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I am trying to construct a context-free grammar for the following language: $$L=\{w_1cw_2:w_1\neq w_2^{R}\},$$ where words are over $\{a,b,c\}$.

I have tried to do this by taking the union of two sub-languages:

  1. The length of $w_1$ is not equal to the length of $w_2$.
  2. The lengths are equal.

For 1, I've already written the grammar. But I'm having difficulty with 2. In terms of a PDA I think non-determinism can solve this. However, what is the CFG for 2? If I can find it, I will union 1 and 2 to get the final solution.

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Here's a hint for part (2): work from the inside out. A string of the form $w_1cw_2$ where $|w_1|=|w_2|$ is "matched" ($w_1=w_2^R$) if and only if it is of the form

  • $c$, or
  • (any character $x$) ("matched" string) (character $x$)

Similarly, such a string is "unmatched" if and only it is of the form

  • (any character) ("unmatched" string) (any character), or
  • (any character $x$) ("matched" string) (any character $y\ne x$)

Thus, we can make a grammar where the "matched" strings are generated by $$ M\rightarrow c\mid aMa\mid bMb\mid cMc $$ We can make the "unmatched" strings ($w_1\ne w_2$) generated by similar productions, which I'll let you do, using the variable $U$. Finally, you want to generate "unmatched" strings, so $U$ will be the start variable for your grammar.

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