Question

For reference, here is the CFL pumping lemma game (for language A): 1. Adversary picks a number $p \ge 0$. 2. You pick a string $s \in A$, such that $|s| \ge p$. 3. Adversary breaks $s$ into $s = uvwxy$, such that $|vwx| \le p$ and $|vx| > 0$. 4. You pick a number $i \ge 0$. If $uv^iw^ix^iy \notin A$, then you win. If you can describe a strategy in which you always win, then A is not context-free. 1. Show that the following languages are not context-free. You can use pumping lemma or closure properties or a combination. (a) \{xcy | x, y \in \{0, 1\}^* and bin(x) + 1 = bin(y)\} Example: a string like 100111c101000 would be in this language. (b) \{a$^n$b$^m$c$^k$ | k = mn\}

          For reference, here is the CFL pumping lemma game (for language A):
1. Adversary picks a number $p \ge 0$.
2. You pick a string $s \in A$, such that $|s| \ge p$.
3. Adversary breaks $s$ into $s = uvwxy$, such that $|vwx| \le p$ and $|vx| > 0$.
4. You pick a number $i \ge 0$. If $uv^iw^ix^iy \notin A$, then you win.
If you can describe a strategy in which you always win, then A is not context-free.
1. Show that the following languages are not context-free. You can use pumping lemma or
closure properties or a combination.
(a) \{xcy | x, y \in \{0, 1\}^* and bin(x) + 1 = bin(y)\}
Example: a string like 100111c101000 would be in this language.
(b) \{a$^n$b$^m$c$^k$ | k = mn\}

$For reference, here is the CFL pumping lemma game (for language A): 1. Adversary picks a number p ≥ 0. 2. You pick a string s ∈ A, such that |s| ≥ p. 3. Adversary breaks s into s = uvwxy, such that |vwx| ≤ p and |vx| > 0. 4. You pick a number i ≥ 0. If uv^iw^ix^iy ∉ A, then you win. If you can describe a strategy in which you always win, then A is not context-free. 1. Show that the following languages are not context-free. You can use pumping lemma or closure properties or a combination. (a) {xcy | x, y ∈{0, 1}^* and bin(x) + 1 = bin(y)} Example: a string like 100111c101000 would be in this language. (b) {a^nb^mc^k | k = mn}$

Added by Jacqueline P.

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