A grammar for the complement of language $L=\{a^{t+3}b^t:t \ge 0\}$ Assume that the  language $L=\{a^{t+3}b^t:t \ge 0\}$ is given.
Q1 : How can we write a grammar for the complement of this language?
Q2 : Assume that $L'=\{a^nb^m:n\ge0,m\gt n\}$ is given. Can you write a grammar for $L'\cap L$?
Note: I can easily write a grammar like this for the language $L$ itself.  
S->aaaP
P->aPb
P->E

Also, I can easily write the grammar for $L'$:
S->PbX
X->bX
X->E
P->aPb
P->E

In these grammars, by E , i mean the empty string.
The first problem is the complementation and the second one is that i don't know how strings are made in the language $L' \cap L$ .  
Thanks in advance.
 A: Corrected HINT: Everything matching the regular expression 
$$b^*+ab^*+aab^*+aaaaa^*+(a+b)^*ba(a+b)^*\tag{1}$$
is in the complement of $L$, and it’s easy to write a grammar for this part of the complement. Every string in the rest of the complement begins with $aaa$, contains at least one $b$, and does not contain the string $ba$, so it must be $a^nb^m$ for some $n\ge 3$ and $m\ge 1$. You can handle this part of the complement by breaking it into two parts: 


*

*strings of the form $a^{n+3}b^m$ such that $1\le m<n$, those that have too few $b$s; and  

*strings that begin $a^{n+3}b^m$ such that $0\le n<m$, those that have too many $b$s.  


Those in the first part must begin with at least $5$ $a$s; you can get them with
$$\begin{align*}
&S\to aaaaaFbX\\
&F\to aFb\mid A\\
&A\to aA\mid\epsilon\\
&X\to aY\mid a\mid\epsilon\\
&Y\to aY\mid bY\mid\epsilon
\end{align*}$$
See if you can work out a grammar for the other part and combine it with a grammar for the regular expression $(1)$ to get the desired grammar for the complement of $L$.
