Parsing pregroup grammars using partial composition echet (1) , - PowerPoint PPT Presentation

Parsing pregroup grammars using partial composition echet (1) , Annie Foret (2) and Isabelle Tellier (3) Denis B´ (1) LINA, University of Nantes (2) IRISA, University of Rennes (3) LIFL, University of Lilles III Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 1

Parsing with word separation Parsing of “whom have you seen ?” ( q ′ ≤ Pr s ) whom have you seen q ′ o ll q l qp l 2 π l p 2 o l π 2 2 [2] [1] [1] q ′ o ll q l , qp l 2 π l , p 2 o l 2 , π 2 Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 2

PLAN Introduction Background Free pregroup Pregroup grammar and language Parsing Parsing with word separation Rewriting Partial composition Majority partial composition Conclusion Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 3

Free pregroup X ≤ Y Y ≤ Z ( CUT ) ( Id ) p ( n ) ≤ p ( n ) X ≤ Z XY ≤ Z X ≤ Y Z ( A G ) ( A D ) Xp ( n ) p ( n +1) Y ≤ Z X ≤ Y p ( n +1) p ( n ) Z Xp ( k ) Y ≤ Z ( IND G ) X ≤ Y p ( k ) Z ( IND D ) Xq ( k ) Y ≤ Z X ≤ Y q ( k ) Z q ≤ Pr p if k is even or p ≤ Pr q if k is odd Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 4

Free pregroup grammar and language A grammar G = (Σ , ( Pr, ≤ Pr ) , I, s ) : Σ finite alphabet ( Pr, ≤ ) finite partially ordered set (primitive types) that defines free pregroup ( Tp, ≤ Tp ) I ⊂ Σ × Tp , a lexicon, assigns a finite set of types to each c ∈ Σ s ∈ Pr is a primitive type for correct sentences The language L ( G ) ∈ Σ + : v 1 · · · v n ∈ L ( G ) iff for 1 ≤ i ≤ n , ∃ X i ∈ I ( v i ) such that X 1 · · · X n ≤ Tp s Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 5

Parsing using rewriting (1) Because s is a primitive type, v 1 · · · v n ∈ L ( G ) iff for 1 ≤ i ≤ n , ∃ X i ∈ I ( v i ) and ∃ s ′ ∈ Pr such that: � (1) ∗ X 1 · · · X n − → s ′ s ′ ≤ Pr s (1) ∗ (1) − → : the reflexive and transitive closure of − → : (1) Xp ( n ) q ( n +1) Y − → XY if q ≤ Pr p and n is even or if p ≤ Pr q and n is odd Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 6

Parsing using rewriting (1) : example Parsing of “whom have you seen ?” ( q ′ ≤ Pr s ) whom have you seen q ′ o ll q l qp l 2 π l p 2 o l π 2 2 (1) q ′ o ll q l qp l 2 π l 2 π 2 p 2 o l → q ′ o ll p l 2 π l 2 π 2 p 2 o l − (1) → q ′ o ll p l 2 p 2 o l − and q ′ ≤ Pr s (1) → q ′ o ll o l − (1) → q ′ − Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 7

Parsing using rewriting (1) : “proof net” Parsing of “whom have you seen ?” ( q ′ ≤ Pr s ) whom have you seen q ′ o ll q l qp l 2 π l p 2 o l π 2 2 q ′ o ll q l qp l 2 π l 2 π 2 p 2 o l ( q ′ ≤ Pr s ) Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 8

Parsing with word separation (list) Three rewriting rules ( Γ , ∆ ∈ Tp ∗ , X, Y ∈ Tp , p, q ∈ Pr ): M [ M ] (merge) : Γ , X, Y, ∆ − → Γ , XY, ∆ . I [ I ] (internal) : Γ , Xp ( n ) q ( n +1) Y, ∆ − → Γ , XY, ∆ , if q ≤ Pr p and n is even or if p ≤ Pr q and n is odd. E [ E ] (external) : Γ , Xp ( n ) , q ( n +1) Y, ∆ − → Γ , X, Y, ∆ , if q ≤ Pr p and n is even or if p ≤ Pr q and n is odd. Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 9

Parsing with word separation (example) Parsing of “whom have you seen ?” ( q ′ ≤ Pr s ) whom have you seen q ′ o ll q l qp l 2 π l p 2 o l π 2 2 q ′ o ll q l , q p l E 2 π l 2 , π 2 , p 2 o l → q ′ o ll , p l 2 π l 2 , π 2 , p 2 o l − M → q ′ o ll , p l 2 π l 2 π 2 , p 2 o l − I → q ′ o ll , p l 2 , p 2 o l and q ′ ≤ Pr s − E → q ′ o ll , , o l − M → q ′ o ll , o l − E − → q ′ Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 10

Parsing with word separation (lemma) Parsing (for a pregroup grammar) can be done using MIE ∗ − → : v 1 · · · v n ∈ L ( G ) iff for 1 ≤ i ≤ n , ∃ X i ∈ I ( v i ) and ∃ s ′ ∈ Pr such that: � MIE ∗ → s ′ X 1 , · · · , X n − s ′ ≤ Pr s Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 11

Internal before Merge/External (lemma) I M E − → can be performed before − → and − → : v 1 · · · v n ∈ L ( G ) iff for 1 ≤ i ≤ n , ∃ X i ∈ I ( v i ) , ∃ Y i ∈ Tp and ∃ s ′ ∈ Pr such that:  I ∗ for 1 ≤ i ≤ n, X i − → Y i    ME ∗ Y 1 , · · · , Y n − → s ′ s ′ ≤ Pr s    Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 12

Internal before Merge/External (example) q ′ o ll q l , q p l E 2 π l 2 , π 2 , p 2 o l → q ′ o ll , p l 2 π l 2 , π 2 , p 2 o l − M → q ′ o ll , p l 2 π l 2 π 2 , p 2 o l − I → q ′ o ll , p l 2 , p 2 o l − E M → q ′ o ll , o l E → q ′ o ll , , o l → q ′ − − − becomes: q ′ o ll q l , q p l E 2 π l 2 , π 2 , p 2 o l → q ′ o ll , p l 2 π l 2 , π 2 , p 2 o l − E → q ′ o ll , p l 2 , , p 2 o l − M → q ′ o ll , p l 2 , p 2 o l − E M E → q ′ o ll , o l → q ′ o ll , , o l − − − → q ′ Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 13

Partial composition E ∗ M − → and − → corresponding to the same couple of types are C − → : joined together in [ C ] (partial composition) : For k ∈ N , Γ , Xp ( n 1 ) · · · p ( n k ) , q ( n k +1) · · · q ( n 1 +1) E − → Γ , XY, ∆ , if Y, ∆ 1 1 k k q i ≤ Pr p i and n i is even or if p i ≤ Pr q i and n i is odd, for 1 ≤ i ≤ k . Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 14

Partial composition (example) q ′ o ll q l , q p l E 2 π l 2 , π 2 , p 2 o l → q ′ o ll , p l 2 π l 2 , π 2 , p 2 o l − E → q ′ o ll , p l 2 , , p 2 o l − M → q ′ o ll , p l 2 , p 2 o l − E M E → q ′ o ll , o l → q ′ o ll , , o l − − − → q ′ becomes: [1] q ′ o ll q l , q p l E 2 π l 2 , π 2 , p 2 o l → q ′ o ll , p l 2 π l , p 2 o l − 2 , π 2 C → q ′ o ll , p l 2 , p 2 o l − → q ′ o ll , , o l [0] E C → q ′ o ll , o l E − − − → q ′ Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 15

Partial composition (lemma) Lemma: For a list of types Γ and p ∈ Pr , Γ ME ∗ C ∗ − → p iff Γ − → p Corollary: v 1 · · · v n ∈ L ( G ) iff for 1 ≤ i ≤ n , ∃ X i ∈ I ( v i ) , ∃ Y i ∈ Tp and ∃ s ′ ∈ Pr such that:  I ∗ for 1 ≤ i ≤ n, X i − → Y i    C ∗ → s ′ Y 1 , · · · , Y n − s ′ ≤ Pr s    Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 16

Partial composition (example) q ′ o ll q l , q p l E 2 π l 2 , π 2 , p 2 o l → q ′ o ll , p l 2 π l 2 , π 2 , p 2 o l − E → q ′ o ll , p l 2 , , p 2 o l − M → q ′ o ll , p l 2 , p 2 o l − E M E → q ′ o ll , o l → q ′ o ll , , o l − − − → q ′ becomes: [1] [1] C q ′ o ll q l , qp l 2 π l , π 2 , p 2 o l q ′ o ll p l 2 π l , p 2 o l − → 2 , π 2 2 [2] C q ′ o ll p l 2 , p 2 o l − → C → q ′ − Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 17

Partial composition (parse tree) [1] [1] C q ′ o ll q l , qp l 2 π l , π 2 , p 2 o l q ′ o ll p l 2 π l , p 2 o l − → 2 , π 2 2 [2] C q ′ o ll p l 2 , p 2 o l − → C − → q ′ corresponds to the following parse tree: [2] [1] [1] q ′ o ll q l , qp l 2 π l , p 2 o l , π 2 2 Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 18

Parsing using partial composition Partial composition does not give a polynomial parsing algorithm because the result of partial composition is not bounded by the lexicon: [1] C Γ , q ′ o ll q l , qp l 2 π l Γ , q ′ o ll p l 2 π l − → , ∆ 2 , ∆ 2 3 , 3 4 Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 19

Majority partial composition C − → is a majority partial composition A partial composition @ (noted − → ) if the width of the result is less or equal to the maximum of the widths of the arguments A non majoritory partial composition: [1] C → Γ , q ′ o ll p l 2 π l Γ , q ′ o ll q l , qp l 2 π l , ∆ − 2 , ∆ 2 A majoritory partial composition: [2] @ → Γ , q ′ π l Γ , q ′ o ll q l , qo l π l − , ∆ 2 , ∆ 2 Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 20

Parsing using majority composition Main theorem: v 1 · · · v n ∈ L ( G ) iff for 1 ≤ i ≤ n , ∃ X i ∈ R I ∗ ( I )( v i ) , and ∃ s ′ ∈ Pr such that: � @ ∗ X 1 , · · · , X n − → s ′ s ′ ≤ Pr s I ∗ where R I ∗ ( I ) is the completion of I by − → Proof : property of (planar) proof nets. there exists a type in Γ that is only linked to its immediate neighour(s) q ′ o ll q l , qp l 2 π l 2 , π 2 , p 2 o l Workshop on Pregroups and Linear Logic – Chieti, 6-7 May 2005 – p. 21