Reflexive Composition of Elementary State Machines, with an Application to the Reversal of Cellular Automata Rule 90

TL;DR

This paper investigates the complex behaviors of elementary state machines under reflexive composition, revealing connections to cellular automata Rule 90 and its inverse, with implications for understanding reversible dynamics.

Contribution

It introduces a novel formalism for reflexive composition of state machines that naturally produces inverse dynamics of Rule 90, expanding understanding of reversible cellular automata.

Findings

Generated the Sierpinski Triangle from a state machine

Identified machines isomorphic to Rule 90 and its inverse

Demonstrated natural emergence of inverse dynamics from the formalism

Abstract

We explore the dynamics of a one-dimensional lattice of state machines on two states and two symbols sequentially updated via a process of "reflexive composition." The space of 256 machines exhibits a variety of behavior, including substitution, reversible "billiard ball" dynamics, and fractal nesting. We show that one machine generates the Sierpinski Triangle and, for a subset of boundary conditions, is isomorphic to cellular automata Rule 90 in Wolfram's naming scheme. More surprisingly, two other machines follow trajectories that map to Rule 90 in reverse. Whereas previous techniques have been developed to uncover preimages of Rule 90, this is the first study to produce such inverse dynamics naturally from the formalism itself. We argue that the system's symmetric treatment of state and message underlies its expressive power.