Calibrating the complexity of Delta 2 sets via their changes

TL;DR

This paper investigates the relationship between the complexity of Delta 2 sets and the amount of changes in their computable approximations, covering Martin-Loef random sets, c.e. sets, and their combination.

Contribution

It introduces a unified framework for calibrating Delta 2 set complexity through change quantification across different classes of sets.

Findings

Quantifies changes in Martin-Loef random sets' initial segments.

Measures overall changes in c.e. sets via cost functions.

Establishes principles linking complexity and changes.

Abstract

The computational complexity of a Delta 2 set will be calibrated by the amount of changes needed for any of its computable approximations. Firstly, we study Martin-Loef random sets, where we quantify the changes of initial segments. Secondly, we look at c.e. sets, where we quantify the overall amount of changes by obedience to cost functions. Finally, we combine the two settings. The discussions lead to three basic principles on how complexity and changes relate.