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In my exam, i was supposed to write all pumping lemma conditions. that exactly what i did :

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a friend told me that there is some errors but i can't find them... Can some one help please ? what are the errors & why ?

templatetypedef
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Mooh
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  • Can’t your friend (or [Wikipedia](http://en.wikipedia.org/wiki/Pumping_lemma_for_regular_languages)) do that? – Gumbo Feb 06 '11 at 23:03

2 Answers2

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If I remember correctly, the conditions need to be as follows:

  • |xy| ≤ p
  • |y| ≥ 1
  • xyiz ∈ L, i ≥ 0

So y must not be empty and y can be repeated zero or more times.

Gumbo
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You're almost right, but the pumping lemma requires that |xy| ≤ p, not |xz| ≤ p. The idea is that the string is split into some initialization (x), steady-state (y), and tail (z) and that the initialization plus steady state logic is some length.

templatetypedef
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  • the pumping lemma also requires |y| > 0 and i >= 0... why ? – Mooh Feb 07 '11 at 17:17
  • The idea is that you can take any string that's sufficiently long and use it to generate an infinite family of strings that are also in the language. It's based on the fact that if you take a DFA for the language and take any string that's longer than the number of states in the DFA, then the string must go around a cycle in the DFA. Taking the part of the string that goes around the cycle and repeating it arbitrarily many times gives back a family of legal strings. We need |y| > 0 so that the cycle isn't empty and i >= 0 so that we repeat the at string zero or more times. – templatetypedef Feb 07 '11 at 19:59