Proving $L = \{ w : w \neq w^R \}$ over $\Sigma = \{0,1\}$ is CFL
Use induction.

For instance:


Define $\mathcal{L}_i(G) = \{ w \in \mathcal{L}(G) \mid |w| = i \}$ for $i \in \{ 0, 1, 2, \ldots \}$.
Prove that for all $w$ and $i &gt; 1$, $w \in \mathcal{L}_i(G)$ if and only if $\exists x \in \Sigma, v \in \mathcal{L}_{i-2}(G) : xvx = w$.