Let's Make a Difference!

TL;DR

This paper investigates the behavior of difference operators on binary streams, establishing a characterization of eventual periodicity through delta-orbits and exploring generalizations and intriguing connections with well-known streams.

Contribution

It introduces a novel characterization of eventual periodicity in streams via delta-orbits and generalizes the difference operator to blocks of arbitrary length.

Findings

A stream is eventually periodic iff its delta-orbit is eventually periodic.

Generalization to delta_d operators sums blocks modulo 2.

Surprising links found between the Sierpinski stream and the Mephisto Waltz.

Abstract

We study the behaviour of iterations of the difference operator delta on streams over {0,1}. In particular, we show that a stream sigma is eventually periodic if and only if the sequence of differences sigma, delta(sigma), delta(delta(sigma)), ..., the `delta-orbit' of sigma as we call it, is eventually periodic. Moreover, we generalise this result to operations delta_d that sum modulo 2 the elements of each consecutive block of length d+1 in a given 01-stream. Some experimentation with delta-orbits of well-known streams reveals a surprising connexion between the Sierpinski stream and the Mephisto Waltz.