Applying production rules:

• $\Rightarrow$: If we have $w = uxv$ and we also have rule $x \rightarrow y$, then we have $w \Rightarrow uyv$, by replacing $x$ by $y$.
• $\stackrel{\ast}{\Rightarrow}$: If $w_1 \Rightarrow w_2 \Rightarrow \ldots \Rightarrow w_n$, we say that $w_1$ deriving $w_n$ and write it as
  
  $$w_1 \stackrel{\ast}{\Rightarrow} w_n.$$
  
  
  Here $\ast$ indicates that an unspecified number of steps can be taken to derive $w_n$ from $w_1$.

Let $G = (V, T, S, P)$ be a grammar. Then the set

$$L(G) = \{ w \in T^*: S \stackrel{\ast}{\Rightarrow} w \}$$

is the language generated by $G$.



Let $n_a(w)$ denote the number of a's in the string $w$.

Two grammars are equivalent if they generate the same language.