Semantic Modeling with Frames Rainer Osswald & Wiebke Petersen - PowerPoint PPT Presentation

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 2 Formal foundations R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 1

Topics Atribute-value descriptions and formulas Translation into predicate logic Formal definition of frames Frames as models Subsumption and unification Atribute-value constraints Frames versus feature structures Type constraints versus type hierarchy R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 2

Recap Example actor man x house locomotion e z mover manner in-region path endp part-of walking path region region Ingredients Atributes (funct. relations): actor , mover , path , manner , in-region , ... Type symbols: locomotion , man , path , walking , region , ... Proper relation symbols: part-of Node labels (variables, constants): e , x , z Core property Every node is reachable from some labeled “base” node via atributes. R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 3

Atribute-value descriptions Vocabulary / Signature Atr atributes ( = dyadic functional relation symbols) Rel (proper) relation symbols Type type symbols ( = monadic predicates) Nname node names (“nominals”) } Nlabel node labels Nvar node variables R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 4

Atribute-value descriptions Vocabulary / Signature Atr atributes ( = dyadic functional relation symbols) Rel (proper) relation symbols Type type symbols ( = monadic predicates) Nname node names (“nominals”) } Nlabel node labels Nvar node variables Primitive atribute-value descriptions (pAVDesc) t | p : t | p � q | [ p 1 , . . . , p n ] : r | p � k ( t ∈ Type, r ∈ Rel, p , q , p i ∈ Atr ∗ , k ∈ Nlabel) Semantics ⎡ ⎤ ⎢ ⎥ P [ P [ t ] ] P ∶ t P ⎢ ⎥ 1 t [ P , Q ]∶ r P ⎢ ⎥ ⎢ ⎥ r ⎣ Q 2 ⎦ r ( 1 , 2 ) Q P ⎡ ⎤ ⎢ ⎥ P ≐ Q ⎢ ⎥ P 1 ⎢ ⎥ ⎢ ⎥ [ P k [ ] ] P ⎣ ⎦ P ≜ k Q 1 Q k R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 4

Translation into first-order predicate logic Vocabulary / Signature Atr dyadic relation symbols (atributes) Rel relation symbols Type monadic predicates (type symbols) Nname constants (node names) Nvar variables Important Functionality of atributes has to be enforced axiomatically! Primitive atribute-value descriptions as predicates: p : t λ x ∃ y ( p ( x , y ) ∧ t ( y )) λ x ∃ y ( p ( x , y ) ∧ q ( x , y )) p � q [ p 1 , . . . , p n ] : r λ x ∃ y 1 . . . ∃ y n ( p 1 ( x , y 1 ) ∧ · · · ∧ p n ( x , y n ) ∧ r ( y 1 , . . . , y n )) p � k λ x ( p ( x , k )) R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 5

Atribute-value formulas Primitive atribute-value formulas (pAVForm) k · p : t | k · p � l · q | � k 1 · p 1 , . . . , k n · p n � : r ( t ∈ Type, r ∈ Rel, p , q , p i ∈ Atr ∗ , k , l , k i ∈ Nlabel) Semantics P P k [ P [ t ] ] 1 ] k [ P k ⋅ P ∶ t t ⟨ k ⋅ P , l ⋅ Q ⟩∶ r k k r 2 ] l [ Q l r ( 1 , 2 ) 1 ] k [ P Q k ⋅ P ≜ l ⋅ Q P k 1 ] l [ Q l Q R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 6

Atribute-value formulas Primitive atribute-value formulas (pAVForm) k · p : t | k · p � l · q | � k 1 · p 1 , . . . , k n · p n � : r ( t ∈ Type, r ∈ Rel, p , q , p i ∈ Atr ∗ , k , l , k i ∈ Nlabel) Semantics P P k [ P [ t ] ] 1 ] k [ P k ⋅ P ∶ t t ⟨ k ⋅ P , l ⋅ Q ⟩∶ r k k r 2 ] l [ Q l r ( 1 , 2 ) 1 ] k [ P Q k ⋅ P ≜ l ⋅ Q P k 1 ] l [ Q l Q Formal definitions (fairly standard) Set / universe of “nodes” V R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 6

Atribute-value formulas Primitive atribute-value formulas (pAVForm) k · p : t | k · p � l · q | � k 1 · p 1 , . . . , k n · p n � : r ( t ∈ Type, r ∈ Rel, p , q , p i ∈ Atr ∗ , k , l , k i ∈ Nlabel) Semantics P P k [ P [ t ] ] 1 ] k [ P k ⋅ P ∶ t t ⟨ k ⋅ P , l ⋅ Q ⟩∶ r k k r 2 ] l [ Q l r ( 1 , 2 ) 1 ] k [ P Q k ⋅ P ≜ l ⋅ Q P k 1 ] l [ Q l Q Formal definitions (fairly standard) Set / universe of “nodes” V Interpretation function I : Atr → [ V ⇀ V ] , R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 6

Atribute-value formulas Primitive atribute-value formulas (pAVForm) k · p : t | k · p � l · q | � k 1 · p 1 , . . . , k n · p n � : r ( t ∈ Type, r ∈ Rel, p , q , p i ∈ Atr ∗ , k , l , k i ∈ Nlabel) Semantics P P k [ P [ t ] ] 1 ] k [ P k ⋅ P ∶ t t ⟨ k ⋅ P , l ⋅ Q ⟩∶ r k k r 2 ] l [ Q l r ( 1 , 2 ) 1 ] k [ P Q k ⋅ P ≜ l ⋅ Q P k 1 ] l [ Q l Q Formal definitions (fairly standard) Set / universe of “nodes” V Interpretation function I : Atr → [ V ⇀ V ] , Type → ℘ ( V ) , R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 6

Atribute-value formulas Primitive atribute-value formulas (pAVForm) k · p : t | k · p � l · q | � k 1 · p 1 , . . . , k n · p n � : r ( t ∈ Type, r ∈ Rel, p , q , p i ∈ Atr ∗ , k , l , k i ∈ Nlabel) Semantics P P k [ P [ t ] ] 1 ] k [ P k ⋅ P ∶ t t ⟨ k ⋅ P , l ⋅ Q ⟩∶ r k k r 2 ] l [ Q l r ( 1 , 2 ) 1 ] k [ P Q k ⋅ P ≜ l ⋅ Q P k 1 ] l [ Q l Q Formal definitions (fairly standard) Set / universe of “nodes” V Interpretation function I : Atr → [ V ⇀ V ] , Type → ℘ ( V ) , n ℘ ( V n ) , Rel → � R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 6

Atribute-value formulas Primitive atribute-value formulas (pAVForm) k · p : t | k · p � l · q | � k 1 · p 1 , . . . , k n · p n � : r ( t ∈ Type, r ∈ Rel, p , q , p i ∈ Atr ∗ , k , l , k i ∈ Nlabel) Semantics P P k [ P [ t ] ] 1 ] k [ P k ⋅ P ∶ t t ⟨ k ⋅ P , l ⋅ Q ⟩∶ r k k r 2 ] l [ Q l r ( 1 , 2 ) 1 ] k [ P Q k ⋅ P ≜ l ⋅ Q P k 1 ] l [ Q l Q Formal definitions (fairly standard) Set / universe of “nodes” V Interpretation function I : Atr → [ V ⇀ V ] , Type → ℘ ( V ) , n ℘ ( V n ) , Nname → V Rel → � R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 6

Atribute-value formulas Primitive atribute-value formulas (pAVForm) k · p : t | k · p � l · q | � k 1 · p 1 , . . . , k n · p n � : r ( t ∈ Type, r ∈ Rel, p , q , p i ∈ Atr ∗ , k , l , k i ∈ Nlabel) Semantics P P k [ P [ t ] ] 1 ] k [ P k ⋅ P ∶ t t ⟨ k ⋅ P , l ⋅ Q ⟩∶ r k k r 2 ] l [ Q l r ( 1 , 2 ) 1 ] k [ P Q k ⋅ P ≜ l ⋅ Q P k 1 ] l [ Q l Q Formal definitions (fairly standard) Set / universe of “nodes” V Interpretation function I : Atr → [ V ⇀ V ] , Type → ℘ ( V ) , n ℘ ( V n ) , Nname → V Rel → � (Partial) variable assignment g : Nvar ⇀ V R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 6

Satisfaction of AV descriptions and formulas Formal definitions (cont’d) Abbreviation: I g ( k ) = v for k ∈ Nlabel iff I ( k ) = v if k ∈ Nname and g ( k ) = v if k ∈ Nvar ( g ( k ) defined) R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 7

Satisfaction of AV descriptions and formulas Formal definitions (cont’d) Abbreviation: I g ( k ) = v for k ∈ Nlabel iff I ( k ) = v if k ∈ Nname and g ( k ) = v if k ∈ Nvar ( g ( k ) defined) Satisfaction of primitive descriptions � V , I , g � , v � t iff v ∈ I ( t ) R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 7

Satisfaction of AV descriptions and formulas Formal definitions (cont’d) Abbreviation: I g ( k ) = v for k ∈ Nlabel iff I ( k ) = v if k ∈ Nname and g ( k ) = v if k ∈ Nvar ( g ( k ) defined) Satisfaction of primitive descriptions � V , I , g � , v � t iff v ∈ I ( t ) � V , I , g � , v � p : t iff I ( p )( v ) ∈ I ( t ) R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 7

Satisfaction of AV descriptions and formulas Formal definitions (cont’d) Abbreviation: I g ( k ) = v for k ∈ Nlabel iff I ( k ) = v if k ∈ Nname and g ( k ) = v if k ∈ Nvar ( g ( k ) defined) Satisfaction of primitive descriptions � V , I , g � , v � t iff v ∈ I ( t ) � V , I , g � , v � p : t iff I ( p )( v ) ∈ I ( t ) � V , I , g � , v � p � q iff I ( p )( v ) = I ( q )( v ) R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 7

Satisfaction of AV descriptions and formulas Formal definitions (cont’d) Abbreviation: I g ( k ) = v for k ∈ Nlabel iff I ( k ) = v if k ∈ Nname and g ( k ) = v if k ∈ Nvar ( g ( k ) defined) Satisfaction of primitive descriptions � V , I , g � , v � t iff v ∈ I ( t ) � V , I , g � , v � p : t iff I ( p )( v ) ∈ I ( t ) � V , I , g � , v � p � q iff I ( p )( v ) = I ( q )( v ) � V , I , g � , v � [ p 1 , . . . , p n ] : r iff �I ( p 1 )( v ) , . . . , I ( p n )( v ) � ∈ I ( r ) R. Osswald & W. Petersen Semantic Modeling with Frames | Part 2 | ESSLLI 2018 | Sofia 7