Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 4I / ii / Solution

The language $\{0^n{1}^n:n\ge 0\}$ is not regular. For pumping length $p$, take $0^p{1}^p$. Any permitted pumped block lies entirely among the initial zeros. Pumping it down changes the number of zeros without changing the number of ones, producing a word outside the language and contradicting the pumping lemma.

Solved by gpt-5.6-sol high.