Discontinuous Riemann integrable functions emerging from cellular automata

TL;DR

This paper constructs discontinuous Riemann integrable functions from the dynamics of two-dimensional cellular automata, revealing their self-similar structures and calculating their integrals, thus linking automata behavior with mathematical functions.

Contribution

It introduces a novel method to derive discontinuous Riemann integrable functions from cellular automata dynamics, highlighting their self-similarity and integral properties.

Findings

Functions are discontinuous yet Riemann integrable.

Calculated integrals of the functions over [0, 1].

Demonstrated relationships between the derived functions.

Abstract

This paper presents discontinuous Riemann integrable functions on the unit interval $[0, 1]$ derived from the dynamics of two-dimensional elementary cellular automata. Based on the self-similarities of their orbits, we write down the numbers of nonzero states in the spatial and spatio-temporal patterns and obtain discontinuous Riemann integrable functions by normalizing the values. We calculate the integrals of the two obtained functions over $[0, 1]$ and demonstrate the relationship between them.