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Ultimate Cognition à la Gödel

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Abstract

“All life is problem solving,” said Popper. To deal with arbitrary problems in arbitrary environments, an ultimate cognitive agent should use its limited hardware in the “best” and “most efficient” possible way. Can we formally nail down this informal statement, and derive a mathematically rigorous blueprint of ultimate cognition? Yes, we can, using Kurt Gödel’s celebrated self-reference trick of 1931 in a new way. Gödel exhibited the limits of mathematics and computation by creating a formula that speaks about itself, claiming to be unprovable by an algorithmic theorem prover: either the formula is true but unprovable, or math itself is flawed in an algorithmic sense. Here we describe an agent-controlling program that speaks about itself, ready to rewrite itself in arbitrary fashion once it has found a proof that the rewrite is useful according to a user-defined utility function. Any such a rewrite is necessarily globally optimal—no local maxima!—since this proof necessarily must have demonstrated the uselessness of continuing the proof search for even better rewrites. Our self-referential program will optimally speed up its proof searcher and other program parts, but only if the speed up’s utility is indeed provable—even ultimate cognition has limits of the Gödelian kind.

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Notes

  1. Or `Goedel machine’, to avoid the Umlaut. But ‘Godel machine’ would not be quite correct. Not to be confused with what Penrose calls, in a different context, ‘Gödel’s putative theorem-proving machine’ [36]!

  2. Turing reformulated Gödel’s unprovability results in terms of TMs [74] which subsequently became the most widely used abstract model of computation. It is well known that there are universal TMs that in a certain sense can emulate any other TM or any other known computer. Gödel’s integer-based formal language can be used to describe any universal TM, and vice versa.

  3. We see that certain parts of the current s may not be directly observable without changing the observable itself. Sometimes, however, axioms and previous observations will allow the Gödel machine to deduce time-dependent storage contents that are not directly observable. For instance, by analyzing the code being executed through instruction pointer IP in the example above, the value of IP at certain times may be predictable (or postdictable, after the fact). The values of other variables at given times, however, may not be deducible at all. Such limits of self-observability are reminiscent of Heisenberg’s celebrated uncertainty principle [16], which states that certain physical measurements are necessarily imprecise, since the measuring process affects the measured quantity.

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Acknowledgments

Thanks to Alexey Chernov, Marcus Hutter, Jan Poland, Ray Solomonoff, Sepp Hochreiter, Shane Legg, Leonid Levin, Alex Graves, Matteo Gagliolo, Viktor Zhumatiy, Ben Goertzel, Will Pearson, and Faustino Gomez for useful comments on drafts or summaries or earlier versions of this article. I am also grateful to many others who asked questions during Gödel machine talks or sent comments by email.

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  1. IDSIA, Galleria 2, 6928, Manno-Lugano, Switzerland

    Jürgen Schmidhuber

  2. TU München, Boltzmannstr. 3, 85748, Garching bei München, Germany

    Jürgen Schmidhuber

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Schmidhuber, J. Ultimate Cognition à la Gödel. Cogn Comput 1, 177–193 (2009). https://doi.org/10.1007/s12559-009-9014-y

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