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How do I construct a context-free grammar for the following language?

$$L = \{w\#x \mid w, x \in \{0, 1\}^*, |w| = |x|, w \neq x^R \} $$

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A word belongs to your language if it has one of the following forms: $$ \Sigma^i 0 \Sigma^j \# \Sigma^j 1 \Sigma^i \\ \Sigma^i 1 \Sigma^j \# \Sigma^j 0 \Sigma^i $$ Given this, you should be able to construct a context-free grammar for $L$.

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  • $\begingroup$ So would you say something like $ \\ S -> A0B\#B1A | A1B\#B0A \\ A -> 0|1|epsilon \\ B ->0|1|epsilon$ $\endgroup$ – George Walker Apr 10 '20 at 16:45
  • $\begingroup$ Your grammar generates palindromes: $S \to A0B\#B1A \to 10B\#B1A \to 10\#B1A \to 10\#01A \to 10\#01$. It also doesn't guarantee that $|w| = |x|$: $S \to A0B\#B1A \to 0B\#B1A \to 0\#B1A \to 0\#1A \to 0\#10$. $\endgroup$ – Yuval Filmus Apr 10 '20 at 16:51
  • $\begingroup$ From now on, I will not check any more "answers". The hints in my answer should be enough for you to solve the exercise. $\endgroup$ – Yuval Filmus Apr 10 '20 at 16:51

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