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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Endrullis, J., Hendriks, D., Klop, J.: Degrees of streams. J. Integers 11B(A6), 1–40 (2011). Proceedings of the Leiden Numeration Conference 2010
Shallit, J.: Open problems in automata theory: an idiosyncratic view. LMS Keynote Talk in Discrete Mathematics, BCTCS (2014). https://cs.uwaterloo.ca/shallit/Talks/bc4.pdf
Sakarovitch, J.: Elements Of Automata Theory. Cambridge University Press, Cambridge (2003)
Endrullis, J., Grabmayer, C., Hendriks, D., Zantema, H.: The Degree of Squares is an Atom (Extended Version). Technical report 1506.00884, arxiv.org, June 2015
Siefkes, D.: Undecidable extensions of monadic second order successor arithmetic. Math. Logic Q. 17(1), 385–394 (1971)
Seiferas, J., McNaughton, R.: Regularity-preserving relations. Theor. Comput. Sci. 2(2), 147–154 (1976)
Cautis, S., Mignosi, F., Shallit, J., Wang, M., Yazdani, S.: Periodicity, morphisms, and matrices. Theor. Comput. Sci. 295, 107–121 (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Endrullis, J., Grabmayer, C., Hendriks, D., Zantema, H. (2015). The Degree of Squares is an Atom. In: Manea, F., Nowotka, D. (eds) Combinatorics on Words. WORDS 2015. Lecture Notes in Computer Science(), vol 9304. Springer, Cham. https://doi.org/10.1007/978-3-319-23660-5_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-23660-5_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-23659-9
Online ISBN: 978-3-319-23660-5
eBook Packages: Computer ScienceComputer Science (R0)