According to pumping lemma for context free languages : Let L be an infinite context free language, then there exists some positive integer m such that any \(w \in L\) with \(|w| \ge m\) can be decomposed as \(w = uvxyz\)
with \(|vxy| \le m\), and \(|vy| \ge 1\), such that \(uv^i xy^i z \in L\) for all \(i = 0, 1, 2, \ldots\)
The pumping lemma for context free languages comes from the parse tree: a long enough string forces some variable to repeat on one root-to-leaf path, and the part of the tree between the two copies can be repeated or removed.
Choosing the lowest repetition keeps the pumped window short, which gives \(|vxy| \le m\). A long window \(|vxy| \ge m\) is not what the lemma guarantees, so the conditions with \(|vxy| \ge m\) are wrong.
The two repeated pieces cannot both be empty, otherwise pumping would change nothing; this is the condition \(|vy| \ge 1\). The condition \(|vy| \le 1\) has the inequality reversed.
Pumping works for every \(i \ge 0\): \(i = 0\) deletes v and y, \(i = 2, 3, \ldots\) repeats them. Stopping at \(i = 2\) and leaving out \(|vy| \ge 1\) makes the statement incomplete.
So the complete statement is: \(|vxy| \le m\), \(|vy| \ge 1\) and \(uv^i xy^i z \in L\) for all \(i = 0, 1, 2, \ldots\).
Hence, the answer is with \(|vxy| \le m\), and \(|vy| \ge 1\), such that \(uv^i xy^i z \in L\) for all \(i = 0, 1, 2, \ldots\).
Similar Questions
Consider a language A defined over the alphabet \(\Sigma = \{0, 1\}\) as \(A = \left\{0^{\lfloor n/2 \rfloor} 1^{n} : n \gt = 0\right\}\).
The expression \(\lfloor n/2 \rfloor\) means the floor of n/2, or what you get by rounding n/2 down to the nearest integer.
Which of the following is not an example of a string in A ?
Given the following two languages :
\(L_{1} = \{a^{n}\, b\, a^{n} \mid n \gt 0\}\)
\(L_{2} = \{a^{n}\, b\, a^{n}\, b^{n+1} \mid n \gt 0\}\)
Which of the following is correct ?
Which of the following are context free language ?
(A) {wi xj yk zl | i + k = j + l, where i, j, k, l ≥ 0}
(B) {wi xj yk zl | i = j and k = l, where i, j, k, l ≥ 0}
(C) {wi xj yk zl | i = j = k and k ≠ l, where i, j, k, l ≥ 0}
(D) {wi xj yk zl | i = j = k + l, where i, j, k, l ≥ 0}
(E) {wi xj yk zl | i = j = l and k ≠ l, where i, j, k, l ≥ 0}
Choose the correct answer from the options given below :
If L1 and L2 are context free languages, which of the following is True about L1∩L2 ?
In Pumping Lemma for context free languages, to say a language is satisfying pumping lemma, what is the minimum length of ‘vx’ together, if you consider the string as ‘uvwxy’?
Let L = {0n1n|n ≥ 0} be a context free language.
Which of the following is correct ?
Given the following two statements :
A. L = {w|na(w) = nb(w)} is deterministic context free language, but not linear.
B. L = {an bn} ∪ {an b2n} is linear, but not deterministic context free language.
Which of the following options is correct ?
Given the following two languages :
L1 = {an bn | n ≥ 0, n ≠ 100}
L2 = {w ∈ {a, b, c}*| na(w) = nb(w) = nc(w)}
Which of the following options is correct ?
Important Questions from Context Free Languages
- Let $\Sigma = \{a, b, c, d\}$ and let $L = \{a^ib^jc^kd^\ell \mid i, j, k, \ell \ge 0\}$.
Which of the following constraints ensure(s) that the language $L$ is context-free? Consider a language A defined over the alphabet \(\Sigma = \{0, 1\}\) as \(A = \left\{0^{\lfloor n/2 \rfloor} 1^{n} : n \gt = 0\right\}\).
The expression \(\lfloor n/2 \rfloor\) means the floor of n/2, or what you get by rounding n/2 down to the nearest integer.
Which of the following is not an example of a string in A ?
Given the following two languages :
\(L_{1} = \{a^{n}\, b\, a^{n} \mid n \gt 0\}\)
\(L_{2} = \{a^{n}\, b\, a^{n}\, b^{n+1} \mid n \gt 0\}\)
Which of the following is correct ?
Which of the following are context free language ?
(A) {wi xj yk zl | i + k = j + l, where i, j, k, l ≥ 0}
(B) {wi xj yk zl | i = j and k = l, where i, j, k, l ≥ 0}
(C) {wi xj yk zl | i = j = k and k ≠ l, where i, j, k, l ≥ 0}
(D) {wi xj yk zl | i = j = k + l, where i, j, k, l ≥ 0}
(E) {wi xj yk zl | i = j = l and k ≠ l, where i, j, k, l ≥ 0}
Choose the correct answer from the options given below :
If L1 and L2 are context free languages, which of the following is True about L1∩L2 ?
