Skip to main content
SpringerLink
Log in

Decidability and undecidability of the halting problem on Turing machines, a survey

  • Conference paper
  • First Online:
Logical Foundations of Computer Science (LFCS 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1234))

Included in the following conference series:

Abstract

The paper surveys the main results obtained for Turing machines about the frontier between a decidable halting problem and universality. The notion of decidability criterion is introduced. Techniques for decidability proofs and for contracting universal objects are sketchily explained. A new approach for finding very small universal Turing machines is considered in the last part of the paper.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Diekert V. and Kudlek M. Small Deterministic Turing Machines, Papers on Automata and Languages, Dep. of Math., Karl Marx Univ. of Economics, Budapest, 1988–4, p.77–87, (1989)

    Google Scholar 

  2. Hermann G.T. The uniform halting problem for generalised one state Turing machines, Proceedings of the Ninth Annual Switching and Automata Theory Symposium, (1968)

    Google Scholar 

  3. Kudlek M. Small Deterministic Turing Machines, Theoretical Computer Science, 168–2, Special issue on Universal Machines and Computations, 241–255, (1996)

    Google Scholar 

  4. Margenstern M. Turing machines: on the frontier between a decidable halting problem and universality. Lecture Notes in Computer Science, 710, 375–385, in Proceedings of FCT'93, (1993)

    Google Scholar 

  5. Margenstern M. Non-erasing Turing machines: a new frontier between a decidable halting problem and universality. Lecture Notes in Computer Science, 911, 386–397, in Proceedings of LATIN'95), (1995)

    Google Scholar 

  6. Margenstern M. Results on the halting problem, LMPS'95 International Conference, Florence, August 1995

    Google Scholar 

  7. Margenstern M. The laterality problem for non-erasing Turing machines on {0,1} is completely solved. RAIRO, 39p. (1997) (to appear)

    Google Scholar 

  8. Margenstern M. and Pavlotskaya L. Deux machines de Turing universelles à au plus deux instructions gauches. C.R.A.S., Paris, 320, I, 1395–1400, (1995)

    Google Scholar 

  9. Margenstern M. and PavlotskaÏa L. Vers une nouvelle approche de l'universalité concernant les machines de Turing, LITP/IBP research report Nℴ95.58 (1995)

    Google Scholar 

  10. Minsky M.L. Computation: Finite and Infinite Machines. Prentice Hall, Englewood Cliffs, N.J. (1967)

    Google Scholar 

  11. Pavlotskaya L.M. Razreshimost' problemy ostanovki dlja nekotorykh klassov mashin T'juringa. Matematicheskie Zametki, 13, (6), 899–909, Ijun' 1973 899–909. (transl. Solvability of the halting problem for certain classes of Turing machines, Notes of the Acad. Sci. USSR, 13 (6) Nov.1973, 537–541)

    Google Scholar 

  12. Pavlotskaya L.M. O minimal'nom chisle razlichnykh kodov vershin v grafe universal'noj mashiny T'juringa. Diskretnyj analiz, Sbornik trudov instituta matematiki SO AN SSSR, 27, 52–60, (1975) (On the minimal number of distinct codes for the vertices of the graph of a universal Turing machine) (in Russian)

    Google Scholar 

  13. Pavlotskaya L.M. Dostatochnye uslovija razreshimosti problemy ostanovki dlja mashin T'juring, Avtomaty i mashiny, 91–118, 1978 (Sufficient conditions for halting problem decidability of Turing machines) (in Russian)

    Google Scholar 

  14. Pavlotskaya L.M, On machines, universal by extensions, Theoretical Computer Science, 168-2, Special issue on Universal Machines and Computations, pp.257–266, (1996)

    Google Scholar 

  15. Priese L. Towards a precise characterization of the complexity of universal and nonuniversal Turing machines, SIAM Journal of Computation, 8, 4, 508–523, (1979)

    Google Scholar 

  16. Rogozhin Ju.V. Sem' universal'nykh mashin T'juringa. Matematicheskie Issledovanija, 69, 76–90, 1982 (Seven universal Turing machines) (in Russian)

    Google Scholar 

  17. Rogozhin Ju.V. Universal'naja mashina T'juringa s 10 sostojanijami i 3 simvolami. Matematicheskie Issledovanija, 69, 76–90, 1992 (A universal Turing machine with 10 states and 3 symbols) (in Russian)

    Google Scholar 

  18. Rogozhin Ju.V. Small universal Turing machines, Theoretical Computer Science, 168-2, Special issue on Universal Machines and Computations, 215–240, (1996)

    Google Scholar 

  19. Shannon C.E. A universal Turing machine with two internal states. Ann. of Math. Studies, 34, 157–165, (1956)

    Google Scholar 

  20. Turing A.M. On computable real numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc, ser. 2, 42, 230–265 (1936)

    Google Scholar 

  21. Wang H. Tag Systems and Lag Systems, Mat. Annalen, 152, (1963), 65–74

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département d'Informatique, I.U.T. de Metz, Île du Saulcy, 57045, Metz Cedex, France

    Maurice Margenstern

  2. L.I.A.F.A. (Université Paris 7), France

    Maurice Margenstern

  3. L.R.I.M. (Université de Metz), France

    Maurice Margenstern

Authors
  1. Maurice Margenstern
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Sergei Adian Anil Nerode

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Margenstern, M. (1997). Decidability and undecidability of the halting problem on Turing machines, a survey. In: Adian, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 1997. Lecture Notes in Computer Science, vol 1234. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63045-7_23

Download citation

  • DOI: https://doi.org/10.1007/3-540-63045-7_23

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63045-6

  • Online ISBN: 978-3-540-69065-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation