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International Conference on Combinatorics on Words

WORDS 2015: Combinatorics on Words pp 109-121 | Cite as

The Degree of Squares is an Atom

  • Jörg Endrullis
  • Clemens Grabmayer
  • Dimitri Hendriks
  • Hans Zantema
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9304)

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.

Keywords

Infinite Sequence Periodic Sequence Input Word Deterministic Finite Automaton Polynomial Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Endrullis, J., Hendriks, D., Klop, J.: Degrees of streams. J. Integers 11B(A6), 1–40 (2011). Proceedings of the Leiden Numeration Conference 2010MathSciNetGoogle Scholar
  2. 2.
    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
  3. 3.
    Sakarovitch, J.: Elements Of Automata Theory. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
  4. 4.
    Endrullis, J., Grabmayer, C., Hendriks, D., Zantema, H.: The Degree of Squares is an Atom (Extended Version). Technical report 1506.00884, arxiv.org, June 2015Google Scholar
  5. 5.
    Siefkes, D.: Undecidable extensions of monadic second order successor arithmetic. Math. Logic Q. 17(1), 385–394 (1971)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Seiferas, J., McNaughton, R.: Regularity-preserving relations. Theor. Comput. Sci. 2(2), 147–154 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cautis, S., Mignosi, F., Shallit, J., Wang, M., Yazdani, S.: Periodicity, morphisms, and matrices. Theor. Comput. Sci. 295, 107–121 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jörg Endrullis
    • 1
  • Clemens Grabmayer
    • 1
  • Dimitri Hendriks
    • 1
  • Hans Zantema
    • 2
    • 3
  1. 1.Department of Computer ScienceVU University AmsterdamAmsterdamNetherlands
  2. 2.Department of Computer ScienceEindhoven University of TechnologyEindhovenNetherlands
  3. 3.Institute for Computing and Information ScienceRadboud University NijmegenNijmegenNetherlands

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