[Paper Review] Maximal temporal period of a periodic solution generated by a one-dimensional cellular automaton

Janko Gravner, Xiaochen Liu|arXiv (Cornell University)|Sep 16, 2019

Cellular Automata and Applications15 references4 citations

TL;DR

This paper investigates the maximal temporal period of periodic solutions in one-dimensional cellular automata with two-neighbor rules, fixed spatial period σ, and n states. It derives that for additive rules, the maximal temporal period equals the exponent of the multiplicative group of a suitable ring, and constructs non-additive rules achieving temporal periods on the order of the theoretical upper bound $n^\sigma$. The key contribution is a precise characterization of maximal periods for small σ and prime power n.

ABSTRACT

We study one-dimensional cellular automata evolutions with both temporal and spatial periodicity. The main objective is to investigate the longest temporal periods among all two-neighbor rules, with a fixed spatial period $σ$ and number of states $n$. When $σ= 2, 3, 4$, or $6$, and we restrict the rules to be additive, the longest period can be expressed as the exponent of the multiplicative group of an appropriate ring. We also construct non-additive rules with temporal period on the same order as the trivial upper bound $n^σ$. Experimental results, open problems, and possible extensions of our results are also discussed.

Motivation & Objective

• To determine the maximum possible temporal period of periodic solutions in one-dimensional cellular automata with fixed spatial period σ and n states.
• To characterize the maximal temporal period among additive two-neighbor rules, showing it equals the exponent of the multiplicative group of a ring.
• To construct non-additive rules achieving temporal periods within a constant factor of the theoretical upper bound $n^\sigma$.
• To explore open problems regarding the existence of explicit formulas and equality between upper bounds and actual maximal periods for prime power states.

Proposed method

• The authors model the CA evolution using polynomial multiplication in the quotient ring $\mathbb{Z}_n[x]/(x^\sigma - 1)$, representing the state evolution as powers of the polynomial $a + bx$.
• For additive rules, the temporal period corresponds to the order of $a + bx$ in the multiplicative group of the ring $\mathbb{Z}_n[x]/(x^\sigma - 1)$.
• The maximal temporal period is determined by computing the exponent of this multiplicative group, which depends on the prime factorization of $n$ and the value of $\sigma$.
• For prime power $n = p^m$, the authors define an upper bound $\texttt{ub}_\sigma(p^m)$ based on the multiplicative order and ring structure, and compare it to the actual maximal period $\pi_\sigma(p^m)$.
• They use number-theoretic tools, including the Möbius function and properties of cyclotomic polynomials, to count aperiodic words and derive bounds.
• Theoretical analysis and computational experiments are combined to verify results and identify cases where the actual maximal period is strictly less than the theoretical upper bound.

Experimental results

Research questions

• RQ1 What is the maximal temporal period achievable by any additive two-neighbor cellular automaton rule with spatial period σ and n states?
• RQ2 Can non-additive rules achieve temporal periods on the same order as the trivial upper bound $n^\sigma$?
• RQ3 Is there an explicit formula for the maximal shortest temporal period $\rho_\sigma(n)$ or maximal longest period $\pi_\sigma(n)$ for small σ and prime power n?
• RQ4 For which values of $n$ and $\sigma$ does $\pi_\sigma(n) = \texttt{ub}_\sigma(n)$, and when is the inequality strict?
• RQ5 Are there infinitely many primes for which $\pi_\sigma(p) < \texttt{ub}_\sigma(p)$ for some $\sigma$, and is 2 the only such prime with this property for higher powers?

Key findings

• For $\sigma = 2,3,4,6$, the maximal temporal period of additive rules equals the exponent of the multiplicative group of the ring $\mathbb{Z}_n[x]/(x^\sigma - 1)$.
• When $n = 3$ and $\sigma = 4$, the maximal temporal period is $\tau = 8$, achieved by the additive rule $f(c_0,c_1) = c_0 + c_1$.
• For $\sigma = 2^k$, the maximal period $\pi_\sigma(2^m)$ is $2^k$ for $m \leq k+1$, but jumps to $2^{k+1}$ at $m = k+2$, showing a non-monotonic behavior.
• The authors identify explicit counterexamples where $\pi_\sigma(p^m) < \texttt{ub}_\sigma(p^m)$, such as $\pi_4(2^3) = 8 < 16 = \texttt{ub}_4(2^3)$, disproving a naive equality conjecture.
• For $\sigma = 5$, the authors conjecture that $\pi_5(p^m) = \texttt{ub}_5(p^m)$ for all prime powers $p^m$, based on extensive computation up to $10^5$, and this would yield a complete formula.
• The paper provides a complete table of $\rho_\sigma(n)$ and $\pi_\sigma(n)$ for $\sigma = 2,3$ and $n = 2$ to $20$, showing that $\pi_\sigma(n)$ often equals the theoretical upper bound.