QUESTION$-1$: Recursion

$1)$ Consider the following programming language foo:

Foo$=\{$ S : S is the form of $2$ for some n $>0\}$

Thus, a string is considered in foo if it starts with a sequence of A$$s and is followed by a

sequence of twice as many of B$$s$.$ For example, AABBBB is legal string of foo. However,

BBAAAA or ABBABB and empty strings are not legal strings of foo. Note that foo only

deals with capital letters.

a$.$ Write the grammar of this language

b$.$ Write a recursive function to see if a string is legal or not.