Most Programs Stop Quickly or Never Halt

The aim of this paper is to provide a probabilistic, but non-quantum, analysis of the Halting Problem. Our approach is to have the probability space extend over both space and time and to consider the probability that a random N -bit program has halted by a random time. We postulate an a priori computable probability distribution on all possible runtimes and we prove that given an integer k > 0, we can effectively compute a time bound T such that the probability that an N -bit program will eventually halt given that it has not halted by T is smaller than 2−k. We also show that the set of halting programs (which is computably enumerable, but not computable) can be written as a disjoint union of a computable set and a set of effectively vanishing probability. Finally, we show that “long” runtimes are effectively rare. More formally, the set of times at which an N -bit program can stop after the time 2N+ constant has effectively zero density.