Measuring the Computational Power of Finite Patches of Cellular Automata

TL;DR

This paper introduces a method to measure the computational power of finite patches of cellular automata by converting them into algebraic structures and analyzing their hierarchical decompositions.

Contribution

It develops a novel algebraic approach to quantify cellular automata's computational capabilities and provides macro-level descriptions through state space partitioning.

Findings

Conversion of cellular automata into transformation semigroups captures dynamics and interactions.

Hierarchical decompositions reveal macro/micro-state structures in cellular automata.

Method applies broadly to discrete dynamical systems beyond cellular automata.

Abstract

Computational power can be measured by assigning an algebraic structure to a computational device. Here, we convert a small patch of Conway's Game of Life into a transformation semigroup. The conversion captures not only time evolution but also interactive operations. In this way, the cellular automaton becomes directly programmable. Once this measurement is made, we apply hierarchical decompositions to the resulting algebraic object as a way of understanding it. These decompositions are based on a macro/micro-state division inspired by statistical mechanics. However, cellular automata have a large number of global states. Therefore, we focus on partitioning the state space and creating morphic images approximations that can serve as macro-level descriptions. The methods developed here are not limited to cellular automata; they apply more generally to discrete dynamical systems.