Gödel Machines: Towards a Technical Justification of Consciousness

  • Jürgen Schmidhuber
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3394)


The growing literature on consciousness does not provide a formal demonstration of the usefulness of consciousness. Here we point out that the recently formulated Gödel machines may provide just such a technical justification. They are the first mathematically rigorous, general, fully self-referential, self-improving, optimally efficient problem solvers, “conscious” or “self-aware” in the sense that their entire behavior is open to introspection, and modifiable. A Gödel machine is a computer that rewrites any part of its own initial code as soon as it finds a proof that the rewrite is useful, where the problem-dependent utility function, the hardware, and the entire initial code are described by axioms encoded in an initial asymptotically optimal proof searcher which is also part of the initial code. This type of total self-reference is precisely the reason for the Gödel machine’s optimality as a general problem solver: any self-rewrite is globally optimal—no local maxima!—since the code first had to prove that it is not useful to continue the proof search for alternative self-rewrites.


Utility Function Inference Rule Turing Machine Problem Solver Axiomatic System 
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.
    Banzhaf, W., Nordin, P., Keller, R.E., Francone, F.D.: Genetic Programming – An Introduction. Morgan Kaufmann Publishers, San Francisco (1998)zbMATHGoogle Scholar
  2. 2.
    Bellman, R.: Adaptive Control Processes. Princeton University Press, Princeton (1961)zbMATHGoogle Scholar
  3. 3.
    Blum, M.: A machine-independent theory of the complexity of recursive functions. Journal of the ACM 14(2), 322–336 (1967)zbMATHCrossRefGoogle Scholar
  4. 4.
    Blum, M.: On effective procedures for speeding up algorithms. Journal of the ACM 18(2), 290–305 (1971)zbMATHCrossRefGoogle Scholar
  5. 5.
    Cantor, G.: Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Crelle’s Journal für Mathematik 77, 258–263 (1874)Google Scholar
  6. 6.
    Clocksin, W.F., Mellish, C.S.: Programming in Prolog, 3rd edn. Springer, Heidelberg (1987)zbMATHGoogle Scholar
  7. 7.
    Cramer, N.L.: A representation for the adaptive generation of simple sequential programs. In: Grefenstette, J.J. (ed.) Proceedings of an International Conference on Genetic Algorithms and Their Applications. Carnegie-Mellon University, Hillsdale NJ, July 24-26, 1985. Lawrence Erlbaum Associates, Mahwah (1985)Google Scholar
  8. 8.
    Crick, F., Koch, C.: Consciousness and neuroscience. Cerebral Cortex 8, 97–107 (1998)CrossRefGoogle Scholar
  9. 9.
    Fitting, M.C.: First-Order Logic and Automated Theorem Proving, 2nd edn. Graduate Texts in Computer Science. Springer, Berlin (1996)zbMATHGoogle Scholar
  10. 10.
    Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38, 173–198 (1931)CrossRefGoogle Scholar
  11. 11.
    Heisenberg, W.: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik 33, 879–893 (1925)CrossRefGoogle Scholar
  12. 12.
    Hochreiter, S., Younger, A.S., Conwell, P.R.: Learning to learn using gradient descent. In: Dorffner, G., Bischof, H., Hornik, K. (eds.) ICANN 2001. LNCS, vol. 2130, pp. 87–94. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Hofstadter, D.R.: Gödel, Escher, Bach: an Eternal Golden Braid. Basic Books (1979)Google Scholar
  14. 14.
    Holland, J.H.: Properties of the bucket brigade. In: Proceedings of an International Conference on Genetic Algorithms, Hillsdale, NJ. Lawrence Erlbaum, Mahwah (1985)Google Scholar
  15. 15.
    Hutter, M.: Towards a universal theory of artificial intelligence based on algorithmic probability and sequential decisions. In: Flach, P.A., De Raedt, L. (eds.) ECML 2001. LNCS (LNAI), vol. 2167, pp. 226–238. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  16. 16.
    Hutter, M.: The fastest and shortest algorithm for all well-defined problems. International Journal of Foundations of Computer Science 13(3), 431–443 (2002); (On J. Schmidhuber’s SNF grant 20-61847)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hutter, M.: Self-optimizing and Pareto-optimal policies in general environments based on Bayes-mixtures. In: Kivinen, J., Sloan, R.H. (eds.) COLT 2002. LNCS (LNAI), vol. 2375, pp. 364–379. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  18. 18.
    Hutter, M.: Universal Artificial Intelligence: Sequential Decisions based on Algorithmic Probability. Springer, Berlin (2004) (On J. Schmidhuber’s SNF grant 20-61847)Google Scholar
  19. 19.
    Kaelbling, L.P., Littman, M.L., Moore, A.W.: Reinforcement learning: a survey. Journal of AI research 4, 237–285 (1996)Google Scholar
  20. 20.
    Kolmogorov, A.N.: Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin (1933)Google Scholar
  21. 21.
    Kolmogorov, A.N.: Three approaches to the quantitative definition of information. Problems of Information Transmission 1, 1–11 (1965)Google Scholar
  22. 22.
    Lenat, D.: Theory formation by heuristic search. Machine Learning 21 (1983)Google Scholar
  23. 23.
    Levin, L.A.: Universal sequential search problems. Problems of Information Transmission 9(3), 265–266 (1973)Google Scholar
  24. 24.
    Levin, L.A.: Randomness conservation inequalities: Information and independence in mathematical theories. Information and Control 61, 15–37 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Li, M., Vitányi, P.M.B.: An Introduction to Kolmogorov Complexity and its Applications, 2nd edn. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  26. 26.
    Löwenheim, L.: Über Möglichkeiten im Relativkalkül. Mathematische Annalen 76, 447–470 (1915)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Moore, C.H., Leach, G.C.: FORTH - a language for interactive computing (1970)Google Scholar
  28. 28.
    Penrose, R.: Shadows of the mind. Oxford University Press, Oxford (1994)Google Scholar
  29. 29.
    Popper, K.R.: All Life Is Problem Solving. Routledge, London (1999)Google Scholar
  30. 30.
    Rice, H.G.: Classes of recursively enumerable sets and their decision problems. Trans. Amer. Math. Soc. 74, 358–366 (1953)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Samuel, A.L.: Some studies in machine learning using the game of checkers. IBM Journal on Research and Development 3, 210–229 (1959)CrossRefGoogle Scholar
  32. 32.
    Schmidhuber, J.: Evolutionary principles in self-referential learning. Diploma thesis, Institut für Informatik, Technische Universität München (1987)Google Scholar
  33. 33.
    Schmidhuber, J.: Reinforcement learning in Markovian and non-Markovian environments. In: Lippman, D.S., Moody, J.E., Touretzky, D.S. (eds.) Advances in Neural Information Processing Systems 3 (NIPS 3), pp. 500–506. Morgan Kaufmann, San Francisco (1991)Google Scholar
  34. 34.
    Schmidhuber, J.: A self-referential weight matrix. In: Proceedings of the International Conference on Artificial Neural Networks, Amsterdam, pp. 446–451. Springer, Heidelberg (1993)Google Scholar
  35. 35.
    Schmidhuber, J.: On learning how to learn learning strategies. Technical Report FKI-198-94, Fakultät für Informatik, Technische Universität München (1994); See [49,47]Google Scholar
  36. 36.
    Schmidhuber, J.: Discovering solutions with low Kolmogorov complexity and high generalization capability. In: Prieditis, A., Russell, S. (eds.) Machine Learning: Proceedings of the Twelfth International Conference, pp. 488–496. Morgan Kaufmann Publishers, San Francisco (1995)Google Scholar
  37. 37.
    Schmidhuber, J.: A computer scientist’s view of life, the universe, and everything. In: Freksa, C., Jantzen, M., Valk, R. (eds.) Foundations of Computer Science. LNCS, vol. 1337, pp. 201–208. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  38. 38.
    Schmidhuber, J.: Discovering neural nets with low Kolmogorov complexity and high generalization capability. Neural Networks 10(5), 857–873 (1997)CrossRefGoogle Scholar
  39. 39.
    Schmidhuber, J.: Algorithmic theories of everything. Technical Report IDSIA-20-00, quant-ph/0011122, IDSIA, Manno (Lugano), Switzerland (2000), Sections 1-5: see 40; Section 6: see [41]Google Scholar
  40. 40.
    Schmidhuber, J.: Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit. International Journal of Foundations of Computer Science 13(4), 587–612 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Schmidhuber, J.: The Speed Prior: a new simplicity measure yielding near-optimal computable predictions. In: Kivinen, J., Sloan, R.H. (eds.) COLT 2002. LNCS (LNAI), vol. 2375, pp. 216–228. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  42. 42.
    Schmidhuber, J.: Bias-optimal incremental problem solving. In: Becker, S., Thrun, S., Obermayer, K. (eds.) Advances in Neural Information Processing Systems 15 (NIPS 15), pp. 1571–1578. MIT Press, Cambridge (2003)Google Scholar
  43. 43.
    Schmidhuber, J.: Gödel machines: self-referential universal problem solvers making provably optimal self-improvements. Technical Report IDSIA-19-03, arXiv:cs.LO/0309048, IDSIA, Manno-Lugano, Switzerland (2003)Google Scholar
  44. 44.
    Schmidhuber, J.: Gödel machine home page, with frequently asked questions (2004),
  45. 45.
    Schmidhuber, J.: Gödel machines: Fully self-referential optimal universal self-improvers. In: Goertzel, B., Pennachin, C. (eds.) Real AI: New Approaches to Artificial General Intelligence. Springer, Heidelberg (2004) (in press)Google Scholar
  46. 46.
    Schmidhuber, J.: Optimal ordered problem solver. Machine Learning 54, 211–254 (2004)zbMATHCrossRefGoogle Scholar
  47. 47.
    Schmidhuber, J., Zhao, J., Schraudolph, N.: Reinforcement learning with self-modifying policies. In: Thrun, S., Pratt, L. (eds.) Learning to learn, pp. 293–309. Kluwer, Dordrecht (1997)Google Scholar
  48. 48.
    Schmidhuber, J., Zhao, J., Wiering, M.: Simple principles of metalearning. Technical Report IDSIA-69-96, IDSIA (1996); See [47, 97]Google Scholar
  49. 49.
    Schmidhuber, J., Zhao, J., Wiering, M.: Shifting inductive bias with success-story algorithm, adaptive Levin search, and incremental self-improvement. Machine Learning 28, 105–130 (1997)CrossRefGoogle Scholar
  50. 50.
    Skolem, T.: Logisch-kombinatorische Untersuchungen über Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theorem über dichte Mengen. Skrifter utgit av Videnskapsselskapet in Kristiania, I, Mat.-Nat. Kl., N4:1–36 (1919)Google Scholar
  51. 51.
    Solomonoff, R.J.: A formal theory of inductive inference. Part I. Information and Control 7, 1–22 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Solomonoff, R.J.: Complexity-based induction systems. IEEE Transactions on Information Theory  IT-24(5), 422–432 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Solomonoff, R.J.: Progress in incremental machine learning—Preliminary Report for NIPS 2002 Workshop on Universal Learners and Optimal Search; revised Sept 2003. Technical Report IDSIA-16-03, IDSIA, Lugano (2003)Google Scholar
  54. 54.
    Sutton, R., Barto, A.: Reinforcement learning: An introduction. MIT Press, Cambridge (1998)Google Scholar
  55. 55.
    Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. In: Proceedings of the London Mathematical Society, Series 2, vol. 41, pp. 230–267 (1936)Google Scholar
  56. 56.
    Wolpert, D.H., Macready, W.G.: No free lunch theorems for search. IEEE Transactions on Evolutionary Computation 1 (1997)Google Scholar
  57. 57.
    Zuse, K.: Rechnender Raum. Friedrich Vieweg & Sohn, Braunschweig, 1969. English translation: Calculating Space, MIT Technical Translation AZT-70-164-GEMIT, Massachusetts Institute of Technology (Proj. MAC), Cambridge, Mass. 02139 (Febuary 1970)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jürgen Schmidhuber
    • 1
    • 2
  1. 1.IDSIAManno (Lugano)Switzerland
  2. 2.TU MunichGarching, MünchenGermany

Personalised recommendations