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

2025-05-12

AI Summary

This paper investigates the dynamical evolution of one-dimensional, two-state, two-symbol finite-state machines under a “reflective composition” mechanism. We propose a novel reflective composition paradigm that symmetrically models states and messages, and employ discrete dynamical systems analysis combined with cellular automata (CA) isomorphism mapping to uncover emergent phenomena—including Sierpinski triangle patterns, reversible billiard dynamics, and fractal nested structures. Our key contributions are: (i) the first demonstration of *natural emergence*—within a formal system—of the time-reversed trajectory of elementary CA Rule 90, circumventing conventional inverse modeling reliant on preimage enumeration; and (ii) a rigorous proof that a three-machine composite system admits an exact isomorphism to Rule 90 and its time reversal: one machine is equivalent to Rule 90, while the other two precisely implement its time-reversed dynamics; moreover, under specific boundary conditions, the entire system is isomorphic to both Rule 90 and its inverse.

Technology Category

• Multiagent Systems: Agent-Based Simulation and Emergent Behavior
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• Explores dynamics of reflexive composition in state machines
• Identifies machines generating Sierpinski Triangle and Rule 90
• Discovers natural inverse dynamics for Rule 90 reversal

Innovation

• Reflexive composition of state machines
• Reversal of cellular automata Rule 90
• Symmetric treatment of state and message