Skip to main content
SpringerLink
Log in

A machine description and the hierarchy of initial Grzegorczyk classes

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

A type of abstract computing machine and a measure of complexity of computations on this machine are constructed in a natural way with Grzegorczyk's hierarchy. To define this type of machine the recursive construction present in the original definition of Grzegorczyk classes is not needed. All the Grzegorczyk classes

for n=0, 1, 2, 3,... turn out to be complexity classes for the indicated type of machines and the indicated measure of complexity. On the basis of similar machines one also gets a complexity description of Smullyan's class of rudimentary predicates. With the help of the machine studied, one proves the coincidence on predicates with

of certain subclasses of

, based on bounded recursion. For the class of unary functions from

and certain similar classes, the absence is proved of a finite basis with respect to composition. It is proved that the equation

implies the equation

.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature cited

  1. A. Grzegorczyk, “Some classes of recursive functions,” Rozpr. Mat.,4, Warszawa (1957).

  2. R. W. Ritchie, “Classes of predictably computable functions,” Trans. Am. Math, Soc.,106, 139–173 (1963).

    Google Scholar 

  3. Ya. B. Kazanovich, “Classification of primitively recursive functions with the help of Turing machines,” Probl. Kibern.,22, 95–106 (1970).

    Google Scholar 

  4. D. Rödding, “Über die Eliminierbarkeit von der Definitionsschemata in der Theorie der rekursiven Funktionen,” Z. Math. Logik Grundl. Math.,10, No. 4, 315–330 (1964).

    Google Scholar 

  5. S. S. Marchenkov, “Elimination of recursion schemes in Grzegorczyk's class,” Mat. Zametki,5, No. 5, 561–568 (1969).

    Google Scholar 

  6. A. A. Muchnik, “Two approaches to the classification of recursive functions,” in: Problems of Mathematical Logic [in Russian], Moscow (1970), pp. 123–138.

  7. A. P. Bel'tyukov, “Iterative description of a class of Grzegorczyk's hierarchy,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,60, 3–14 (1976).

    Google Scholar 

  8. A. P. Bel'tyukov, “Maximal sequence of classes transforming a primitive recursion in a given class,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,68, 3–18 (1977).

    Google Scholar 

  9. R. E. Stearns, J. Hartmanis, and P. M. Lewis, II, “Hierarchies of memory limited computations,” in: IEEE Conf. Rec. Symp. Switch. Circuit Theory and Logic Design, New York (1965), pp. 170–190.

  10. S. S. Marchenkov, “Skolem elementary functions,” Mat. Zametki,17, No. 1, 133–142 (1975).

    Google Scholar 

  11. V. A. Nepomnyashchii, “Rudimentary simulation of undetermined computations,” Kibernetika, No. 2, 23–29 (1973).

    Google Scholar 

  12. Y. Yu, “Rudimentary relations and formal languages,” Ph. D. Dissertation, University of California, Berkeley (1970).

    Google Scholar 

  13. R. M. Smullyan, Theory of Formal Systems, Princeton (1961).

  14. S. V. Pakhomov, “Proof of the noncoincidence of classes of simple primitive recursive functions,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,68, 115–122 (1977).

    Google Scholar 

Download references

Authors
  1. A. P. Bel'tyukov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 30–46, 1979.

The author expresses gratitude to his scientific instructor N. K. Kosovskii for the interest displayed in the work.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bel'tyukov, A.P. A machine description and the hierarchy of initial Grzegorczyk classes. J Math Sci 20, 2280–2289 (1982). https://doi.org/10.1007/BF01629435

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01629435

Keywords

Search

Navigation