Asymptotic Proportion of Hard Instances of the Halting Problem
Abstract
Although the halting problem is undecidable, imperfect testers that fail on some instances are possible. Such instances are called hard for the tester. One variant of imperfect testers replies "I don't know" on hard instances, another variant fails to halt, and yet another replies incorrectly "yes" or "no". Also the halting problem has three variants: does a given program halt on the empty input, does a given program halt when given itself as its input, or does a given program halt on a given input. The failure rate of a tester for some size is the proportion of hard instances among all instances of that size. This publication investigates the behaviour of the failure rate as the size grows without limit. Earlier results are surveyed and new results are proven. Some of them use C++ on Linux as the computational model. It turns out that the behaviour is sensitive to the details of the programming language or computational model, but in many cases it is possible to prove that the proportion of hard instances does not vanish.
Keywords
Cite
@article{arxiv.1307.7066,
title = {Asymptotic Proportion of Hard Instances of the Halting Problem},
author = {Antti Valmari},
journal= {arXiv preprint arXiv:1307.7066},
year = {2014}
}
Comments
18 pages. The differences between this version and arXiv:1307.7066v1 are significant. They have been listed in the last paragraph of Section 1. Excluding layout, this arXiv version is essentially identical to the Acta Cybernetica version