Asymptotic Proportion of Hard Instances of the Halting Problem

TL;DR

This paper investigates the asymptotic behavior of hard instances of the halting problem, revealing that their proportion often remains significant as instance size grows, with results influenced by computational models.

Contribution

It provides new theoretical results on the asymptotic proportion of hard halting problem instances and explores the impact of computational models on this behavior.

Findings

Failure rate of testers does not vanish for many variants.

Behavior is sensitive to programming language and computational model.

Proportion of hard instances remains significant asymptotically.

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.