Minds and Machines

, Volume 14, Issue 2, pp 133–143

Undecidability in the Imitation Game

  • Yuzuru Sato
  • Takashi Ikegami


This paper considers undecidability in the imitation game, the so-called Turing Test. In the Turing Test, a human, a machine, and an interrogator are the players of the game. In our model of the Turing Test, the machine and the interrogator are formalized as Turing machines, allowing us to derive several impossibility results concerning the capabilities of the interrogator. The key issue is that the validity of the Turing test is not attributed to the capability of human or machine, but rather to the capability of the interrogator. In particular, it is shown that no Turing machine can be a perfect interrogator. We also discuss meta-imitation game and imitation game with analog interfaces where both the imitator and the interrogator are mimicked by continuous dynamical systems.

analog computation dynamical systems imitation game Turing machine undecidability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anderlini, L. (1990), 'Some Notes on Church's Thesis and the Theory of Games', Theory and Decision 29, pp. 19–52.Google Scholar
  2. Blum, L., Cucker, F., Shub, M. and Smale, S. (1998), Complexity and Real Computation, New York: Springer-Verlag.Google Scholar
  3. Bradford, P.G. and Wollowski, M. (1993), 'A Formalization of the Turing Test', Proceedings of 5th Midwest Artificial Intelligence and Cognitive Science Conference, pp. 83–87.Google Scholar
  4. Campagnolo, M.L., Moore, C. and Costa, J.F. (2000), 'Iteration, Inequalities, and Differentiability in Analog Computers', Journal of Complexity 16(4), pp. 642–660.Google Scholar
  5. Crutchfield, J.P. (1998), 'Dynamical Embodiment of Computation in Cognitive Processes', Behavioral and Brain Sciences 21(5), pp. 635–637.Google Scholar
  6. Crutchfield, J.P. and Young, K. (1989), 'Inferring Statistical Complexity', Physical Review Letters 63, pp. 105–108.Google Scholar
  7. Dowling, W.F. (1990), 'Computer Viruses: Diagonalization and Fixed Points', Notice of the AMS 37, pp. 858–861.Google Scholar
  8. Forner, L. (1997), 'Entertaining Agents: A Sociological Case Study', Proceedings of International Conference on Autonomous Agents 97, pp. 122–129.Google Scholar
  9. French, R.M. (2000), 'The Turing Test: The First Fifty Years', Trends in Cognitive Sciences 4(3), pp. 115–121.Google Scholar
  10. Harnad, S. (1990), 'The Symbol Grounding Problem', Physica D 42, pp. 335–346.Google Scholar
  11. Harnad, S. (2001), 'Minds, Machines, and Turing: The Indistinguishability of Indistinguishables', Journal of Logic, Language, and Information 9(4), pp. 425–445.Google Scholar
  12. Hofstadter, D.R. and Dennett, D.C. (1981), The Mind's I, Basic Books.Google Scholar
  13. Mauldin, M. (1994), 'Chatterbots, Tiny MUDs, and the Turing Test', Proceedings of National Conference of Artificial Intelligence 94, pp. 16–21.Google Scholar
  14. Moore, C. (1990), 'Unpredictability and Undecidability in Dynamical Systems', Physical Review Letters 64, pp. 2354–2357.Google Scholar
  15. Moore, C. (1998), 'Dynamical Recognizers: Real-Time Language Recognition by Analog Computers', Theoretical Computer Science 201, pp. 99–136.Google Scholar
  16. Pollack, J.B. (1991), 'The Induction of Dynamical Recognizers', Machine Learning 7, pp. 227–252.Google Scholar
  17. Pour-El, M.B. and Richards, J.I. (1989), 'Computability in Analysis and Physics', Berlin: Springer-Verlag.Google Scholar
  18. Premack, D. and Woodruff, G. (1978), 'Does the Chimpanzee Have a Theory of Mind?', Behavioral and Brain Sciences 1, pp. 516–526.Google Scholar
  19. Rice, H.G. (1953), 'Classes of Recursively Enumerable Sets and Their Decision Problem', AMS 89, pp. 25–59.Google Scholar
  20. Sato, Y. and Ikegami, T. (1999), 'Undecidability of the Imitation Game', Proceedings of Symposium on Imitation in Animals and Artifacts, pp. 157–159.Google Scholar
  21. Sato, Y., Taiji, M. and Ikegami, T. (2001), 'Computation with Switching Map Systems: Nonlinearity and Computational Complexity', Santa Fe Institute working paper, WP01–12–083.Google Scholar
  22. Searle, J.R. (1980), 'Minds, Brains, and Programs', Behavioral and Brain Sciences 3, pp. 417–424.Google Scholar
  23. Siegelmann, H.T. (1999), Neural Networks and Analog Computation, Boston: Birkhauser.Google Scholar
  24. Turing, A.M. (1936), 'On Computable Numbers, with an Application to the Enentscheidungsproblem', Proceedings of the London Mathematical Society 2(42), pp. 230–265.Google Scholar
  25. Turing, A.M. (1948), 'Intelligent Machinery', in B. Meltzer and D. Michie, eds., National Physical Laboratory Report, Machine Intelligence 5, pp. 3–23, 1969.Google Scholar
  26. Turing, A.M. (1950), 'Computing Machinery and Intelligence', Mind 59(236), pp. 433–460.Google Scholar
  27. Tversky, A. and Kahneman, D. (1974), 'Judgement under Uncertainty: Heuristics and Biases', Science 185, pp. 1124–1131.Google Scholar
  28. van Gelder, T. (1998), 'The Dynamical Hypothesis in Cognitive Science', Behavioral and Brain Science 21, pp. 1–14.Google Scholar
  29. Wagenaar, W.A. (1972), 'Generation of Random Sequences by Human Subjects: A Critical Survey of Literature', Psychological Bulletin 77, pp. 65–72.Google Scholar
  30. Watt, S.N.K. (1996), 'Naive Psychology and the Inverse Turing Test', Psycholoquy 14(7), p. 1.Google Scholar
  31. Weizenbaum, J. (1966), 'ELIZA — A Computer Program for the Study of Natural Language Communication Between Man and Machine', Communication of the ACM 9, pp. 36–45.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Yuzuru Sato
    • 1
  • Takashi Ikegami
    • 2
  1. 1.Brain Science InstituteInstitute of Physical and Chemical Research (RIKEN)Wako, SaitamaJapan
  2. 2.Graduate School of Arts and ScienceUniversity of TokyoMeguro-ku, TokyoJapan

Personalised recommendations