Tag Systems and the Complexity of Simple Programs

  • Turlough NearyEmail author
  • Damien Woods
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9099)


In this mini-survey we discuss time complexity and program size results for universal Turing machines, tag systems, cellular automata, and other simple models of computation. We discuss results that show that many of the simplest known models of computation including the smallest known universal Turing machines and the elementary cellular automaton Rule 110 are efficient simulators of Turing machines. We also recall a recent result where the halting problem for tag systems with only 2 symbols (the minimum possible) is proved undecidable. This result has already yielded applications including a significant improvement on previous undecidability bounds for the Post correspondence problem and the matrix mortality problem.


Cellular Automaton Turing Machine Universal Machine Universal Turing Machine Deterministic Turing 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.
    Cassaigne, J., Karhumäki, J.: Examples of undecidable problems for 2-generator matrix semigroups. Theoretical Computer Science 204(1–2), 29–34 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cocke, J., Minsky, M.: Universality of tag systems with \(P = 2\). Journal of the Association for Computing Machinery 11(1), 15–20 (1964)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Cook, M.: Universality in elementary cellular automata. Complex Systems 15(1), 1–40 (2004)zbMATHMathSciNetGoogle Scholar
  5. 5.
    De Mol, L.: Tag systems and Collatz-like functions. Theoretical Computer Science 390(1), 92–101 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    De Mol, L.: On the complex behavior of simple tag systems - an experimental approach. Theoretical Computer Science 412(1–2), 97–112 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ehrenfeucht, A., Karhumäki, J., Rozenberg, G.: The (generalized) Post correspondence problem with lists consisting of two words is decidable. Theoretical Computer Science 21(2), 119–144 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Halava, V., Harju, T.: Mortality in matrix semigroups. American Mathematical Monthly 108(7), 649–653 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Halava, V., Harju, T., Hirvensalo, M.: Undecidability bounds for integer matrices using Claus instances. International Journal of Foundations of Computer Science 18(5), 931–948 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Harju, T., Margenstern, M.: Splicing systems for universal Turing machines. In: Ferretti, C., Mauri, G., Zandron, C. (eds.) DNA 2004. LNCS, vol. 3384, pp. 149–158. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  11. 11.
    Hermann, G.T.: The uniform halting problem for generalized one state Turing machines. In: Proceedings of the Ninth Annual Symposium on Switching and Automata Theory (FOCS), pp. 368–372, IEEE Computer Society Press, Schenectady, New York, Oct. 1968Google Scholar
  12. 12.
    Hooper, P.: Some small, multitape universal Turing machines. Information Sciences 1(2), 205–215 (1969)CrossRefGoogle Scholar
  13. 13.
    Kudlek, M.: Small deterministic Turing machines. Theoretical Computer Science 168(2), 241–255 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    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. 149–158. Springer, Heidelberg (2002) CrossRefGoogle Scholar
  15. 15.
    Lindgren, K., Nordahl, M.G.: Universal computation in simple one-dimensional cellular automata. Complex Systems 4(3), 299–318 (1990)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Margenstern, M.: Non-erasing Turing machines: A new frontier between a decidable halting problem and universality. In: Baeza-Yates, R.A., Poblete, P.V., Goles, E. (eds.) LATIN. LNCS, vol. 911, pp. 386–397. Springer, Heidelberg (1995)Google Scholar
  17. 17.
    Margenstern, M.: Frontier between decidability and undecidability: a survey. Theoretical Computer Science 231(2), 217–251 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Matiyasevich, Y., Sénizergues, G.: Decision problems for semi-Thue systems with a few rules. Theoretical Computer Science 330(1), 145–169 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Minsky, M.: Recursive unsolvability of Post’s problem of “tag" and other topics in theory of Turing machines. Annals of Mathematics 74(3), 437–455 (1961)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Minsky, M.: Size and structure of universal Turing machines using tag systems. In: Recursive Function Theory: Proceedings, Symposium in Pure Mathematics, vol. 5, pp. 229–238, AMS, Provelence (1962)Google Scholar
  21. 21.
    Neary, T.: Small universal Turing machines. Ph.D thesis, National University of Ireland, Maynooth (2008)Google Scholar
  22. 22.
    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
  23. 23.
    Neary, T., Woods, D.: Small fast universal Turing machines. Theoretical Computer Science 362(1–3), 171–195 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Neary, T.: Undecidability in binary tag systems and the Post correspondence problem for five pairs of words. In: Mayr, Ernst W., Ollinger, Nicolas (eds.) 32nd International Symposium on Theoretical Aspects of Computer Science, (STACS 2015), vol. 30 of LIPIcs, pp. 649–661 (2015)Google Scholar
  25. 25.
    Neary, T., Woods, D.: Small weakly universal turing machines. In: Kutyłowski, M., Charatonik, W., Gębala, M. (eds.) FCT 2009. LNCS, vol. 5699, pp. 262–273. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  26. 26.
    Neary, T., Woods, D.: Four small universal Turing machines. Fundamenta Informaticae 91(1), 123–144 (2009)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Neary, T., Woods, D.: The complexity of small universal turing machines: a survey. In: Bieliková, M., Friedrich, G., Gottlob, G., Katzenbeisser, S., Turán, G. (eds.) SOFSEM 2012. LNCS, vol. 7147, pp. 385–405. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  28. 28.
    Pavlotskaya, L.: Solvability of the halting problem for certain classes of Turing machines. Mathematical Notes (Springer) 13(6), 537–541 (1973). (Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 899–909, June, 1973)zbMATHCrossRefGoogle Scholar
  29. 29.
    Pavlotskaya, L.: Dostatochnye uslovija razreshimosti problemy ostanovki dlja mashin T’juring. Avtomaty i Mashiny, pp. 91–118 (1978). (Sufficient conditions for the halting problem decidability of Turing machines. In Russian)Google Scholar
  30. 30.
    Post, E.L.: Formal reductions of the general combinatorial decision problem. American Journal of Mathmatics 65(2), 197–215 (1943)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Post, E.L.: Absolutely unsolvable problems and relatively undecidable propositions - account of an anticipation. In: Davis, M. (ed.) The undecidable: basic papers on undecidable propositions, unsolvable problems and computable functions, pages 340–406. Raven Press, New York (1965). (Corrected republication, Dover publications, New York, 2004)Google Scholar
  32. 32.
    Rogozhin, Y.: Small universal Turing machines. Theoretical Computer Science 168(2), 215–240 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Rogozhin, Y., Verlan, S.: On the rule complexity of universal tissue P systems. In: Freund, R., Păun, G., Rozenberg, G., Salomaa, A. (eds.) WMC 2005. LNCS, vol. 3850, pp. 356–362. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  34. 34.
    Rothemund, P.W.K.: A DNA and restriction enzyme implementation of Turing Machines, In: Lipton, R.J., Baum, E.B. (eds.) DNA Based Computers: Proceeding of a DIMACS Workshop, vol. 2055 of DIMACS, pp. 75–119. AMS, Princeton University (1996)Google Scholar
  35. 35.
    Shannon, C.E.: A universal Turing machine with two internal states. Automata Studies, Annals of Mathematics Studies 34, 157–165 (1956)MathSciNetGoogle Scholar
  36. 36.
    Siegelmann, H.T., Margenstern, M.: Nine switch-affine neurons suffice for Turing universality. Neural Networks 12(4–5), 593–600 (1999)CrossRefGoogle Scholar
  37. 37.
    Wang, H.: Tag systems and lag systems. Mathematical Annals 152(4), 65–74 (1963)zbMATHCrossRefGoogle Scholar
  38. 38.
    Watanabe, S.: 4-symbol 5-state universal Turing machine. Information Processing Society of Japan Magazine 13(9), 588–592 (1972)Google Scholar
  39. 39.
    Wolfram, S.: A new kind of science. Wolfram Media Inc (2002)Google Scholar
  40. 40.
    Woods, D., Neary, T.: On the time complexity of 2-tag systems and small universal Turing machines. In: 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 439–446. IEEE, Berkeley, California, Oct. 2006Google Scholar
  41. 41.
    Woods, D., Neary, T.: Small semi-weakly universal Turing machines. Fundamenta Informaticae 91(1), 179–195 (2009)zbMATHMathSciNetGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2015

Authors and Affiliations

  1. 1.Institute of NeuroinformaticsUniversity of Zürich and ETH ZürichZürichSwitzerland
  2. 2.Division of Engineering and Applied ScienceCalifornia Institute of TechnologyPasadenaUSA

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