What Percentage of Programs Halt?

摘要

Fix an optimal Turing machine U and for each n consider the ratio ρ_n~U 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 ρ_n~U. 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.