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It is well known that any language or function computable by a probabilistic algorithm is also computable deterministically. Here, we require that with probability $>1/2$, the algorithm outputs the correct answer (and therefore halts), but we allow the existence of infinite runs where the algorithm uses infinitely many random bits. Indeed, by $\sigma$-...
The standard proof that the halting problem $L$ is undecidable also gives an efficient algorithm for constructing an instance on which a given Turing machine $H$ fails to solve the halting problem. For any Turing machine $H$, let $M_H$ be a Turing machine implementing the following algorithm: "On input $\langle P \rangle$ where $P$ is a Turing machine, ...