What's grammar for a^n b^n c^n d^n

Question

What wiil be grammar rules for the language L={a^n b^n c^n d^n; n>0}

Answer 1

May I propose the following indexed grammar $G= ( \{S,A,C\}, \{a,b,c,d\}, \{f\}, S )$:

• $S[\sigma] \to S[f\sigma]$

• $S[\sigma] \to A[\sigma]\, C[\sigma]$

• $A[f\sigma] \to a\, A[\sigma]\, b$, $\quad A[\,] \to \varepsilon$

• $C[f\sigma] \to c\, C[\sigma]\, d$, $\quad C[\,] \to \varepsilon$