Theory of One Tape Linear Time Turing Machines

  • Kohtaro Tadaki
  • Tomoyuki Yamakami
  • Jack C. H. Lin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2932)


A theory of one-tape linear-time Turing machines is quite different from its polynomial-time counterpart. This paper discusses the computational complexity of one-tape Turing machines of various machine types (deterministic, nondeterministic, reversible, alternating, probabilistic, counting, and quantum Turing machines) that halt in time O(n), where the running time of a machine is defined as the height of its computation tree. We also address a close connection between one-tape linear-time Turing machines and finite state automata.


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  1. 1.
    Adleman, L.M., DeMarrais, J., Huang, M.A.: Quantum Computability. SIAM J. Comput. 26, 1524–1540 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bennett, C.H.: Logical Reversibility of Computation. IBM J. Res. Develop. 17, 525–532 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bernstein, E., Vazirani, U.: Quantum Complexity Theory. SIAM J. Comput. 26, 1411–1473 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brodsky, A., Pippenger, N.: Characterizations of 1-Way Quantum Finite Automata. SIAM J. Comput. 31, 1456–1478 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Damm, C., Holzer, M.: Automata that Take Advice. In: Hájek, P., Wiedermann, J. (eds.) MFCS 1995. LNCS, vol. 969, pp. 149–158. Springer, Heidelberg (1995)Google Scholar
  6. 6.
    Dwork, C., Stockmeyer, L.J.: A Time Complexity Gap for Two-Way Probabilistic Finite State Automata. SIAM J. Comput. 19, 1011–1023 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dwork, C., Stockmeyer, L.: Finite State Verifiers I: The Power of Interaction. J. ACM 39, 800–828 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hennie, F.C.: One-Tape, Off-Line Turing Machine Computations. Inform. Control 8, 553–578 (1965)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)zbMATHGoogle Scholar
  10. 10.
    Karp, R.M.: Some Bounds on the Storage Requirements of Sequential Machines and Turing Machines. J. ACM 14, 478–489 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kaneps, J., Freivalds, R.: Minimal Nontrivial Space Complexity of Probabilistic One-Way Turing Machines. In: Rovan, B. (ed.) MFCS 1990. LNCS, vol. 452, pp. 355–361. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  12. 12.
    Kobayashi, K.: On the Structure of One-Tape Nondeterministic Turing Machine Time Hierarchy. Theor. Comput. Sci. 40, 175–193 (1985)zbMATHCrossRefGoogle Scholar
  13. 13.
    Kondacs, A., Watrous, J.: On the Power of Quantum Finite State Automata. In: Proc. 38th FOCS, pp. 66–75 (1997)Google Scholar
  14. 14.
    Macarie, I.I.: Space-Efficient Deterministic Simulation of Probabilistic Automata. SIAM J. Comput. 27, 448–465 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Michel, P.: An NP-Complete Language Accepted in Linear Time by a One-Tape Turing Machine. Theor. Comput. Sci. 85, 205–212 (1991)zbMATHCrossRefGoogle Scholar
  16. 16.
    Rabin, M.O.: Probabilistic Automata. Inform. Control 6, 230–245 (1963)CrossRefGoogle Scholar
  17. 17.
    Turakainen, P.: On Stochastic Languages. Inform. Control 12, 304–313 (1968)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Turakainen, P.: On Languages Representable in Rational Probabilistic Automata. Annales Academiae Scientiarum Fennicae, Ser. A 439, 4–10 (1969)MathSciNetGoogle Scholar
  19. 19.
    Turakainen, P.: Generalized Automata and Stochastic Languages. Proc. Amer. Math. Soc. 21, 303–309 (1969)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Yamakami, T.: Average Case Complexity Theory. Ph.D. Dissertation, University of Toronto. Technical Report 307/97, University of Toronto. See also ECCC Thesis Listings (1997)Google Scholar
  21. 21.
    Yamakami, T.: A Foundation of Programming a Multi-Tape Quantum Turing Machine. In: Kutyłowski, M., Wierzbicki, T., Pacholski, L. (eds.) MFCS 1999. LNCS, vol. 1672, pp. 430–441. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  22. 22.
    Yamakami, T.: Analysis of Quantum Functions. To appear in: International Journal of Foundations of Computer Science; A preliminary version appeared in: Pandu Rangan, C., Raman, V., Sarukkai, S. (eds.): FST TCS 1999. LNCS, vol. 1738, pp. 407–419. Springer, Heidelberg (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Kohtaro Tadaki
    • 1
  • Tomoyuki Yamakami
    • 2
  • Jack C. H. Lin
    • 2
  1. 1.ERATO Quantum Computation and Information ProjectJapan Science and Technology CorporationTokyoJapan
  2. 2.School of Information Technology and EngineeringUniversity of OttawaOttawaCanada

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