Abstract
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.
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References
Banzhaf, W., Nordin, P., Keller, R.E., Francone, F.D.: Genetic Programming – An Introduction. Morgan Kaufmann Publishers, San Francisco (1998)
Bellman, R.: Adaptive Control Processes. Princeton University Press, Princeton (1961)
Blum, M.: A machine-independent theory of the complexity of recursive functions. Journal of the ACM 14(2), 322–336 (1967)
Blum, M.: On effective procedures for speeding up algorithms. Journal of the ACM 18(2), 290–305 (1971)
Cantor, G.: Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Crelle’s Journal für Mathematik 77, 258–263 (1874)
Clocksin, W.F., Mellish, C.S.: Programming in Prolog, 3rd edn. Springer, Heidelberg (1987)
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)
Crick, F., Koch, C.: Consciousness and neuroscience. Cerebral Cortex 8, 97–107 (1998)
Fitting, M.C.: First-Order Logic and Automated Theorem Proving, 2nd edn. Graduate Texts in Computer Science. Springer, Berlin (1996)
Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38, 173–198 (1931)
Heisenberg, W.: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik 33, 879–893 (1925)
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)
Hofstadter, D.R.: Gödel, Escher, Bach: an Eternal Golden Braid. Basic Books (1979)
Holland, J.H.: Properties of the bucket brigade. In: Proceedings of an International Conference on Genetic Algorithms, Hillsdale, NJ. Lawrence Erlbaum, Mahwah (1985)
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)
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)
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)
Hutter, M.: Universal Artificial Intelligence: Sequential Decisions based on Algorithmic Probability. Springer, Berlin (2004) (On J. Schmidhuber’s SNF grant 20-61847)
Kaelbling, L.P., Littman, M.L., Moore, A.W.: Reinforcement learning: a survey. Journal of AI research 4, 237–285 (1996)
Kolmogorov, A.N.: Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin (1933)
Kolmogorov, A.N.: Three approaches to the quantitative definition of information. Problems of Information Transmission 1, 1–11 (1965)
Lenat, D.: Theory formation by heuristic search. Machine Learning 21 (1983)
Levin, L.A.: Universal sequential search problems. Problems of Information Transmission 9(3), 265–266 (1973)
Levin, L.A.: Randomness conservation inequalities: Information and independence in mathematical theories. Information and Control 61, 15–37 (1984)
Li, M., Vitányi, P.M.B.: An Introduction to Kolmogorov Complexity and its Applications, 2nd edn. Springer, Heidelberg (1997)
Löwenheim, L.: Über Möglichkeiten im Relativkalkül. Mathematische Annalen 76, 447–470 (1915)
Moore, C.H., Leach, G.C.: FORTH - a language for interactive computing (1970)
Penrose, R.: Shadows of the mind. Oxford University Press, Oxford (1994)
Popper, K.R.: All Life Is Problem Solving. Routledge, London (1999)
Rice, H.G.: Classes of recursively enumerable sets and their decision problems. Trans. Amer. Math. Soc. 74, 358–366 (1953)
Samuel, A.L.: Some studies in machine learning using the game of checkers. IBM Journal on Research and Development 3, 210–229 (1959)
Schmidhuber, J.: Evolutionary principles in self-referential learning. Diploma thesis, Institut für Informatik, Technische Universität München (1987)
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)
Schmidhuber, J.: A self-referential weight matrix. In: Proceedings of the International Conference on Artificial Neural Networks, Amsterdam, pp. 446–451. Springer, Heidelberg (1993)
Schmidhuber, J.: On learning how to learn learning strategies. Technical Report FKI-198-94, Fakultät für Informatik, Technische Universität München (1994); See [49,47]
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)
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)
Schmidhuber, J.: Discovering neural nets with low Kolmogorov complexity and high generalization capability. Neural Networks 10(5), 857–873 (1997)
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]
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)
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)
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)
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)
Schmidhuber, J.: Gödel machine home page, with frequently asked questions (2004), http://www.idsia.ch/~juergen/goedelmachine.html
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)
Schmidhuber, J.: Optimal ordered problem solver. Machine Learning 54, 211–254 (2004)
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)
Schmidhuber, J., Zhao, J., Wiering, M.: Simple principles of metalearning. Technical Report IDSIA-69-96, IDSIA (1996); See [47, 97]
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)
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)
Solomonoff, R.J.: A formal theory of inductive inference. Part I. Information and Control 7, 1–22 (1964)
Solomonoff, R.J.: Complexity-based induction systems. IEEE Transactions on Information Theory IT-24(5), 422–432 (1978)
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)
Sutton, R., Barto, A.: Reinforcement learning: An introduction. MIT Press, Cambridge (1998)
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)
Wolpert, D.H., Macready, W.G.: No free lunch theorems for search. IEEE Transactions on Evolutionary Computation 1 (1997)
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)
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Schmidhuber, J. (2005). Gödel Machines: Towards a Technical Justification of Consciousness. In: Kudenko, D., Kazakov, D., Alonso, E. (eds) Adaptive Agents and Multi-Agent Systems II. AAMAS AAMAS 2004 2003. Lecture Notes in Computer Science(), vol 3394. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-32274-0_1
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