Four Small Universal Turing Machines

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


We present small polynomial time universal Turing machines with state-symbol pairs of (5,5), (6,4), (9,3) and (18,2). These machines simulate our new variant of tag system, the bi-tag system and are the smallest known universal Turing machines with 5, 4, 3 and 2-symbols respectively. Our 5-symbol machine uses the same number of instructions (22) as the smallest known universal Turing machine by Rogozhin.


Turing Machine Computation Step Production Simulation 47th Annual IEEE Symposium Universal Machine 
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  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 ACM 11(1), 15–20 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Hermann, G.: The uniform halting problem for generalized one state Turing machines. In: Proceedings, Ninth Annual Symposium on Switching and Automata Theory, New York, October 1968, pp. 368–372. IEEE, Los Alamitos (1968)CrossRefGoogle Scholar
  4. 4.
    Kudlek, M.: Small deterministic Turing machines. Theoretical Computer Science 168(2), 241–255 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    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
  6. 6.
    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
  7. 7.
    Minsky, M.: Size and structure of universal Turing machines using tag systems. In: Recursive Function Theory, Symposium in Pure Mathematics, Provelence, vol. 5, pp. 229–238. AMS (1962)Google Scholar
  8. 8.
    Minsky, M.: Computation, finite and infinite machines. Prentice-Hall, Englewood Cliffs (1967)zbMATHGoogle Scholar
  9. 9.
    Neary, T.: Small polynomial time universal Turing machines. In: Hurley, T., Seda, A., et al. (eds.) 4th Irish Conference on the Mathematical Foundations of Computer Science and Information Technology(MFCSIT), Cork, Ireland, August 2006, pp. 325–329 (2006)Google Scholar
  10. 10.
    Neary, T., Woods, D.: A small fast universal Turing machine. Technical Report NUIM-CS-TR-2005-12, National university of Ireland, Maynooth (2005)Google Scholar
  11. 11.
    Neary, T., Woods, D.: Small fast universal Turing machines. Theoretical Computer Science 362(1–3), 171–195 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Pavlotskaya, L.: Solvability of the halting problem for certain classes of Turing machines. Mathematical Notes (Springer) 13(6), 537–541 (1973)zbMATHCrossRefGoogle Scholar
  13. 13.
    Pavlotskaya, L.: Dostatochnye uslovija razreshimosti problemy ostanovki dlja mashin T’juring. Problemi kibernetiki, 91–118 (1978) (Sufficient conditions for the halting problem decidability of Turing machines) (in Russian))Google Scholar
  14. 14.
    Robinson, R.: Minsky’s small universal Turing machine. International Journal of Mathematics 2(5), 551–562 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rogozhin, Y.: Small universal Turing machines. Theoretical Computer Science 168(2), 215–240 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Shannon, C.E.: A universal Turing machine with two internal states. Automata Studies, Annals of Mathematics Studies 34, 157–165 (1956)MathSciNetGoogle Scholar
  17. 17.
    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. 132–143. IEEE, Los Alamitos (2006)Google Scholar
  18. 18.
    Woods, D., Neary, T.: The complexity of small universal Turing machines. In: Cooper, S.B., Lowe, B., Sorbi, A. (eds.) Computability in Europe 2007. CIE, Sienna, Italy, June 2007. LNCS, vol. 4497, pp. 791–798. Springer, Heidelberg (2007)Google Scholar
  19. 19.
    Woods, D., Neary, T.: Small semi-weakly universal Turing machines. In: Durand-Lose, J., Margenstern, M. (eds.) Machines, Computations, and Universality (MCU), Orélans, France, September 2007. LNCS, vol. 4664, pp. 306–323. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

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

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