From the authors’ abstract: “We introduce the notion of $ε$-universal prefix-free Turing machine ($ε$ is a computable real in $(0,1]$) and study its halting probability. The main result is the extension of the representability theorem for left-computable random reals to the case of $ϵ$-random reals: a real is left-computable $ε$-random iff it is the halting probability of an $ε$-universal prefix-free Turing machine. We also show that left-computable $ε$-random reals are provable $ε$-random in Peano arithmetic."
Reviewer: Liang Yu (Nanjing)