How local rules generate emergent structure in cellular automata

TL;DR

This paper introduces a novel method to analyze cellular automata by extracting their causal architecture from look-up tables, revealing how local rules generate emergent structures and regions of causal independence.

Contribution

It formalizes the causal interactions in cellular automata using a tiling transducer, enabling classification of rules based on their capacity to sustain decoupled regions.

Findings

The look-up table encodes the distribution of causal coupling in cellular automata.

The proposed framework predicts the prevalence of decoupled regions with high accuracy.

Structural properties of the transducer classify rules by their ability to sustain independent regions.

Abstract

Cellular automata generate spatially extended, temporally persistent emergent structures from local update rules. No general method derives the mechanisms of that generation from the rule itself; existing tools reconstruct structure from observed dynamics. This paper shows that the look-up table contains a readable causal architecture and introduces a forward model to extract it. The key observation in elementary cellular automata (ECA) is that adjacent cells share input positions, so the prime implicants of neighbouring transitions overlap. That overlap can couple the transitions causally or leave them independent. We formalize each pairwise interaction as a tile. A finite-state, tiling transducer, $\mathcal{T}$, composes tiles across the CA lattice, tracking how coupling and independence propagate from one cell pair to the next. Structural properties of $\mathcal{T}$ are used to classify ECA rules that can sustain regions of causal independence across space and time. We find that, in the 88 ECA equivalence classes, the number of local configurations at which coupling is structurally impossible -- computable from the look-up table -- predicts the prevalence of dynamically decoupled regions with Spearman $\rho = 0.89$ ($p < 10^{-31}$). The look-up table encodes not just what a rule computes but where it distributes causal coupling across the lattice; the framework reads that distribution forward, from local logical redundancy to emergent mesoscopic organization.