Past exam of the computer science course of the University of Oxford / 2015 / Models of computation / 3 / c / i

The formal language is a context-free language. A context-free grammar with start nonterminal symbol $S$ isThe production rule for $A$ nests one $a$ on the left and one $b$ on the right on each recursive use, so $A$ generates exactly $a^m{b}^m$ for $m\ge 0$. Similarly, $B$ generates exactly $b^n{c}^n$ for $n\ge 0$. The start production rule concatenates these words, yieldingConversely, every choice of nonnegative integers $m,n$ is achieved by using the recursive production rules exactly $m,n$ times before their epsilon productions. Hence this context-free grammar generates precisely the required formal language, including all cases where either count is zero.