[Paper Review] Most Programs Stop Quickly or Never Halt

Cristian S. Calude, Michael Stay|ArXiv.org|Oct 26, 2006

Computability, Logic, AI Algorithms11 references4 citations

TL;DR

This paper proposes a probabilistic analysis of the Halting Problem by extending the probability space over both program space and time, using an a priori computable runtime distribution. It proves that for any k > 0, a time bound T can be effectively computed such that the probability an N-bit program halts after T, given it hasn't halted by then, is less than 2^{-k}, showing that long runtimes are effectively rare.

ABSTRACT

Since many real-world problems arising in the fields of compiler optimisation, automated 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. 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 2^{N+constant} has effectively zero density.

Motivation & Objective

• To provide a non-quantum probabilistic analysis of the Halting Problem by extending probability over both programs and time.
• To demonstrate that for any k > 0, a time bound T can be effectively computed such that the probability of halting after T is less than 2^{-k} given non-halting by T.
• To show that the set of halting programs can be decomposed into a computable set and a set of effectively vanishing probability.
• To argue that runtimes significantly longer than 2^{N+constant} have effectively zero density for N-bit programs.

Proposed method

• Postulate an a priori computable probability distribution over all possible runtimes to define a joint probability space over programs and time.
• Use the product measure of uniform program distribution and the chosen runtime distribution to compute conditional halting probabilities.
• Establish that the probability of halting after time T, given non-halting by T, is bounded by 2^{-k} for effectively computable T.
• Leverage algorithmic randomness to show that nonrandom, long runtimes are effectively rare.
• Prove that the set of halting programs is a disjoint union of a computable set and a set of probability < 2^{-k}.
• Demonstrate that programs not halting by time 2^{N+constant} cannot halt at later random times, due to algorithmic nonrandomness of such runtimes.

Experimental results

Research questions

• RQ1 Can we effectively compute a time bound T such that the probability of a non-halting N-bit program halting after T is less than 2^{-k}?
• RQ2 Can the set of halting programs be decomposed into a computable part and a part of effectively vanishing probability?
• RQ3 Are runtimes much longer than 2^{N+constant} effectively rare for N-bit programs?
• RQ4 Does the choice of runtime distribution matter in the limit, given computable distributions all yield similar rareness of long times?

Key findings

• For any k > 0, a time bound T can be effectively computed such that the conditional probability of halting after T, given non-halting by T, is less than 2^{-k}.
• The set of halting programs can be written as a disjoint union of a computable set and a set of probability effectively smaller than 2^{-k} for any k > 0.
• Runtimes longer than 2^{N+constant} have effectively zero density for N-bit programs, regardless of the choice of computable runtime distribution.
• The probability that an N-bit program never halts on a convergent Turing machine tends to one as N increases.
• Programs that have not halted by time 2^{N+constant} cannot halt at a later time that is algorithmically nonrandom.
• The result holds independently of the specific choice of computable runtime distribution, as long as most long times are effectively rare.