Which language is produced by this grammar?

Question

I was asked to define the language which is produced by the following grammar:

$G = (V, \Sigma, S, P)$

$V = \{S, A, B\}$

$\Sigma = \{a, b, c\}$

\begin{equation} P= \begin{cases} S \rightarrow cA | bB\\ A \rightarrow c \\ B \rightarrow aB|b \end{cases} \end{equation}

Obviously, you can produce $c^n$ and $b^n$, but since we can also produce $ccbab$ for example, I do not know how to find a formular for the language $L(G)$

Any help would be really appreciated!

Answer 1

Let's start with the nonterminal $B$. Since all productions of the form $B \to \alpha$ involve only $B$ (and no other nonterminal), we can easily find $L(B) = a^*b$ using Arden's rule. We also have $L(A) = c$. Finally, $$ L(S) = L(cA) \cup L(bB) = cc + ba^*b. $$