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How to simplify this context-free grammar into a similar one with no unit productions and no useless symbols?

$$ S \to ABA \\ A \to aAA \mid aBC \mid bB \\ B \to A \mid bB \mid Cb \\ C \to CC \mid cC $$

Can the simplification result in this CFG?

$$ S \to ABA \\ A \to aAA \\ A \to Bb \\ B \to b $$

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  • $\begingroup$ The first grammar can produce a $c$… $\endgroup$ – greybeard Jun 26 '20 at 22:30
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The context free language at the top seems to be empty, as there always will be a variable at any time while trying to derive a word! (this is because every transition is from a variable to some a few letters and at least one variable)

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