2015/06/02 by Jörg Endrullis, Endrullis, Jörg, Clemens Grabmayer +5
Computer Science · Physics and Astronomy · #Advanced Mathematical Theories and Applications #FOS: Computer and information sciences #Formal Languages and Automata Theory (cs.FL) #cs.FL
paper · pdf · doi:10.48550/arxiv.1506.00884
arxiv created 2015/06/02 · openalex publication_date 2015/06/02 · arxiv updated 2015/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (Endrullis, Hendriks, Klop, 2011), 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 00 1 01 1 02 1 03 .... 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' 1 00 1 01 1 04 1 09 1 016 ... 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.