Numbers Defined by Turing Machines

Conference paper
Part of the Collegium Logicum book series (COLLLOGICUM, volume 2)


We consider three types of Turing machines defining functions on infinite words and investigate some characteristic properties of these types of Turing machine mappings. Using the interpretation of infinite words as the expansions of numbers we obtain three classes of real respectively complex numbers. We prove that the three classes of complex numbers form algebraically closed subfields of the field of complex numbers.


Recursive Relation Turing Machine Recursive Function Real Polynomial Input Tape 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Eilenberg: Automata, Languages and Machines, Vol. A ( Academic Press, New York, 1974 ).zbMATHGoogle Scholar
  2. [2]
    R. Freund: Real Functions and Numbers Defined by Turing Machines, Theoret. Comp. Sci. 23 (1983) 287–304.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    M. L. Minsky: Berechnung: Endliche und unendliche Maschinen (Verlag Berliner Union, Stuttgart, 1971 ).zbMATHGoogle Scholar
  4. [4]
    H. Rogers: Theory of Recursive Functions and Effective Computability (McGraw Hill, 1967).Google Scholar
  5. [5]
    A. Salomaa: Formal Languages ( Academic Press, New York, 1973 ).zbMATHGoogle Scholar
  6. [6]
    L. Staiger: Hierarchies of Recursive w-Languages, J. Inform. Process. Cybernet. EIK 22 (1986) 5 /6, 219–241.MathSciNetGoogle Scholar
  7. [7]
    L. Staiger: Sequential Mappings of w-Languages, RAIRO Infor. Théor. Appl. 21 (1987) 2, 147–173.MathSciNetGoogle Scholar
  8. [8]
    L. Staiger and K. Wagner, Rekursive Folgenmengen I, Z. Math Logik Grund-lag. Math. 24 (1978) 523–538.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    K. Wagner: Arithmetische Operatoren, Z. Math Logik Grundlag. Math. 22 (1976) 553–570.zbMATHCrossRefGoogle Scholar
  10. [10]
    K. Weihrauch: Computability ( Springer-Verlag, Berlin, 1987 ).zbMATHGoogle Scholar
  11. [11]
    K. Weihrauch: A simple Introduction to Computable Analysis, Technical Report 171, Fernuniversität Hagen, 1995.Google Scholar

Copyright information

© Springer-Verlag/Wien 1996

Authors and Affiliations

  1. 1.Institut für ComputersprachenTechnische Universität WienWienAustria
  2. 2.Institut für InformatikMartin-Luther-Universität Halle-WittenbergHalle (Saale)Germany

Personalised recommendations