Proving that a CFG generates a language

Question

Is a suitable way to prove that any given CFG generates (or not) any given language to draw its total language tree?

What if the tree is infinite? What would then be a better way to prove that a given CFG generates a given language?

Answer 1

Suppose that we have a context-free grammar $G$ and a set of words $S$, and we would like to prove that the language $L(G)$ generated by $G$ is precisely the set $S$. The most direct way of doing this is to prove $L(G) \subseteq S$ and $S \subseteq L(G)$, which amounts to showing:

1. For every $x \in L(G)$ we have $x \in S$.
2. For every $y \in S$ we have $y \in L(G)$.

How this is done depends on how the set $S$ is given, so there is no further, more specific advice that I can offer, except that drawing trees is not really going to result in a proof.

Answer 2

If your language $L$ is, as you state in the comment, given as a regular expression, you can prove that the grammar $G$ generates it in two steps.

1. Prove that $G$ is regular
2. Show that the two automatons (from $G$ and from the expression for $L$) recognizing the language are the same.