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Recognizer Algorithm Example (execution of Recognizer) Input string: abecd Input TAG: (α) S (β) a b e S NA S S ∗ NA c Gen. language: d {a n b n e c n d n | n ≥ 0} Input # Item added to chart Operation read [ γ,adr,pos,i, j, k, l,adj] 1. [ α, 0, la, 0,−,−,0, ⊥ ] initialization 2. [ β, 0, la, 0,−,−,0, ⊥ ] Pred 1 (1) 3. [ α, 1, la, 0,−,−,0, ⊥ ] Pred 2 (1) 4. [ β, 1, la, 0,−,−,0, ⊥ ] Pred 2 (2) a 5. [ β, 2, la, 0,−,−,1, ⊥ ] Scan 1 (4) a 6. [ β, 0, la, 1,−,−,1, ⊥ ] Pred 1 (5) a 7. [ β,2.1, la, 1,−,−,1, ⊥ ] Pred 2 (5) a 8. [ β, 1, la, 1,−,−,1, ⊥ ] Pred 2 (6) ab 9. [ β,2.2, la, 1,−,−,2, ⊥ ] Scan 1 (7) ab 10. [ β,2.2, lb, 2,−,−,2, ⊥ ] Pred 2 (9) ab 11. [ α, 1, la, 2,−,−,2, ⊥ ] Pred 3 (10) ab 12. [ β,2.1, la, 2,−,−,2, ⊥ ] Pred 3 (10) abe 13. [ α, 0, rb, 2,−,−,3, ⊥ ] Scan 1 (11) abe 14. [ β,2.2, rb, 2, 2, 3, 3, ⊥ ] Comp 1 (13, 10) abe 15. [ β,2.3, la, 1, 2, 3, 3, ⊥ ] Comp 2 (14, 9) abec 16. [ β, 2, rb, 1, 2, 3, 4, ⊥ ] Scan 1 (15) abec 17. [ β, 3, la, 0, 2, 3, 4, ⊥ ] Comp 2 (16, 5) abecd 18. [ β, 0, rb, 0, 2, 3, 5, ⊥ ] Scan 1 (17) abecd 19. [ β, 0, ra, 0, 2, 3, 5, ⊥ ] Comp 2 (18, 2) abecd 20. [ α, 0, rb, 0,−,−,5, ⊤ ] Adj(19, 13) abecd 21. [ α, 0, ra, 0,−,−,5, ⊥ ] Comp 3 (20, 1) 42 / 52
Complexity of Recognizer Given: n: length of the input string G = (N, T , I , A, S): input TAG m: maximal number of internal nodes per tree in I ∪ A Worst-case complexity can be reached by the Adjoin operation: [β, 0, ra, i, j, k, l, ⊥] [γ, adr, rb, j, p, q, k, ⊥] β ∈ SA(γ, adr) [γ, adr, rb, i, p, q, l, ⊤] At most: |A| possibilities for β |I ∪ A| possibilities for γ m possibilities for adr ( ) n + 2 possibilities per index i, · · · , q ∈ {0, · · · , n} ∪ {−} ⇒ Adjoin can be applied at most |A|·|I ∪ A|·m·(n + 2) 6 times. ⇒ Time complexity of Recognizer: O(|A|·|I ∪ A|·m·n 6 ) ⇒ For a specific grammar: O(n 6 ) 43 / 52
Page 1 and 2: Lexicalized Tree-Adjoining Grammars
Page 3 and 4: Context-Free Grammars Definition (C
Page 5 and 6: Tree-Substitution Grammars Example
Page 7 and 8: Outline 1 Why CFGs are not enough (
Page 9 and 10: Cross-Serial Dependencies Example (
Page 11 and 12: Lexicalization Example (Lexicalized
Page 13 and 14: Lexicalization Example (CFG which i
Page 15 and 16: Outline 1 Why CFGs are not enough (
Page 17 and 18: Initial Trees Definition (Initial T
Page 19 and 20: Substitution Definition (Substituti
Page 21 and 22: Adjunction Definition (Adjunction)
Page 23 and 24: Adjunction Constraints Given TAG G
Page 25 and 26: Lexicalization Example (strong lexi
Page 27 and 28: Further Formal Properties of TAL Tr
Page 29 and 30: TAG Parsing Parser: Given a string
Page 31 and 32: Recognizing Adjunction But the algo
Page 33 and 34: Dotted Tree We introduce the notion
Page 35 and 36: Chart Items The algorithm stores in
Page 37 and 38: Scan Operations Input string: c 1
Page 39 and 40: Complete Operations 1 2 3 [γ, adr,
Page 41: Recognizer Algorithm Algorithm (Rec
Page 45 and 46: Recognizing Substitution Recognizer
Page 47 and 48: LTAG-Spinal Parser LTAG-spinal: Rou
Page 49 and 50: LTAG-Spinal Parser Graphical repres
Page 51 and 52: Conclusion TAG: a grammar formalism

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