The Degree of Squares is an Atom

Abstract

We answer an open question in the theory of degrees of infinite sequences with respect to transducibilityby finite-state transducers. An initial study of this partial order of degrees was carried out in [1], but many basic questions remain unanswered.One of the central questions concerns the existence of atom degrees, other than the degree of the ‘identity sequence’ $1 0^0 1 0^1 1 0^2 1 0^3 \cdots$. A degree is called an ‘atom’ if below it there is only the bottom degree $\varvec{0}$, which consists of the ultimately periodic sequences. We show that also the degree of the ‘squares sequence’ $1 0^0 1 0^1 1 0^4 1 0^9 1 0^{16}\cdots$ is an atom.

As the main tool for this result we characterise the transducts of ‘spiralling’ sequences and their degrees. We use this to show that every transduct of a ‘polynomial sequence’ either is in $\varvec{0}$ or can be transduced back to a polynomial sequence for a polynomial of the same order.