Universal Computation Is 'Almost Surely' Chaotic

TL;DR

This paper demonstrates that when considering rational state spaces, universal Turing machines are almost certainly chaotic with random inputs, extending chaos theory to computational models.

Contribution

It establishes a new framework for chaos in Turing machines by shifting from integer to rational state spaces, showing universal machines are almost surely chaotic.

Findings

Universal Turing machines are almost surely chaotic with random inputs.

Chaos can be rigorously defined for Turing machines in rational state spaces.

Fixed point iterations can generate chaos, and Turing machines are a special case.

Abstract

Fixed point iterations are known to generate chaos, for some values in their parameter range. It is an established fact that Turing Machines are fixed point iterations. However, as these Machines operate in integer space, the standard notions of a chaotic system is not readily applicable for them. Changing the state space of Turing Machines from integer to rational space, the condition for chaotic dynamics can be suitably established, as presented in the current paper. Further it is deduced that, given random input, computation performed by a Universal Turing Machine would be 'almost surely' chaotic.