Categorial Grammar Raffaella Bernardi Contents First Last Prev - PowerPoint PPT Presentation

Categorial Grammar Raffaella Bernardi Contents First Last Prev Next ◭

Contents 1 Recognition Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Classical Categorial Grammar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Classical Categorial Grammar. Examples . . . . . . . . . . . . . . . . . . . . . 5 4 Logic Grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5 Lambek calculus. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 6 Lambek calculus. Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 7 Lambek calculus. Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 8 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 9 Residuated and Galois Connected Functions . . . . . . . . . . . . . . . . . . 14 10 Interpretation of the Constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 11 Nonveridical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 12 Dutch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 13 Classification of NPIs in Dutch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 14 Antilicensing Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Contents First Last Prev Next ◭

1. Recognition Device ◮ Aim: To build a language recognition device. ◮ Who: Lesniewski (1929), Ajdukiewicz (1935), Bar-Hillel (1953). ◮ How: Linguistic strings are seen as the result of function applications starting from the categories assigned to lexicon items. Contents First Last Prev Next ◭

2. Classical Categorial Grammar ◮ Language: Given a set of basic categories ATOM , the set of categories CAT is the smallest set such that: ⊲ if X ∈ ATOM , then X ∈ CAT ; ⊲ if X, Y ∈ ATOM , then X/Y, Y \ X ∈ CAT ◮ Rules: The above categories can be composed by means of functional appli- cation rules X/Y, Y ⇒ X MPr Y, Y \ X ⇒ X MPl X/Y Y Y Y \ X [MPl] [MPr] X X Contents First Last Prev Next ◭

3. Classical Categorial Grammar. Examples Given ATOM = { np, s, n } , we can build the following lexicon: Lexicon John, Mary ∈ np the ∈ np/n student ∈ n some ∈ ( s/ ( np \ s )) /n walks ∈ np \ s sees ∈ ( np \ s ) /np Analysis John walks ∈ s ? ❀ np, np \ s ⇒ s ? Yes np np \ s [MPl] s John sees Mary ∈ s ? np, ( np \ s ) /np, np ⇒ s ? Yes ❀ ( np \ s ) /np np [MPr] np np \ s [MPl] s Contents First Last Prev Next ◭

who knows Lori ∈ n \ n ? ( n \ n ) / ( np \ s ) , ( np \ s ) /np, np ⇒ n \ n ? ❀ knows Lori ( np \ s ) /np np who [MPr] ( n \ n ) / ( np \ s ) np \ s [MPr] n \ n which Sara wrote [ . . . ] ∈ n \ n ? Modus ponens corresponds to functional application. X/Y : t Y : r Y : r Y \ X : t [MPl] [MPr] X : t ( r ) X : t ( r ) Example np : john np \ s : walk [MPl] s : walk ( john ) np \ s : λx. walk ( x ) ( λx. walk ( x ))( john ) ❀ λ − conv. walk ( john ) Contents First Last Prev Next ◭

( np \ s ) /np : know np : mary [MPr] np : john np \ s : know ( mary ) [MPl] s : know ( mary )( john ) Contents First Last Prev Next ◭

4. Logic Grammar ◮ Aim: To define the logic behind CG. ◮ How: Considering categories as formulae; \ , / as logic connectives. ◮ Who: Jim Lambek [1958] Lambek Calculus (Rules): Natural Deduction proof format [Elimination and Introduction rules] Besides functional applications rules – which correspond to the elimination of \ , / – we have their introduction rules. Γ ⊢ A means that A derives from Γ; Γ , ∆ stand for structures, A, B, C for logic formulae. ∆ ⊢ B/A Γ ⊢ A Γ ⊢ A ∆ ⊢ A \ B [ / E] [ \ E] ∆ , Γ ⊢ B Γ , ∆ ⊢ B ∆ , B ⊢ C B, ∆ ⊢ C ∆ ⊢ C/B [ / I] ∆ ⊢ B \ C [ \ I] Contents First Last Prev Next ◭

5. Lambek calculus. Examples which Sara wrote ∈ n \ n ? [ np ⊢ np ] 1 wrote ⊢ ( np \ s ) /np [ / E] Sara ⊢ np wrote np ⊢ np \ s [ \ E] Sara wrote np ⊢ s Sara wrote ⊢ s/np [ / I] 1 which ⊢ ( n \ n ) / ( s/np ) [ / E] which Sara wrote ⊢ n \ n The logical formulas built from ( \ , • / ) are interpreted using Kripke Models as below: { z |∃ x ∃ y [ R 3 zxy & x ∈ V ( A ) & y ∈ V ( B )] } V ( A • B ) = { x |∀ y ∀ z [( R 3 zxy & y ∈ V ( B )) ⇒ z ∈ V ( C )] } V ( C/B ) = { y |∀ x ∀ z [( R 3 zxy & x ∈ V ( A )) ⇒ z ∈ V ( C )] } V ( A \ C ) = NL is sound and complete with respect to Kripke models. Extractions are accounted for by means of introduction rules. Contents First Last Prev Next ◭

john ∈ np Lex np ⊢ np john ⊢ np ❀ Contents First Last Prev Next ◭

6. Lambek calculus. Semantics [ P ⊢ np \ s : P ] 1 john ⊢ np : john [ \ E] john P ⊢ s : P ( john ) john ⊢ s/ ( np \ s ) : λP.P ( john ) [ / I] 1 [ z ⊢ np : z ] 1 knows ⊢ ( np \ s ) /np : know [ / E] np ⊢ np : john john knows z ⊢ np \ s : know ( z )( john ) [ \ E] john knows z ⊢ s : know ( z )( john ) john knows ⊢ s/np : λz. know ( z )( john ) [ / I] 1 ⇓ The introduction rules correspond to λ -abstraction. Contents First Last Prev Next ◭

7. Lambek calculus. Advantages ◮ Hypothetical reasoning: Having added [ \ I] , [ / I] gives the system the right expressiveness to reason about hypothesis and abstract over them. ◮ Curry Howard Correspondence: Curry-Howard correspondence holds be- tween proofs and terms. This means that parsed structures are assigned an interpretation into a model via the connection ‘categories-terms’. ◮ Logic: We have moved from a grammar to a logic. Hence its behavior can be studied. The system is sound, complete and decidable. Contents First Last Prev Next ◭

8. Derivations A ⊢ A A ⊢ B � ✷ ↓ A � ⊢ A [ ✷ ↓ L] � A � ⊢ ✸ B [ ✸ R] ✸✷ ↓ A ⊢ A [ ✸ L] ✸ A ⊢ ✸ B [ ✸ L] A ⊢ B A ⊢ A � ✷ ↓ A � ⊢ B [ ✷ ↓ L] � ✷ ↓ A � ⊢ A [ ✸ R] ✷ ↓ A ⊢ ✷ ↓ B [ ✷ ↓ R] [ ✷ ↓ R] A ⊢ A A ⊢ A A ⊢ A ( A ) 0 ⊢ ♯A [( · ) 0 L] 0 ( A ) ⊢ ♭A [ 0 ( · )L] A ⊢ 0 (( A ) 0 ) [ 0 ( · )R] A ⊢ ( 0 ( A )) 0 [( · ) 0 R] Contents First Last Prev Next ◭

9. Residuated and Galois Connected Functions Remark 2 Let B ′ be a poset s.t. B ′ = ( B, ⊑ ′ def B ) where x ⊑ ′ B y = y ⊑ B x , and h : B → A . If ( f, h ) is a residuated pair with respect to ⊑ A and ⊑ ′ B , then it’s Galois connected with respect to ⊑ A and ⊑ B . f ( a ) ⊑ ′ b ⊑ B f ( a ) iff B b iff a ⊑ A h ( b ) Recall Consider two posets A = ( A, ⊑ A ) and B = ( B, ⊑ B ), and functions f : A → B, g : B → A . The pair ( f, g ) is said to be residuated iff ∀ a ∈ A, b ∈ B [ RES 1 ] f ( a ) ⊑ B b iff a ⊑ A g ( b ) The pair ( f, g ) is said to be Galois connected iff ∀ a ∈ A, b ∈ B [ GC 1 ] b ⊑ B f ( a ) iff a ⊑ A g ( b ) Contents First Last Prev Next ◭

10. Interpretation of the Constants { x | ∃ y ( R 2 V ( ✸ A ) = ✸ xy & y ∈ V ( A ) } { x | ∀ y ( R 2 V ( ✷ ↓ A ) = ✸ yx ⇒ y ∈ V ( A ) } { x | ∀ y ( y ∈ V ( A ) ⇒ ¬ R 2 V ( 0 A ) = 0 yx } { x | ∀ y ( y ∈ V ( A ) ⇒ ¬ R 2 V ( A 0 ) = 0 xy } { z |∃ x ∃ y [ R 3 zxy & x ∈ V ( A ) & y ∈ V ( B )] } V ( A • B ) = { x |∀ y ∀ z [( R 3 zxy & y ∈ V ( B )) ⇒ z ∈ V ( C )] } V ( C/B ) = { y |∀ x ∀ z [( R 3 zxy & x ∈ V ( A )) ⇒ z ∈ V ( C )] } V ( A \ C ) = Contents First Last Prev Next ◭

11. Nonveridical Functions definition [(Non)veridical functions (II)] → Let ( a n , t ) stand for a boolean type ( a 1 , ( . . . ( a n , t ) . . . )) where a 1 , . . . , a n are arbitrary types and 0 ≤ n . Let f ( a ,t ) be a constant. → 1. The expression represented by f is veridical in its i -argument, if a i is a boolean → type, i.e . a i = ( b, t ), and ∀M , g → → y → y → [ [ f ( x a 1 , . . . , x a i − 1 , x ( b ,t ) , x a i +1 , . . . , x a n )] ] M ,g = 1 entails [ [ ∃ b .x ( b ,t ) ( b )] ] M ,g = 1 . → → Otherwise f is nonveridical. 2. A nonveridical function represented by f ( a ,t ) is antiveridical in its i -argument, → → if a i = ( b, t ) and ∀M , g → → y → y → [ [ f ( x a 1 , . . . , x a i − 1 , x ( b ,t ) , x a i +1 , . . . , x a n )] ] M ,g = 1 entails [ [ ¬∃ . b x ( b ,t ) ( b )] ] M ,g = 1 . → → Contents First Last Prev Next ◭

→ y empty. Notice that the base case of a i = t is obtained by taking Contents First Last Prev Next ◭

12. Dutch In [van Wouden] it is shown that in Dutch polarity items are sensitive to downward monotonicity. Among downward monotone functions we can distinguish the sets below: antimorphic antiadditive downward monotone f ( X ∩ Y ) = f ( X ) ∪ f ( Y ) f ( X ) ∪ f ( Y ) ⊆ f ( X ∩ Y ) f ( X ) ∪ f ( Y ) ⊆ f ( X ∩ Y ) f ( X ∪ Y ) = f ( X ) ∩ f ( Y ) f ( X ∪ Y ) = f ( X ) ∩ f ( Y ) f ( X ∪ Y ) ⊆ f ( X ) ∩ f ( Y ) not nobody, never, nothing few, seldom, hardly Contents First Last Prev Next ◭