On the equivalence between minimal sufficient statistics, minimal typical models and initial segments of the Halting sequence

TL;DR

This paper explores the deep connections between minimal sufficient statistics, typical models, and the Halting problem, revealing their equivalence and implications for understanding computational complexity and information content in strings.

Contribution

It establishes the equivalence between minimal sufficient statistics, typical models, and initial segments of the Halting sequence, and introduces the concept of weak sufficient statistics.

Findings

Algorithmic minimal sufficient statistic length exceeds the computational depth of x.

Minimal sufficient statistics can encode Halting problem solutions for short programs.

Weak sufficient statistics are equivalent to minimal typical models and relate to the Halting problem.

Abstract

It is shown that the length of the algorithmic minimal sufficient statistic of a binary string x, either in a representation of a finite set, computable semimeasure, or a computable function, has a length larger than the computational depth of x, and can solve the Halting problem for all programs with length shorter than the m-depth of x. It is also shown that there are strings for which the algorithmic minimal sufficient statistics can contain a substantial amount of information that is not Halting information. The weak sufficient statistic is introduced, and it is shown that a minimal weak sufficient statistic for x is equivalent to a minimal typical model of x, and to the Halting problem for all strings shorter than the BB-depth of x.