On the Running Time of the Shortest Programs

TL;DR

This paper investigates the running time of the shortest programs that produce a given word, exploring the relationship between program length and execution duration within the framework of Kolmogorov complexity.

Contribution

It introduces the problem of analyzing the runtime of shortest programs in the context of Kolmogorov complexity, a novel aspect not extensively studied before.

Findings

Shortest programs can have varying runtimes depending on the word.

There are bounds on the running time of shortest programs for certain classes of words.

The study provides insights into the computational resources needed for minimal descriptions.

Abstract

The Kolmogorov complexity of the word w is equal to the length of the shortest concatenation of program Z and its input x with which the word w is computed by the universal turing machine U. The question introduced in this paper is the following: How long do the shortest programs run for?