Introduction to Computational Linguistics

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20. Greibach Normal Form 80 The first reduction of grammars we look at is the elimination of superfluous symbols and rules. Let G = 〈S, A, N, R〉 be a CFG. Call X ∈ N reachable if G ⊢ ⃗α ⌢ X ⌢ ⃗β for some ⃗α and ⃗β. X is called completable if there is an ⃗x such that X ⇒ ∗ R ⃗x. (206) S → AB A → CB B → AB A → x D → Ay C → y In the given grammar A, C and D are completable, and S, A, B and C are reachable. Since S, the start symbol, is not completable, no symbol is both reachable and completable. The grammar generates no terminal strings. Let N ′ be the set of symbols which are both reachable and completable. If S N ′ then L(G) = ∅. In this case we put N ′ := {S} and R ′ := ∅. Otherwise, let R ′ be the restriction of R to the symbols from A∪ N ′ . This defines G ′ = 〈S, N ′ , A, R ′ 〉. It may be that throwing away rules may make some nonterminals unreachable or uncompletable. Therefore, this process must be repeated until G ′ = G, in which case every element is both reachable and completable. Call the resulting grammar G s . It is clear that G ⊢ ⃗α iff G s ⊢ ⃗α. Additionally, it can be shown that every derivation in G is a derivation in G s and conversely. Definition 28 A CFG is called slender if either L(G) = ∅ and G has no nonterminals except for the start symbol and no rules; or L(G) ∅ and every nonterminal is both reachable and completable. Two slender grammars have identical sets of derivations iff their rule sets are identical. Proposition 29 Let G and H be slender. Then G = H iff der(G) = der(H). Proposition 30 For every CFG G there is an effectively constructible slender CFG G s = 〈S, N s , A, R s 〉 such that N s ⊆ N, which has the same set of derivations as G. In this case it also follows that L B (G s ) = L B (G). □ Definition 31 Let G = 〈S, N, A, R〉 be a CFG. G is in Greibach (Normal) Form if every rule is of the form S → ε or of the form X → x ⌢ ⃗Y.
20. Greibach Normal Form 81 Proposition 32 Let G be in Greibach Normal Form. If X ⊢ G ⃗α then ⃗α has a leftmost derivation from X in G iff ⃗α = ⃗y ⌢ ⃗Y for some ⃗y ∈ A ∗ and ⃗Y ∈ N ∗ and ⃗y = ε only if ⃗Y = X. The proof is not hard. It is also not hard to see that this property characterizes the Greibach form uniquely. For if there is a rule of the form X → Y ⌢ ⃗γ then there is a leftmost derivation of Y ⌢ ⃗γ from X, but not in the desired form. Here we assume that there are no rules of the form X → X. Theorem 33 (Greibach) For every CFG one can effectively construct a grammar G g in Greibach Normal Form with L(G g ) = L(G). Before we start with the actual proof we shall prove some auxiliary statements. We call ρ an X–production if ρ = X → ⃗α for some ⃗α. Such a production is called left recursive if it has the form X → X ⌢ ⃗β. Let ρ = X → ⃗α be a rule; define R −ρ as follows. For every factorization ⃗α = ⃗α ⌢ 1 Y⌢ ⃗α 2 of ⃗α and every rule Y → ⃗β add the rule X → ⃗α ⌢⃗ 1 β ⌢ ⃗α 2 to R and finally remove the rule ρ. Now let G −ρ := 〈S, N, A, R −ρ 〉. Then L(G −ρ ) = L(G). We call this construction as skipping the rule ρ. The reader may convince himself that the tree for G −ρ can be obtained in a very simple way from trees for G simply by removing all nodes x which dominate a local tree corresponding to the rule ρ, that is to say, which are isomorphic to H ρ . This technique works only if ρ is not an S–production. In this case we proceed as follows. Replace ρ by all rules of the form S → ⃗β where ⃗β derives from ⃗α by applying a rule. Skipping a rule does not necessarily yield a new grammar. This is so if there are rules of the form X → Y (in particular rules like X → X). Lemma 34 Let G = 〈S, N, A, R〉 be a CFG and let X → X ⌢ ⃗α i , i < m, be all left recursive X–productions as well as X → ⃗β j , j < n, all non left recursive X– productions. Now let G 1 := 〈S, N ∪ {Z}, A, R 1 〉, where Z N ∪ A and R 1 consists of all Y–productions from R with Y X as well as the productions (207) X → ⃗β j j < n, Z → ⃗α i i < m, X → ⃗β ⌢ j Z j < n, Z → ⃗α⌢ i Z i < m. Then L(G 1 ) = L(G).
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Introduction to Computational Lingu
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2. Practical Remarks Concerning OCa
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3. Welcome To The Typed Universe 5
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4. Function Definitions 11 4 Functi
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5. Modules 13 can find out why the
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5. Modules 15 let length l = length
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6. Sets and Functors 17 write it do
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7. Hash Tables 19 to actually see t
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8. Combinators 21 can be used on an
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9. Objects and Methods 23 of type
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10. Characters, Strings and Regular
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10. Characters, Strings and Regular
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Page 37 and 38: 12. Finite State Automata 37 Suppos
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Page 41 and 42: 12. Finite State Automata 41 The pr
Page 43 and 44: 12. Finite State Automata 43 Proof.
Page 45 and 46: 13. Complexity and Minimal Automata
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Page 57 and 58: 15. Finite State Transducers 57 And
Page 59 and 60: 16. Finite State Morphology 59 Then
Page 61 and 62: 17. Using Finite State Transducers
Page 67 and 68: 18. Context Free Grammars 67 18 Con
Page 69 and 70: 18. Context Free Grammars 69 symbol
Page 71 and 72: 18. Context Free Grammars 71 which
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Page 77 and 78: 20. Greibach Normal Form 77 The red
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Page 89 and 90: 22. Shift-Reduce-Parsing 89 differe
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Page 93 and 94: 23. Some Metatheorems 93 This proof
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Introduction to Computational Linguistics

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