Abstract:
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.