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$L=\{uvw \mid u,v,w \in\{0,1\}^*,|u|=|v|=|w| \wedge u\neq w\} $

Any help would be appreciated.

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The idea is that a word $uvw\in L$ can be written as $x\alpha y\beta z$ with $x, y, z \in \{0,1\}^*$, $\alpha, \beta\in \{0,1\}$, verifying the following conditions:

  • $\alpha \neq \beta$
  • $|x\alpha z| = |u| = |v| = |w|$. This implies $|x| + |z| + 1 = |u|$ and then $|y| = 3|u| - 2 - |x|-|z| = 2|u| - 1 = 2 |x| + 2|z| + 1$.

That means that if we use a grammar symbol $X$ to create $x$ and part of $y$, centered on the position of $\alpha$, informally we will need to push "one symbol on the left and two on the right" of $X$. The converse is done for creating part of $y$ and $z$, centered on $\beta$. And for the ending $+1$, we will add an additional terminal symbol from the starting rule.

Now we only need to do it for $\alpha = 0$ and $\beta = 1$, and the other way around.

In details, we can use the following grammar:

$S \rightarrow X_1AZ_0 \mid X_0AZ_1$
$X_0 \rightarrow AX_0AA \mid 0$
$X_1 \rightarrow AX_1AA \mid 1$
$Z_0 \rightarrow AAZ_0A \mid 0$
$Z_1 \rightarrow AAZ_1A \mid 1$
$A\rightarrow 0\mid 1$

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