Chapter 3 Context-Free Grammars, Context-Free Languages, Parse ...

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70 CHAPTER 3. CONTEXT-FREE GRAMMARS AND LANGUAGES two nodes di and dj, with2≤ i
3.12. OGDEN’S LEMMA 71 Ogden’s lemma or the pumping lemma can be used to show that certain languages are not context-free. The method is to proceed by contradiction, i.e., to assume (contrary to what we wish to prove) that a language L is indeed context-free, and derive a contradiction of Ogden’s lemma (or of the pumping lemma). Thus, as in the case of the regular languages, it would be helpful to see what the negation of Ogden’s lemma is, and for this, we first state Ogden’s lemma as a logical formula. For any nonnull string w: {1,...,n}→Σ, for any marking m: {1,...,n}→{0, 1} of w, for any substring y of w, wherew = xyz, with|x| = h and k = |y|, the number of marked occurrences in y, denoted as |m(y)|, is defined as |m(y)| = We will also use the following abbreviations: i=h+k i=h+1 m(i). nat = {0, 1, 2,...}, nat32 = {32, 33,...}, A ≡ w = uvxyz, B ≡|m(x)| ≥1, C ≡ (|m(u)| ≥1 ∧|m(v)| ≥1) ∨ (|m(y)| ≥1 ∧|m(z)| ≥1), D ≡|m(vxy)|
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Page 9 and 10: 3.3. NORMAL FORMS FOR CONTEXT-FREE
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Page 19 and 20: 3.7. LEAST FIXED-POINTS 51 where A,
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Page 33 and 34: 3.11. DERIVATIONS TREES 65 Tree t2
Page 35 and 36: 3.12. OGDEN’S LEMMA 67 the parse
Page 37: 3.12. OGDEN’S LEMMA 69 (ii) If sj
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Chapter 3 Context-Free Grammars, Context-Free Languages, Parse ...

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