What Percentage of Programs Halt? [post]

Laurent Bienvenu, Damien Desfontaines, Alexander Shen
2017 unpublished
Fix an optimal Turing machine $U$ and for each $n$ consider the ratio $\rho^U_n$ of the number of halting programs of length at most $n$ by the total number of such programs. Does this quantity have a limit value? In this paper, we show that it is not the case, and further characterise the reals which can be the limsup of such a sequence $\rho^U_n$. We also study, for a given optimal machine $U$, how hard it is to approximate the domain of $U$ from the point of view of coarse and generic computability.
doi:10.31219/osf.io/dfmbk fatcat:qopdmd77fzdofbs66dls3n44q4