Measuring the Computational Power of Finite Patches of Cellular Automata

2026-04-16

AI Summary

This work proposes an algebraic approach to quantifying the computational capacity of finite cellular automata, such as Conway’s Game of Life. By encoding local update rules into transformation semigroups, the framework uniformly characterizes both temporal evolution and interaction operations. Inspired by statistical mechanics, it introduces a hierarchical decomposition of the state space through a distinction between macroscopic and microscopic states. The key innovation lies in integrating algebraic structure with interactive modeling to construct, for the first time, a macroscopic description framework based on homomorphic image approximation. This framework not only enables quantitative analysis of computational power but also offers a generalizable method for macroscopic approximation applicable to a broad class of discrete dynamical systems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• computational power
• cellular automata
• state space partitioning
• macro-level description
• discrete dynamical systems

Innovation

• transformation semigroup
• cellular automata
• hierarchical decomposition
• macro/micro-state
• discrete dynamical systems