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.

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    $\begingroup$ Oh, that's a good one! <3 $\endgroup$ – Raphael Mar 13 '12 at 13:01

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 $(x,y)$. Now, $w_2$ and $v_1$ commute (intuitively speaking, $w_2$ and $v_1$ can exchange symbols because both contain symbols chosen independently from the rest of the word). 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 $\{xyz \mid |x|=|y|=|z|, x\neq y \lor y \neq z \lor x \neq z\}$?

  • $\begingroup$ If we use this grammar, we can generate a string like: $ S \rightarrow AB $ $ A \rightarrow a $ $ B \rightarrow bBa, then B \rightarrow b $ After that, we got a S as abba! This is not equal to the raw language L, is there a mistake here? $\endgroup$ – George.Zhao Mar 15 at 7:28
  • $\begingroup$ @George.Zhao I don't follow. Cleary, $abba \in L$ with $x = ab$ and $y=ba$? $\endgroup$ – Raphael Mar 15 at 9:14

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