I'm tasked with transforming this regular expression $((0+1)(0+1)^*(0+1))^*$ into a context free grammar. As an added constraint I'm must do so with a maximum of 2 variables. This is what I did :
S -> VWV | S | ɛ
W -> 0 | 1 | W | ɛ
V -> 0 | 1
Is this considered as using 2 variables or 3 variables? Do we count the starting S as a variable?
Is the context free grammar that I derived correct? If not what would be a correct way of going about it?
Note that the written regular expression is equivalent to $$\varepsilon+(0+1)^2(0+1)^*$$
So, a valid context free grammar is $$S\to\varepsilon\mid00X\mid01X\mid10X\mid11X$$$$X\to\varepsilon\mid X0\mid X1$$
Your example is $3$ variables.
Edit:
To see the equivalence of the regular expressions, it's easy to see that the empty string is accepted by both. Moreover, if a given string is not empty and is one character long, it is not accepted by either regular expression.
If a given string is 2 or more characters long, then it clearly is accepted by $(0+1)(0+1)^*(0+1)$, and also by $(0+1)^2(0+1)^*$.
So, the expressions agree on all inputs.