Fixed Point Constructions in Tilings and Cellular Automata

TL;DR

This paper discusses a fixed point construction method for creating tile sets and cellular automata with complex dynamics, enabling hierarchical simulation of systems through embedded Turing machine computations.

Contribution

It introduces a hierarchical fixed point construction technique that allows for the design of systems with intricate computational and dynamical properties, expanding previous methods.

Findings

Hierarchical simulation of systems via fixed points

Embedded Turing machine computations in tilings and automata

Applications in complex dynamical systems literature

Abstract

The fixed point construction is a method for designing tile sets and cellular automata with highly nontrivial dynamical and computational properties. It produces an infinite hierarchy of systems where each layer simulates the next one. The simulations are implemented entirely by computations of Turing machines embedded in the tilings or spacetime diagrams. We present an overview of the construction and list its applications in the literature.