Degrees of Infinite Words, Polynomials and Atoms

  • Jörg Endrullis
  • Juhani Karhumäki
  • Jan Willem Klop
  • Aleksi Saarela
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9840)

Abstract

Our objects of study are finite state transducers and their power for transforming infinite words. Infinite sequences of symbols are of paramount importance in a wide range of fields, from formal languages to pure mathematics and physics. While finite automata for recognising and transforming languages are well-understood, very little is known about the power of automata to transform infinite words.

We use methods from linear algebra and analysis to show that there is an infinite number of atoms in the transducer degrees, that is, minimal non-trivial degrees.

References

  1. 1.
    Allouche, J.P., Shallit, J.: Automatic Sequences: Theory, Applications Generalizations. Cambridge University Press, New York (2003)CrossRefMATHGoogle Scholar
  2. 2.
    Belov, A.: Some algebraic properties of machine poset of infinite words. ITA 42(3), 451–466 (2008)MathSciNetGoogle Scholar
  3. 3.
    Berstel, J., Boasson, L., Carton, O., Petazzoni, B., Pin, J.E.: Operations preserving regular languages. Theor. Comput. Sci. 354(3), 405–420 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Endrullis, J., Grabmayer, C., Hendriks, D., Zantema, H.: The degree of squares is an atom. In: Manea, F., Nowotka, D. (eds.) WORDS 2015. LNCS, vol. 9304, pp. 109–121. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  5. 5.
    Endrullis, J., Hansen, H.H., Hendriks, D., Polonsky, A., Silva, A.: A coinductive framework for infinitary rewriting and equational reasoning. In: Proceedings of Conference on Rewriting Techniques and Applications (RTA 2015). Schloss Dagstuhl (2015)Google Scholar
  6. 6.
    Endrullis, J., Hendriks, D.: Lazy productivity via termination. Theor. Comput. Sci. 412(28), 3203–3225 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Endrullis, J., Hendriks, D., Klop, J.W.: Degrees of streams. J. Integers 11B(A6), 1–40 (2011). Proceedings of the Leiden Numeration Conference 2010MathSciNetMATHGoogle Scholar
  8. 8.
    Endrullis, J., Karhumäki, J., Klop, J., Saarela, A.: Degrees of infinite words, polynomials and atoms (extended version). CoRR (2016)Google Scholar
  9. 9.
    Endrullis, J., Klop, J.W., Saarela, A., Whiteland, M.: Degrees of transducibility. In: Manea, F., Nowotka, D. (eds.) WORDS 2015. LNCS, vol. 9304, pp. 1–13. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  10. 10.
    Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1988). Reprint of the 1952 editionMATHGoogle Scholar
  11. 11.
    Löwe, B.: Complexity hierarchies derived from reduction functions. In: Löwe, B., Piwinger, B., Räsch, T. (eds.) Classical and New Paradigms of Computation and their Complexity Hierarchies. Trends in Logic, vol. 23, pp. 1–14. Springer, Amsterdam (2004)CrossRefGoogle Scholar
  12. 12.
    Odifreddi, P.: Classical Recursion Theory. Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam (1999)MATHGoogle Scholar
  13. 13.
    Rayna, G.: Degrees of finite-state transformability. Inf. Control 24(2), 144–154 (1974)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Sakarovitch, J.: Elements of Automata Theory. Cambridge University Press, Cambridge (2003)MATHGoogle Scholar
  15. 15.
    Shoenfield, J.R.: Degrees of Unsolvability. North-Holland, Elsevier, New York (1971)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Jörg Endrullis
    • 1
  • Juhani Karhumäki
    • 2
  • Jan Willem Klop
    • 1
    • 3
  • Aleksi Saarela
    • 2
  1. 1.Department of Computer ScienceVU University AmsterdamAmsterdamThe Netherlands
  2. 2.Department of Mathematics and Statistics & FUNDIMUniversity of TurkuTurkuFinland
  3. 3.Centrum Wiskunde & Informatica (CWI)AmsterdamThe Netherlands

Personalised recommendations