Number of nonzero states in prefractal sets generated by cellular automata

Abstract

We count the number of nonzero states in the spatio-temporal or spatial patterns of cellular automata. By observing self-similar structures, we determine the number of nonzero states in the pattern. For Rule 90 and Rule 150 of one-dimensional elementary cellular automata, we provide an overview of previous studies on the number of nonzero states in spatio-temporal patterns until the finite time step 2n − 1. In this study, we calculate the numbers in spatial patterns for each time step t∈Z≥0 that allow us to compute future activities more efficiently than that by simulating the entire spatio-temporal pattern. We obtain results regarding the numbers for Rule 90, Rule 150, a generalization of Rule 90 to a two-dimensional case, and a modified Ulam’s model, which is a two-dimensional crystal growth model. In addition, we examine the relationship between spatio-temporal patterns generated by cellular automata and a singular function. Representing the number of nonzero states by specific numeric values, we show that they are described by a difference form of the singular function.

Singular function emerging from one-dimensional elementary cellular automaton Rule 150

“…The limit set of the orbit of Rule 150 from the single site seed x o , lim k→∞ (S 150 (2 k − 1)/2 k ), is a fractal having the Hausdorff dimension log(1 + √ 5)/ log 2 (for example, see [13]).…”

“…In this case, the number of nonzero states in the spatial pattern for a finite time step cannot be represented by simple functional equations, and we cannot apply the same constructing method as in the previous works. For Rule 150, we previously calculated the number of nonzero states in the spatial and spatio-temporal pattern [13]. Thus, by normalizing and limiting the dynamics of the numbers, we provide the size of a self-similar set and express the function by an infinite sum of the sizes of the self-similar sets.…”

Singular function emerging from one-dimensional elementary cellular automaton Rule 150

“…We introduce the previous results about the number of nonzero states in the spatio-temporal and spatial patterns according to self-similar structures. 16,17,18]). We introduce the transition matrix M and the vector v 0 as the initial values;…”

“…Figure 1 shows the spatio-temporal pattern of Rule 150 from time step 0 to 31, and Figure 2 shows its limit set. The authors have calculated the number of nonzero states in the spatial and spatio-temporal pattern of Rule 150 [16]. By normalizing the dynamic of the numbers, we provide a function.…”

Discontinuous Riemann integrable functions emerging from cellular automata

“…A singular function is a function that is monotonically increasing (or decreasing) and continuous everywhere, with a zero derivative almost everywhere; for example, Salem's singular function [1,2,3,4]. We studied the relationship between Salem's singular function and elementary cellular automata, Rule 90 and two two-dimensional elementary cellular automata [5,6,7,8,9], and that between another new singular function and Rule 150 [10]. This paper presents new Riemann integrable functions with countable discontinuous points from two-dimensional elementary cellular automata.…”