Equality of cycle lengths in one- and two-dimensional $\sigma$ automata

TL;DR

This paper investigates the cycle lengths of multi-dimensional sigma automata, revealing their equality in 1D and 2D cases and establishing bounds for higher dimensions, with implications for cellular automata models.

Contribution

It proves the equality of cycle lengths in 1D and 2D sigma automata and establishes bounds for cycle lengths in higher dimensions, advancing understanding of automata dynamics.

Findings

Cycle lengths of 1D and 2D sigma automata are equal.

Cycle lengths in higher dimensions are bounded and saturate the upper limit.

Abstract

When the game Lights Out is played according to an algorithm specifying the player's sequence of moves, it can be modeled using deterministic cellular automata. One such model reduces to the $\sigma$ automaton, which evolves according to the 2-dimensional analog of Rule 90. We consider how the cycle lengths of multi-dimensional $\sigma$ automata depend on their dimension. We find that the cycle lengths of 1-dimensional $\sigma$ automata and 2-dimensional $\sigma$ automata (of the same size) are equal, and we prove this by relating the eigenvalues and Jordan blocks of their respective adjacency matrices. We also discover that cycle lengths of higher-dimensional $\sigma$ automata are bounded (despite the number of lattice sites increasing with dimension) and eventually saturate the upper bound.