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Show that { xy ∣ |x| = |y|, x ≠ y } is context-free
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Problem
I remember coming across the following question about a language that supposedly is context-free, but I was unable to find a proof of the fact. Have I perhaps misremembered the question?
Anyway, here's the question:
Show that the language $L = \{xy \mid |x| = |y|, x\neq y\}$ is context free.
Anyway, here's the question:
Show that the language $L = \{xy \mid |x| = |y|, x\neq y\}$ is context free.
Solution
Claim: $L$ is context-free.
Proof Idea: There has to be at least one difference between the first and second half; we give a grammar that makes sure to generate one and leaves the rest arbitrary.
Proof: For sake of simplicity, assume a binary alphabet $\Sigma = \{a,b\}$. The proof readily extends to other sizes. Consider the grammar $G$:
$\qquad\begin{align}
S &\to AB \mid BA \\
A &\to a \mid aAa \mid aAb \mid bAa \mid bAb \\
B &\to b \mid aBa \mid aBb \mid bBa \mid bBb
\end{align}$
It is quite clear that it generates
$\qquad \mathcal{L}(G) = \{ \underbrace{w_1}_k x \underbrace{w_2v_1}_{k+l}y\underbrace{v_2}_l \mid |w_1|=|w_2|=k, |v_1|=|v_2|=l, x\neq y \} \subseteq \Sigma^*;$
the suspicious may perform a nested induction over $k$ and $l$ with case distinction over pairs $(x,y)$.
The length of a word in $\mathcal{L}(G)$ is $2(k+l+1)$. The letters $x$ and $y$ occur on positions $k+1$ and $2k+l+2$, respectively. When we split the word in half, i.e. after $(k+l+1)$ letters, then the first half contains the letter $x$ on position $k+1$ and the second half has the letter $y$ on position $k+1$.
Therefore, $x$ and $y$ have the same position (in their respective half), which implies $\mathcal{L}(G) = L$ because $G$ imposes no other restrictions on its language.
The interested reader may enjoy two follow-up problems:
Exercise 1: Come up with a PDA for $L$!
Exercise 2: What about $\{xyz \mid |x|=|y|=|z|, x\neq y \lor y \neq z \lor x \neq z\}$?
Proof Idea: There has to be at least one difference between the first and second half; we give a grammar that makes sure to generate one and leaves the rest arbitrary.
Proof: For sake of simplicity, assume a binary alphabet $\Sigma = \{a,b\}$. The proof readily extends to other sizes. Consider the grammar $G$:
$\qquad\begin{align}
S &\to AB \mid BA \\
A &\to a \mid aAa \mid aAb \mid bAa \mid bAb \\
B &\to b \mid aBa \mid aBb \mid bBa \mid bBb
\end{align}$
It is quite clear that it generates
$\qquad \mathcal{L}(G) = \{ \underbrace{w_1}_k x \underbrace{w_2v_1}_{k+l}y\underbrace{v_2}_l \mid |w_1|=|w_2|=k, |v_1|=|v_2|=l, x\neq y \} \subseteq \Sigma^*;$
the suspicious may perform a nested induction over $k$ and $l$ with case distinction over pairs $(x,y)$.
The length of a word in $\mathcal{L}(G)$ is $2(k+l+1)$. The letters $x$ and $y$ occur on positions $k+1$ and $2k+l+2$, respectively. When we split the word in half, i.e. after $(k+l+1)$ letters, then the first half contains the letter $x$ on position $k+1$ and the second half has the letter $y$ on position $k+1$.
Therefore, $x$ and $y$ have the same position (in their respective half), which implies $\mathcal{L}(G) = L$ because $G$ imposes no other restrictions on its language.
The interested reader may enjoy two follow-up problems:
Exercise 1: Come up with a PDA for $L$!
Exercise 2: What about $\{xyz \mid |x|=|y|=|z|, x\neq y \lor y \neq z \lor x \neq z\}$?
Context
StackExchange Computer Science Q#307, answer score: 46
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