Question

If a Context-Free Grammar is given, is there a systematic way to find out the generated language and express it as a set using a descriptive and not analytic way, like L(G)={0^n.1^n|n?=1} (and not L(G)={01,0011,000111,…}) ?

I actually ask because if a CFG is given and there is a question like: “Find the language of the grammar.Prove/Justify your answer.” , then how can someone prove/justify his/her answer otherwise?

Answer 1

In general, no. For example, for an arbitrary context free grammar, the question of whether the language is equivalent to Sigma* is undecidable — and that’s about the simplest
description of a CFL one might imagine. Another undecidable question is whether
two context free grammars A and B define the same language, which doesn’t bode well
for the more general question of whether a grammar and some other alternate presentation define the same language.

In specific cases, such questions may be decidable — fortunately for formal language theory students! But in light of the above decidability results, you’re not going to find
a simple algorithm that gets you from a grammar, to a concise description of the sort usually presented in language theory textbooks. It’s more of a trial and error process, where
you use some intuition to think up a candidate description, then apply the more formal methods like building parse trees, or using closure properties or pumping lemmas, to prove or disprove the equivalence.