Turing Patterns with Turing Machines: Emergence and Low-level Structure Formation

TL;DR

This paper explores the connection between Turing's work on morphogenesis and universal computation by using algorithmic probability to model pattern formation, providing experimental results on one-dimensional patterns.

Contribution

It introduces a novel approach linking Turing patterns with Turing machines through algorithmic probability, addressing a fundamental question in biological pattern formation and computation.

Findings

Experimental results on one-dimensional patterns demonstrate the approach.

The method generalizes to n-dimensional pattern formation.

Bridges Turing's theories of morphogenesis and computation.

Abstract

Despite having advanced a reaction-diffusion model of ODE's in his 1952 paper on morphogenesis, reflecting his interest in mathematical biology, Alan Turing has never been considered to have approached a definition of Cellular Automata. However, his treatment of morphogenesis, and in particular a difficulty he identified relating to the uneven distribution of certain forms as a result of symmetry breaking, are key to connecting his theory of universal computation with his theory of biological pattern formation. Making such a connection would not overcome the particular difficulty that Turing was concerned about, which has in any case been resolved in biology. But instead the approach developed here captures Turing's initial concern and provides a low-level solution to a more general question by way of the concept of algorithmic probability, thus bridging two of his most important contributions to science: Turing pattern formation and universal computation. I will provide experimental results of one-dimensional patterns using this approach, with no loss of generality to a n-dimensional pattern generalisation.