Goedel Machines: Self-Referential Universal Problem Solvers Making Provably Optimal Self-Improvements
Juergen Schmidhuber
TL;DR
Gödel machines provide a rigorous framework for self-improvement by embedding a provably correct proof searcher inside a self-modifying program that rewrites itself only when a formal proof shows the rewrite increases expected utility. A Global Optimality Theorem guarantees that the chosen self-change is globally optimal within the given axioms and utility, while a bias-optimal online proof searcher (BIOPS) guides the discovery of useful rewrites through online universal search. The approach differentiates itself from prior non-self-referential methods by allowing provably beneficial optimizations to any component of the system and by formalizing optimality beyond traditional asymptotic measures. The paper also discusses intrinsic limitations arising from Gödelian incompleteness and outlines a concrete axiomatic framework for hardware, environment, and uncertainty to enable computable, self-contained self-improvement.
Abstract
We present the first class of mathematically rigorous, general, fully self-referential, self-improving, optimally efficient problem solvers. Inspired by Kurt Goedel's celebrated self-referential formulas (1931), such a problem solver 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.