Why it is sufficient to consider only the case where the seed of linear cellular automata is $1$

TL;DR

This paper proves that for linear cellular automata, analyzing the orbit starting from the seed with state 1 is sufficient, simplifying the study of their fractal-generating behavior regardless of the number of states.

Contribution

It establishes that in linear CAs, only the seed with state 1 needs to be considered, regardless of the number of states, streamlining analysis of their dynamics.

Findings

Single seed with state 1 suffices for linear CAs analysis

Reduces complexity in studying multi-state linear CAs

Simplifies fractal generation analysis

Abstract

When using a cellular automaton (CA) as a fractal generator, consider orbits from the single site seed, an initial configuration that gives only a single cell a positive value. In the case of a two-state CA, since the possible states of each cell are $0$ or $1$, the "seed" in the single site seed is uniquely determined to be the state $1$. However, for a CA with three or more states, there are multiple candidates for the seed. For example, for a $3$-state CA, the possible states of each cell are $0$, $1$, and $2$, so the candidates for the seed are $1$ and $2$. For a $4$-state CA, the possible states of each cell are $0$, $1$, $2$, and $3$, so the candidates for the seed are $1$, $2$, and $3$. Thus, as the number of possible states of a CA increases, the number of seed candidates also increases. In this paper, we prove that for linear CAs it is sufficient to consider only the orbit from the single site seed with the seed $1$.