What Percentage of Programs Halt?

International audienceFix an optimal Turing machine U and for each n consider the ratio ρ^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 ρUn. 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