Generic algorithms for halting problem and optimal machines revisited

TL;DR

This paper revisits the halting problem within the framework of optimal machines and Kolmogorov complexity, providing new insights into its asymptotic properties and related computational problems.

Contribution

It demonstrates how many results about the halting problem can be proven using optimal machines and Kolmogorov complexity, and explores asymptotic and probabilistic aspects.

Findings

The fraction of terminating programs has no limit, with all limit points being Martin-Löf random reals.

The fraction of programs requiring long time for termination relates to the busy beaver function.

Approximate solutions and probabilistic algorithms for halting-related problems are analyzed with both positive and negative results.

Abstract

The halting problem is undecidable --- but can it be solved for "most" inputs? This natural question was considered in a number of papers, in different settings. We revisit their results and show that most of them can be easily proven in a natural framework of optimal machines (considered in algorithmic information theory) using the notion of Kolmogorov complexity. We also consider some related questions about this framework and about asymptotic properties of the halting problem. In particular, we show that the fraction of terminating programs cannot have a limit, and all limit points are Martin-L\"of random reals. We then consider mass problems of finding an approximate solution of halting problem and probabilistic algorithms for them, proving both positive and negative results. We consider the fraction of terminating programs that require a long time for termination, and describe this fraction using the busy beaver function. We also consider approximate versions of separation problems, and revisit Schnorr's results about optimal numberings showing how they can be generalized.