长度为3的字符串的CFG

Question

I'm trying to create context free grammar for the following language:

L={w∈{a,b}* | w is of the form a^nxb^n+2, where x is any string of length 3 in {a,b}*}

(a is raised to the power of n, and b is raised to the power of n+2)

My solution this far is ( _ is epsilon):

S -> aSbb | T | _

T -> aT | bT | _

However this is not correct as it for example accepts string abbb. My problem is that I don't understand how to make the T part of the grammar, how can I limit the string to be size of 3? It would not make any sense to list all the strings of length 3 in {a,b}*, or is it the only way?

Also I'm not 100% confident if the aSbb is correct, because I think my solution is now a^nxb^2n, or am I wrong?

Answer 1

The correct CFG for the language is:

S -> aSb | T  
T -> UUUbb  
U -> a | b

Thanks everybody for help!