Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 12J / e / Solution

Suppose a zero-register machine decided $L=\{a^n{b}^n:n>0\}$, and choose $k\ne \ell$ as in part (d). Take $m\ge \max \{k,\ell \}$ and setThenwhereasPart (d) puts both computations into the identical configuration $(q,x)$. Determinism forces their subsequent computations and outputs to agree, a contradiction. ThereforeThis is the one-register separation for equal block lengths.