The Degree of Squares is an Atom

Abstract

We answer an open question in the theory of degrees of infinite sequences with respect to transducibility by finite-state transducers. An initial study of this partial order of degrees was carried out in [5], 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' 10 0 10 1 10 2 10 3 • • •. A degree is called an 'atom' if below it there is only the bottom degree 0, which consists of the ultimately periodic sequences. We show that also the degree of the 'squares sequence' 10 0 10 1 10 4 10 9 10 16 • • • 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 0 or can be transduced back to a polynomial sequence for a polynomial of the same order.

Ordering sequences by permutation transducers

“…A key notion of [3,2] is the pre-order ≥ defined on sequences by σ ≥ τ ⇐⇒ ∃ finite state transducer T : τ = T (σ ).…”

“…Equivalence under transducers organizes infinite sequences into a hierarchy with interesting properties, as ongoing research is revealing, see for example [2,4]. Under the stricter notion of equivalence under permutation transducers a finer hierarchy of one-sided sequences arises, as well as a novel hierarchy of bi-infinite sequences (in fact, two distinct versions of them, see below).…”

Classifying Non-periodic Sequences by Permutation Transducers

“…We use ≥, ∼, > for the similar relations on sequences based on ordinary finite state transducers, that is, without the additional bijectivity requirement. These were studied extensively in [3][4][5][6]. To see the effect of the additional requirement of permutation transducers, throughout the paper in presenting properties of ≥ p , ∼ p , > p we often present the corresponding properties of ≥, ∼, >.…”

“…A main result of [4] states that the class containing the sequences of the shape f for f linear is atomic, that is, if f ≥ σ for f linear, then either σ ≥ f or σ is ultimately periodic. In [3] it was shown that a similar result holds for quadratic functions, while in [5,6] it was shown that for higher degree it does not hold. Here we are interested in considering ≥ p and ∼ p instead.…”

“…Equivalence under transducers organizes infinite sequences into a hierarchy with interesting properties, as ongoing research is revealing, see for example [3,6] and the conference paper [5] at DLT 2016. In this setting the main definition is that for two sequences σ, τ we have that σ ≥ τ if and only if there exists a transducer T that produces τ when consuming σ.…”