BNF grammar that generates language L

Question

I spend a lot of time but still can't understand how to solve this. I have the answer. Can you tell me how he come up with it?

Is there a specific rules or standard construct I can follow? I know the rules for union and concatenation but not when it's reversed.

Answer 1

If x is a (possibly improper) substring of y, then x^r c y can be written as x^R c wxz where w and z are arbitrary strings over (a+b)*.

Now we can try to "slice" the strings in this language to see how they can be generated by CFGs. The non-regular part of this is making sure that x^R and x come out right; the rest is trivial. So we need some rules like:

S -> aXa | bYb | c

This gives us x^R c x. We are missing the w and the z. The w we can add by letting any arbitrary string come after c:

S -> aSa | bSb | cA
A -> aA | bA | e

The z we can get by changing S to T and adding a new S that produces T followed by anything:

S -> TA
T -> aTa | bTb | cA
A -> aA | bA | e

All we neglected to ensure was that |x| > 1. We do that by changing T -> cA to two productions T -> acAa | bcAb. This yields the suggested grammar (Not the only right one, just one grammar that works) and explains how we got there.