Computing Aggregate Properties of Preimages for 2D Cellular Automata

TL;DR

This paper introduces an incremental aggregation algorithm that significantly speeds up the computation of precursor properties in 2D cellular automata, enabling new insights into complex automaton behaviors.

Contribution

The paper presents a novel incremental aggregation algorithm that computes precursor properties of 2D cellular automata exponentially faster than traditional methods.

Findings

Efficient computation of precursor count distributions in 2D Game of Life.

Calculation of higher-order mean field theory coefficients.

New results previously infeasible with naive approaches.

Abstract

Computing properties of the set of precursors of a given configuration is a common problem underlying many important questions about cellular automata. Unfortunately, such computations quickly become intractable in dimension greater than one. This paper presents an algorithm --- incremental aggregation --- that can compute aggregate properties of the set of precursors exponentially faster than na{\"i}ve approaches. The incremental aggregation algorithm is demonstrated on two problems from the two-dimensional binary Game of Life cellular automaton: precursor count distributions and higher-order mean field theory coefficients. In both cases, incremental aggregation allows us to obtain new results that were previously beyond reach.