International Conference on Combinatorics on Words

WORDS 2015: Combinatorics on Words pp 1-13 | Cite as

Degrees of Transducibility

  • Jörg Endrullis
  • Jan Willem Klop
  • Aleksi Saarela
  • Markus Whiteland
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9304)

Abstract

Our objects of study are infinite sequences and how they can be transformed into each other. As transformational devices, we focus here on Turing Machines, sequential finite state transducers and Mealy Machines. For each of these choices, the resulting transducibility relation \(\ge \) is a preorder on the set of infinite sequences. This preorder induces equivalence classes, called degrees, and a partial order on the degrees.

For Turing Machines, this structure of degrees is well-studied and known as degrees of unsolvability. However, in this hierarchy, all the computable streams are identified in the bottom degree. It is therefore interesting to study transducibility with respect to weaker computational models, giving rise to more fine-grained structures of degrees. In contrast with the degrees of unsolvability, very little is known about the structure of degrees obtained from finite state transducers or Mealy Machines.

References

  1. 1.
    Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications Generalizations. Cambridge University Press, New York (2003)CrossRefGoogle Scholar
  2. 2.
    Belov, A.: Some algebraic properties of machine poset of infinite words. ITA 42(3), 451–466 (2008)Google Scholar
  3. 3.
    Endrullis, J., Grabmayer, C., Hendriks, D., Zantema, H.: The Degree of Squares is an Atom (2015)Google Scholar
  4. 4.
    Endrullis, J., Hendriks, D., Klop, J.W.: Degrees of streams. J. integers 11B(A6), 1–40 (2011). Proceedings of the Leiden Numeration Conference 2010MathSciNetGoogle Scholar
  5. 5.
    Endrullis, J., Hendriks, D., Klop, J.W.: Highlights in infinitary rewriting and lambda calculus. Theor. Comput. Sci. 464, 48–71 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Endrullis, J., Hendriks, D., Klop, J.W.: Streams are forever. Bull. EATCS 109, 70–106 (2013)Google Scholar
  7. 7.
    Jacobs, K.: Invitation to mathematics. Princeton University Press (1992)Google Scholar
  8. 8.
    Kleene, S.C., Post, E.L.: The upper semi-lattice of degrees of recursive unsolvability. Ann. Math. 59(3), 379–407 (1954)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Odifreddi, P.: Classical Recursion Theory. Studies in logic and the foundations of mathematics. North-Holland, Amsterdam (1999)MATHGoogle Scholar
  10. 10.
    Rayna, G.: Degrees of finite-state transformability. Inf. Control 24(2), 144–154 (1974)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Sakarovitch, J.: Elements of Automata Theory. Cambridge (2003)Google Scholar
  12. 12.
    Shoenfield, J.R.: Degrees of Unsolvability. North-Holland, Elsevier (1971)MATHGoogle Scholar
  13. 13.
    Shore, R.A.: Conjectures and questions from Gerald Sacks’s degrees of unsolvability. Arch. Math. Logic 36(4–5), 233–253 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Siefkes, D.: Undecidable extensions of monadic second order successor arithmetic. Math. Logic Q. 17(1), 385–394 (1971)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Soare, R.I.: Recursively enumerable sets and degrees. Bull. Am. Math. Soc. 84(6), 1149–1181 (1978)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Spector, C.: On degrees of recursive unsolvability. Ann. Math. 64, 581–592 (1956)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

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

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