Cellular Automaton Reducibility as a Measure of Complexity for Infinite Words

TL;DR

This paper introduces cellular automaton reducibility as a novel way to measure the complexity of infinite words, creating a hierarchy with unique algebraic properties and implications for stream classification.

Contribution

It defines a new reducibility measure for infinite words based on cellular automata, establishing a complexity hierarchy and analyzing its algebraic structure.

Findings

Hierarchy is not well-founded or dense

Ultimately periodic streams are ordered by period divisibility

Sparse streams are atoms in the hierarchy

Abstract

Infinite words, also known as streams, hold significant interest in computer science and mathematics, raising the natural question of how their complexity should be measured. We introduce cellular automaton reducibility as a measure of stream complexity: {\sigma} is at least as complex as {\tau} when there exists a cellular automaton mapping {\sigma} to {\tau}. This enables the categorization of streams into degrees of complexity, analogous to Turing degrees in computability theory. We investigate the algebraic properties of the hierarchy that emerges from the partial ordering of degrees, showing that it is not well-founded and not dense, that ultimately periodic streams are ordered by divisibility of their period, that sparse streams are atoms, that maximal streams have maximal subword complexity, and that suprema of sets of streams do not generally exist. We also provide a pseudo-algorithm for classifying streams up to this reducibility.