Cellular automata that generate symmetrical patterns give singular functions

TL;DR

This paper investigates symmetrical pattern-generating cellular automata, deriving singular functions from their limit sets, including Salem's functions, revealing deep mathematical properties of these automata.

Contribution

It introduces a method to derive singular functions from cellular automata patterns, linking automata behavior to well-known mathematical functions like Salem's functions.

Findings

Derived singular functions from cellular automata limit sets.

Identified Salem's singular functions in Rule 90 and 2D automata.

Showed different rules can produce the same singular functions.

Abstract

In this paper, we mainly study linear one-dimensional and two-dimensional elementary cellular automata that generate symmetrical spatio-temporal patterns. For spatio-temporal patterns of cellular automata from the single site seed, we normalize the number of nonzero states of the patterns, take the limits, and give one-variable functions for the limit sets. We can obtain a one-variable function for each limit set and show that the resulting functions are singular functions, which are non-constant, are continuous everywhere, and have a zero derivative almost everywhere. We show that for Rule 90, a one-dimensional elementary cellular automaton (CA), and a two-dimensional elementary CA, the resulting functions are Salem's singular functions. We also discuss two nonlinear elementary CAs, Rule 22, and Rule 126. Although their spatio-temporal patterns are different from that of Rule 90, their resulting functions from the number of nonzero states equal the function of Rule 90.