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Jürgen Schmidhuber's
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last update 2012
Original preprint cs.LO/0309048
(2003, revised 2006):

(old v4 html)
Journal (PDF, 2009)

Goedel machine logo
Goedel machines are self-referential universal problem solvers making provably optimal self- improvements.
They formalize I. J. Good's informal remarks (1965) on an "intelligence explosion" through self-improving "super-intelligences".
Related links:
Optimal scientist
Meta- Learning
Universal AI
History converging?
Inspired by Goedel's self-referential formulas (1931) showing that math is either flawed in a certain sense, or contains unprovable truths.

Scholarpedia on Gödel machines: in the articles on Metalearning (2010) & Universal Search (2007)

Goedel portrait .
Gödel machines overcome certain limitations of OOPS and of non- self- referential Universal learning algorithms and AIXI and of Universal search and the asymptotically "fastest" algorithm for all problems and of our previous metalearners.

Disclaimer: Gödel machines can be implemented on traditional computers! No hypothetical super- Turing capabilities and the like.

Is the GM conscious? Is consciousness the ability to execute unlimited self- inspection and provably useful self-change (modulo limits of computability and provability)? Then the Goedel machine's Global Optimality Theorem provides the first technical justification of consciousness in the context of general problem solving.

Abstract: We present the first class of mathematically rigorous, general, fully self-referential, self-improving, optimally efficient problem solvers. Inspired by Kurt Gödel's celebrated self-referential formulas (1931), a Gödel machine (or `Goedel machine' but not `Godel machine') rewrites any part of its own code as soon as it has found a proof that the rewrite is useful, where the problem-dependent utility function and the hardware and the entire initial code are described by axioms encoded in an initial proof searcher which is also part of the initial code. The searcher systematically and efficiently tests computable proof techniques (programs whose outputs are proofs) until it finds a provably useful, computable self-rewrite. We show that such a 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. Unlike previous non-self-referential methods based on hardwired proof searchers, ours not only boasts an optimal order of complexity but can optimally reduce any slowdowns hidden by the O()-notation, provided the utility of such speed-ups is provable at all. (FAQ)

Gödel machine papers:
9. B. R. Steunebrink, J. Schmidhuber. Towards an Actual Gödel Machine Implementation. In P. Wang, B. Goertzel, eds., Theoretical Foundations of Artificial General Intelligence. Springer, 2012. PDF.
8. B. Steunebrink, J. Schmidhuber. A Family of Gödel Machine Implementations. In Proc. Fourth Conference on Artificial General Intelligence (AGI-11), Google, Mountain View, California, 2011. PDF.
7. J. Schmidhuber. Ultimate Cognition à la Gödel. Cognitive Computation 1(2):177-193, 2009. PDF. (Springer.)
6. J. Schmidhuber. Completely Self-Referential Optimal Reinforcement Learners. In W. Duch et al. (Eds.): Proc. Intl. Conf. on Artificial Neural Networks ICANN'05, LNCS 3697, pp. 223-233, Springer-Verlag Berlin Heidelberg, 2005 (plenary talk). PDF.
5. J. Schmidhuber. Gödel machines: Fully Self-Referential Optimal Universal Self-Improvers. In B. Goertzel and C. Pennachin, eds.: Artificial General Intelligence, p. 119-226, 2006. PDF.
4. J. Schmidhuber. A Technical Justification of Consciousness. 9th annual meeting of the Association for the Scientific Study of Consciousness ASSC, Caltech, Pasadena, CA, 2005.
3. J. Schmidhuber. Goedel machines: Towards a Technical Justification of Consciousness. In D. Kudenko, D. Kazakov, and E. Alonso, eds.: Adaptive Agents and Multi-Agent Systems III LNCS 3394, p. 1-23, Springer, 2005. PDF.
2. Section on Goedel machines on page 235 of: J. Schmidhuber, OOPS, Machine Learning, 54, 211-254, 2004. PDF. (The editors also offered to publish the entire original Goedel machine paper instead of the OOPS paper, but the latter came first chronologically).
1. arXiv: cs.LO/0309048 (2003, revised Dec 2006)

Invited talks on Gödel machine and general reinforcement learners and New AI:

15 Jan 2011: Winter Intelligence Conference, Oxford (on universal AI and theory of fun). See video.
3 Oct 2009: Singularity Summit, New York City. Videos: 10min, 40min.
6 Mar 2009: Keynote for Artificial General Intelligence AGI-09
27 July 2006: Artificial Life, Jena
27 May 2006: Turing Days, Istanbul
31 Jan 2006: Dagstuhl Castle: Kolmogorov complexity
Sept 11-15 2005: Plenary talk at ICANN'05, Warsaw, Poland
Dec 9 2004: Keynote talk at Neuro-IT'04, Munich, Germany
Nov 9 2004: Plenary talk at ANNIE'04, St. Louis, US
Jun 4 2004: ETH Zuerich, Switzerland
Apr 1 2004: Univ. Birmingham, UK
Mar 30 2004: Keynote talk at 4th Symposium on Adaptive Agents and Multi-Agent Systems, Univ. Leeds, UK
Feb 19 2004: Univ. Newcastle upon Tyne, UK
Jan 30 2004: Univ. Amsterdam, NL

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robot on robot horse riding off into the sunset, steered by a Goedel machine
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