Cellular Automaton Reducibility as a Measure of Complexity for Infinite Words

TL;DR

This work introduces 1D cellular automaton (1CA) reducibility as a fine-grained measure of complexity for infinite streams, defining $σ \\triangleright^{\\mathrm{C}} τ$ via a 1CA with $f_M(σ) = τ$. It analyzes the induced degree structure, proving the hierarchy is not well-founded and not dense, identifying atoms among sparse streams, and showing that suprema of arbitrary sets generally do not exist. It completely classifies ultimately periodic streams, showing $\\mathcal{T}_n \\leq^{\\mathrm{C}} \\mathcal{T}_m$ iff $n \\mid m$, and demonstrates the existence of infinite ascending and descending chains, while leaving maximal degrees as an open problem. The paper also provides a pseudo-algorithm for classifying streams under 1CA reducibility and contrasts 1CA reducibility with FST reducibility, highlighting fundamental differences arising from the locality and memoryless nature of 1CAs.

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: σ is at least as complex as τ when there exists a cellular automaton mapping σ to τ. 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.