CFG for the given language $L = \{~ a^\ell a^n b^m c^m d^n e^\ell ~|~ \ell,n,m \geq 0 \}$

Question

I am writing this CFG to solve the problem:

$S \to ASBSC$

$A \to aAe ~|~ ε$

$B \to aBd ~|~ ε$

$C \to bcC ~|~ ε$

Is this correct or not?

Answer 1

Strings in your grammar could potentially terminate with a sequence of $bc$s because of the following, which is not desired.

$S \rightarrow ASBSC$

$C \rightarrow bcC \ | \ \epsilon$

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The correct grammar in this case would be:

$S \rightarrow aSe \ | \ A \ | \ \epsilon$

$A \rightarrow aAd \ | \ B \ | \ \epsilon$

$B \rightarrow bBc \ | \ \epsilon$