Simplicity via Provability for Universal Prefix-free Turing Machines

TL;DR

This paper explores new criteria for simplicity in universal prefix-free Turing machines based on provability within formal theories, revealing that some machines are provably simple while others are not.

Contribution

It introduces three novel criteria of simplicity grounded in formal provability, expanding the understanding of universality and simplicity in Turing machines.

Findings

Some machines are provably simple under the new criteria

Not all universal machines meet the provability criteria

The criteria relate to properties like universality being provable in PA or ZFC

Abstract

Universality is one of the most important ideas in computability theory. There are various criteria of simplicity for universal Turing machines. Probably the most popular one is to count the number of states/symbols. This criterion is more complex than it may appear at a first glance. In this note we review recent results in Algorithmic Information Theory and propose three new criteria of simplicity for universal prefix-free Turing machines. These criteria refer to the possibility of proving various natural properties of such a machine (its universality, for example) in a formal theory, PA or ZFC. In all cases some, but not all, machines are simple.