Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 124 / 4 / ii

First suppose $L\in \mathbf{RP}\cap \mathbf{co}\text{-}\mathbf{RP}$. Run the RP algorithm for $L$ and the RP algorithm for its complement with fresh random bits. If the first accepts, output one; if the second accepts, output zero; otherwise repeat. Neither output can be wrong, and on every input the appropriate algorithm accepts in each round with probability at least $1/2$. The number of rounds is dominated by a geometric distribution of mean two, so this is an always-correct algorithm with polynomial expected running time.

Conversely, let $T$ be always correct when it halts and have expected running time at most $p(n)$. Run it for $2p(n)$ steps. Markov inequality givesAccept exactly when $T$ halts and outputs one; this is an RP algorithm for $L$. Accepting exactly when it halts and outputs zero is an RP algorithm for the complement. Thereforeis equivalent to zero-error expected polynomial time.