De-Quantising Non-Halting Programs

Since many real-world problems arising in the fields of compiler optimisation, automatised software engineering, formal proof systems, and so forth are equivalent to the Halting Problem—the most notorious undecidable problem—there is a growing interest, not only academically, in understanding the problem better and in providing alternative solutions. Halting computations can be recognised by simply running them; the main difficulty is to detect non-halting programs. For each program length on a given machine, there is an uncomputable “critical time” after which no more programs of that length will halt. A quantum algorithm [7, 1] has been shown to solve the halting problem to any degree of certainty less than one and various experimental studies have proposed heuristics that apply to a majority of programs [4, 15]. Is it possible to have a classical effective way to describe this phenomenon? The aim of this paper is to provide a non-quantum analysis; 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.