equivalence) Two frames F 1 and F 2 are equivalent ( F 1 ∼ F 2 ), if F 1 ⊑ F 2 and F 2 ⊑ F 1 . Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 4
Definition (Subsumption) A frame F 1 = � Q 1 , ¯ q 1 , θ 1 , δ 1 � subsumes a frame F 2 = � Q 2 , ¯ q 2 , θ 2 , δ 2 � ( F ⊑ F ′ ) iff there is a total injective function h : Q 1 → Q 2 with h ( ¯ q 1 ) = ¯ q 2 , ∀ q ∈ Q 1 : θ 1 ( q ) ⊑ θ 2 ( h ( q )) , if δ 1 ( f , q ) is defined, then h ( δ 1 ( f , q )) = δ 2 ( f , h ( q )) . Definition (Equivalence) Two frames F 1 and F 2 are equivalent ( F 1 ∼ F 2 ), if F 1 ⊑ F 2 and F 2 ⊑ F 1 . Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 4
Adaption of subsumption relation child FATHER ⊑ FATHER child person person R E H T A F child Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 5
Adaption of subsumption relation child F A T H E R ⊒ F A T H E R child person person FATHER child Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 5
concept classification person, pope, house, verb, sun, Mary, wood, brother, mother, meaning, distance, spouse, argument, entrance Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 6
concept classification: relationality non-relationalperson, pope, house, verb, sun, Mary, wood brother, mother, meaning, distance, spouse, relational argument, entrance Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 7
concept classification: uniqueness of reference non-unique unique reference reference non-relational person, house, verb, Mary, pope, sun wood brother, argument, mother, meaning, relational entrance distance, spouse Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 8
concept classification non-unique unique reference reference individual non-relational sortal concept concept λ x . x = ι u . P ( u ) λ x . P ( x ) proper relational functional relational concept concept λ y λ x . x f ( y ) = λ y λ x . R ( x , y ) L¨ obner (2011): ‘Concept Types and Determination’ Journal of Semantics 28: 279-333 Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 9
concept classification non-unique unique reference reference individual non-relational sortal concept concept λ x . x = ι u . P ( u ) λ x . P ( x ) ι u . P ( u ) proper relational functional relational concept concept λ y λ x . x f ( y ) = λ y λ x . R ( x , y ) λ y . f ( y ) L¨ obner (2011): ‘Concept Types and Determination’ Journal of Semantics 28: 279-333 Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 9
frames and functional concepts atributes describe functional crown relations, i.e., they represent n w o r c functions atributes correspond to tree bark trunk bark functional concepts ⇒ frames decompose concepts trunk diameter into functional concepts ⇒ functional concepts embody numeric the concept type on which categorization is based Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 10
sortal concepts tree-Frame tree trunk-Frame crown bark crown k r a b trunk tree tree trunk bark d trunk i bark a m e t e r trunk numeric diameter λ x . trunk ( ε u . x = trunk ( u )) ∧ numeric bark ( bark ( x )) ∧ numeric ( diameter ( x )) ∧ . . . λ x ∃ u . x = trunk ( u ) ∧ bark ( bark ( x )) ∧ λ x . crown ( crown ( x )) ∧ numeric ( diameter ( x )) ∧ . . . bark ( bark ( trunk ( x ))) ∧ numeric ( diameter ( trunk ( x ))) ∧ . . . Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 11
individual concepts pope-frame Mary-frame predicate constant ‘pope’: predicate constant ‘Mary’: head Mary RCC λ x . x = ι y . ( y = Mary ) λ x . x = head ( ι y . y = RCC ) Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 12
individual concepts pope-frame Mary-frame predicate constant ‘pope’: predicate constant ‘Mary’: head Mary RCC λ x . x = ι y . ( y = Mary ) λ x . x = head ( ι y . y = RCC ) individual constant ‘Mary’: individual constant ‘pope’: head Mary RCC ι x . x = Mary ι x . x = head ( RCC ) Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 12
Summary: non-relational concepts sortal concepts one open argument individual concepts one open argument there is a direct path from a definite node to the central node Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 13
proper relational concepts classmate-frame child-frame mother person person class class class person person λ y λ x . y = mother ( x ) ∧ person ( x ) ∧ person ( y ) λ y λ x . class ( x ) = class ( y ) ∧ . . . Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 14
functional concepts head-frame haircolor-frame predicate constant ‘head’: predicate constant ‘haircolor’: head hair color λ y λ x . x = head ( y ) λ y λ x . x = color ( hair ( y )) Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 15
functional concepts head-frame haircolor-frame predicate constant ‘head’: predicate constant ‘haircolor’: head hair color λ y λ x . x = head ( y ) λ y λ x . x = color ( hair ( y )) function constant ‘head’: function constant ‘haircolor’: head hair color λ y . head ( y ) λ y . color ( hair ( y )) Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 15
Summary: relational concepts proper relational concepts two open arguments no direct path from the other open argument to the central node functional concepts two open arguments there is a direct path from the other open argument to the central node Semantic Modeling with Frames | Part 5 | ESSLLI 2018 | Sofia R. Osswald & W. Petersen 16
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