Coxeter Groups and Asynchronous Cellular Automata

The dynamics group of an asynchronous cellular automaton (ACA) relates properties of its long term dynamics to the structure of Coxeter groups. The key mathematical feature connecting these diverse fields is involutions. Group-theoretic results in the latter domain may lead to insight about the dynamics in the former, and vice-versa. In this article, we highlight some central themes and common structures, and discuss novel approaches to some open and open-ended problems. We introduce the state automaton of an ACA, and show how the root automaton of a Coxeter group is essentially part of the state automaton of a related ACA.

💡 Research Summary

The paper establishes a deep connection between asynchronous cellular automata (ACA) and Coxeter groups, showing that the long‑term dynamics of an ACA can be understood through the algebraic structure of a Coxeter group. The authors begin by defining an ACA as a collection of binary vertex functions applied in an arbitrary (asynchronous) order, and they introduce the notion of a π‑independent ACA: a system whose set of periodic points does not depend on the permutation π used to schedule updates. For such systems each local update function acts as a permutation on the periodic set, and the group generated by these permutations is called the dynamics group D​G(F).

A Coxeter system (W,S) consists of involutive generators s_i (s_i²=1) with relations (s_i s_j)^{m_{ij}}=1 for i≠j, where m_{ij}≥2 encodes the edge label of the Coxeter graph Γ. The authors show that D​G(F) is always a quotient of the Coxeter group presented by the same generators, the extra relations coming from the specific Boolean functions used in the ACA. In other words, the dynamics group is a “reflection group over F₂”.

The paper then parallels two classic decision problems. In Coxeter theory the word problem—determining whether two words represent the same group element—is solvable via Matsumoto’s theorem, which says any two reduced expressions differ only by braid moves. For SDS (the generalization of ACA with a fixed update sequence) the analogous question asks when two update words w and w′ produce the same global map or the same phase‑space structure. Although the function space is finite (2ⁿ possible global maps), a naïve exhaustive check is impractical; the authors point out the need for an efficient algorithmic solution.

A major contribution is the identification of several equivalence relations that translate between the two domains. The authors define a κ‑equivalence on acyclic orientations of Γ: converting a source vertex into a sink corresponds to conjugating a Coxeter element by its first generator. They prove that κ‑equivalence classes of orientations are in bijection with conjugacy classes of Coxeter elements. In the SDS setting, a permutation π of the vertices defines a map F_π; when two permutations are related by a graph automorphism γ∈Aut(Γ), the corresponding SDS maps are topologically conjugate and therefore cycle‑equivalent. This yields a coarser equivalence ¯κ on the quotient Aut(Γ)/∼κ, which mirrors the notion of spectral class in Coxeter theory—two Coxeter elements are spectrally equivalent if their geometric representations have the same multiset of eigenvalues. The Tutte polynomial T_Γ(x,y) appears as a unifying counting tool: T_Γ(2,0) counts Coxeter elements (or π‑independent SDS maps) while T_Γ(1,0) counts κ‑equivalence (or cycle‑equivalence) classes.

The authors apply the framework to elementary cellular automata (ECA) on a circular graph Z_n with identical local rules. They show that for many of the 256 possible binary 3‑cell rules, the resulting ACA is π‑independent, and they classify the corresponding dynamics groups. Notable examples include DG(ECA₆₀)=SL_n(F₂) and DG(ECA₂₈)=DG(ECA₂₉)=DG(ECA₅₁)=Z_n². Other rules give rise to symmetric or alternating groups whose orders are governed by Fibonacci or Lucas numbers, leading to several conjectures about the pattern of group sizes.

Finally, the paper introduces the root automaton of a Coxeter group. Using the standard geometric representation ρ:W→GL(V), each generator s_i acts as a reflection across a hyperplane. The set of all roots Φ={w·α_i | w∈W, i∈