Past exam of the computer science course of the University of Oxford / 2015 / Models of computation / 3 / b / ii / Solution

The formal language is not a regular language. Let $K$ denote it and consider the infinitely many prefixesFor any $i\ne j$, append the word $z_i=(bc{)}^i$. Then $u_i{z}_i\in K$, whereas $u_j{z}_i\notin K$: the latter word has $j$ occurrences of $a$ and $i$ occurrences of $c$, and every word of $K$ has equal numbers of those two symbols.

Thus $u_i$ and $u_j$ are not Myhill-Nerode equivalent. There are infinitely many Myhill-Nerode equivalence classes, so the Myhill-Nerode theorem rules out a finite-state automaton recognizing $K$. Equivalently, if two such prefixes left a deterministic finite automaton in the same automaton state, appending $z_i$ could not distinguish their acceptance, giving the same contradiction.