Small Semi-weakly Universal Turing Machines

  • Damien Woods
  • Turlough Neary
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4664)


We present two small universal Turing machines that have 3 states and 7 symbols, and 4 states and 5 symbols respectively. These machines are semi-weak which means that on one side of the input they have an infinitely repeated word and on the other side there is the usual infinitely repeated blank symbol. This work can be regarded as a continuation of early work by Watanabe on semi-weak machines. One of our machines has only 17 transition rules making it the smallest known semi-weakly universal Turing machine. Interestingly, our two machines are symmetric with Watanabe’s 7-state and 3-symbol, and 5-state and 4-symbol machines, even though we use a different simulation technique.


Turing Machine Transition Rule Input Symbol Index Cycle Small Machine 
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.
    Baiocchi, C.: Three small universal Turing machines. In: Margenstern, M., Rogozhin, Y. (eds.) MCU 2001. LNCS, vol. 2055, pp. 1–10. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. 2.
    Cocke, J., Minsky, M.: Universality of tag systems with P= 2. Journal of the Association for Computing Machinery 11(1), 15–20 (1964)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Cook, M.: Universality in elementary cellular automata. Complex Systems 15(1), 1–40 (2004)MathSciNetGoogle Scholar
  4. 4.
    Hermann, G.T.: The uniform halting problem for generalized one state Turing machines. In: FOCS. Proceedings of the ninth annual Symposium on Switching and Automata Theory, Schenectady, New York, October 1968, pp. 368–372. IEEE Computer Society Press, Los Alamitos (1968)Google Scholar
  5. 5.
    Kudlek, M.: Small deterministic Turing machines. Theoretical Computer Science 168(2), 241–255 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kudlek, M., Rogozhin, Y.: A universal Turing machine with 3 states and 9 symbols. In: Kuich, W., Rozenberg, G., Salomaa, A. (eds.) DLT 2001. LNCS, vol. 2295, pp. 311–318. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Margenstern, M.: Frontier between decidability and undecidability: a survey. Theoretical Computer Science 231(2), 217–251 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Margenstern, M., Pavlotskaya, L.: On the optimal number of instructions for universality of Turing machines connected with a finite automaton. International Journal of Algebra and Computation 13(2), 133–202 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Michel, P.: Small Turing machines and generalized busy beaver competition. Theoretical Computer Science 326, 45–56 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Minsky, M.: A 6-symbol 7-state universal Turing machines. Technical Report 54-G-027, MIT (August 1960)Google Scholar
  11. 11.
    Minsky, M.: Size and structure of universal Turing machines using tag systems. In: Recursive Function Theory: Proceedings, Symposium in Pure Mathematics, Provelence, vol. 5, pp. 229–238. AMS (1962)Google Scholar
  12. 12.
    Neary, T.: Small polynomial time universal Turing machines. In: MFCSIT’06. Fourth Irish Conference on the Mathematical Foundations of Computer Science and Information Technology, Ireland, pp. 325–329. University College Cork (2006)Google Scholar
  13. 13.
    Neary, T., Woods, D.: P-completeness of cellular automaton Rule 110. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 132–143. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Neary, T., Woods, D.: Small fast universal Turing machines. Theoretical Computer Science 362(1–3), 171–195 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Neary, T., Woods, D.: Four small universal Turing machines. In: Margenstern, M., Rogozhin, Y. (eds.) MCU 2007. LNCS, vol. 4664, pp. 242–254. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Pavlotskaya, L.: Solvability of the halting problem for certain classes of Turing machines. Mathematical Notes (Springer) 13(6), 537–541 (June 1973) (Translated from Matematicheskie Zametki 13(6), 899–909 (June 1973))Google Scholar
  17. 17.
    Pavlotskaya, L.: Dostatochnye uslovija razreshimosti problemy ostanovki dlja mashin T’juring. Avtomaty i Mashiny (Sufficient conditions for the halting problem decidability of Turing machines) (in Russian), 91–118 (1978)Google Scholar
  18. 18.
    Priese, L.: Towards a precise characterization of the complexity of universal and nonuniversal Turing machines. SIAM J. Comput. 8(4), 508–523 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Robinson, R.M.: Minsky’s small universal Turing machine. International Journal of Mathematics 2(5), 551–562 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Rogozhin, Y.: Sem’ universal’nykh mashin T’juringa. Systems and theoretical programming (Seven universal Turing machines, in Russian). Mat. Issled. 69, 76–90 (1982)Google Scholar
  21. 21.
    Rogozhin, Y.: Small universal Turing machines. Theoretical Computer Science 168(2), 215–240 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Shannon, C.E.: A universal Turing machine with two internal states. Automata Studies, Annals of Mathematics Studies 34, 157–165 (1956)MathSciNetGoogle Scholar
  23. 23.
    Watanabe, S.: On a minimal universal Turing machines. Technical report, MCB Report, Tokyo (August 1960)Google Scholar
  24. 24.
    Watanabe, S.: 5-symbol 8-state and 5-symbol 6-state universal Turing machines. Journal of the ACM 8(4), 476–483 (1961)zbMATHCrossRefGoogle Scholar
  25. 25.
    Watanabe, S.: 4-symbol 5-state universal Turing machines. Information Processing Society of Japan Magazine 13(9), 588–592 (1972)Google Scholar
  26. 26.
    Wolfram, S.: A new kind of science. Wolfram Media, Inc. (2002)Google Scholar
  27. 27.
    Woods, D., Neary, T.: On the time complexity of 2-tag systems and small universal Turing machines. In: FOCS. 47th Annual IEEE Symposium on Foundations of Computer Science, Berkeley, California, October 2006, pp. 439–446. IEEE, Los Alamitos (2006)Google Scholar
  28. 28.
    Woods, D., Neary, T.: The complexity of small universal Turing machines. In: CiE 2007. Computation and Logic in the Real World: Third Conference of Computability in Europe, Siena, Italy, June 2007. LNCS, vol. 4497, Springer, Heidelberg (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Damien Woods
    • 1
  • Turlough Neary
    • 2
  1. 1.Department of Computer Science, University College CorkIreland
  2. 2.TASS, Department of Computer Science, National University of Ireland MaynoothIreland

Personalised recommendations