Cycle-lengths of Multi-Dimensional $σ$ Automata

TL;DR

The paper investigates cycle-lengths of Lights Out–style automata, showing that the 1D Φ automaton and the 2D σ automaton share the same cycle length for essentially all sizes, with only small exceptional cases. It combines geometric constructions (Green's functions, reflections) with algebraic tools (Jordan normal form over GF(2), characteristic polynomials, and Kronecker sums) to relate the spectra and block structures of the corresponding transition matrices. A key technical contribution is a general formula for the size of the largest Jordan block in the Kronecker sum of two blocks over GF(2), derived via Pascal's triangle modulo 2. The results imply that higher-dimensional cycle-lengths are bounded and, for large dimensions, saturate at a maximal odd factor determined by n+1, offering insights into the dynamics of linear, translationally symmetric automata and suggesting directions for analyzing other reduced or non-periodic systems.

Abstract

When the game Lights Out is played according to an algorithm specifying the player's exact sequence of moves, it can be modeled using deterministic cellular automata. One such model reduces to the $σ$ automaton, which evolves according to the 2-dimensional analog of Rule 90. We consider how the cycle lengths of multi-dimensional $σ$ automata depend on their dimension. The main result of this work is that the cycle-lengths of 1-dimensional $σ$ automata and 2-dimensional $σ$ automata (of the same size) are equal, and we prove this by relating the eigenvalues and Jordan blocks of their respective transition matrices. We also find that cycle-lengths of higher-dimensional $σ$ automata are bounded (despite the number of lattice sites increasing with dimension) and eventually saturate the upper bound. On the way, we derive a general formula for the size of the largest Jordan block of the Kronecker sum of two matrices over $GF(2)$ using properties of Pascal's Triangle.

Figures (5)

• Figure 1: Lights Out patterns with frameworks. Some patterns are shown that evolve from a starting point (a) with one light on, on a 63 $\times$ 63 board. (b) is the 21st step, showing a pattern and its framework, which are parallel lines that have the lights that are on at their intersections. The framework lines evolve according to one dimensional lights out rules on an infinite grid. Since the lights have not reached the boundary yet, it is as if the grid is infinite. (c) is the 27th step, and shows that once the pattern reaches the boundary (the bottom side), it reflects, so the lights do not stay on the framework any more. The patterns become much more interesting after the reflections have occurred many times, as shown in (d) (the 73rd step). The light that is on in the first picture has $(x,y)=(31,25)$.
• Figure 2: The patterns for the neighbors of a light that can be derived from grids of lines. The first and third can also be rotated. The state of the center light is not shown, since the state of this light at the next step does not depend on it.
• Figure 3: First $32$ rows of Pascal's Triangle modulo $2$.
• Figure 4: The "V-shaped" region comprised of the sequences of even entries for different potential values of $h$. Here $\binom{a + b - 2}{a - 1} = \binom{20}{13}$ (the entry circled in a red box).
• Figure 5: Example of an algorithmic reduction of a triangle $T$ of size $3$ with base $n = 28$. The full reduction of $T \rightarrow T_0$ only requires two steps after initialization of $T$ (shown in the highlighted triangles).

Theorems & Definitions (44)

• Lemma 2.1
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Lemma 2.2