The "Goedel Machine" is an interesting definition, but wildly impractical (though I wouldn't say it's impossible, since it only has to find some improvement, not "the best" improvement; e.g. it could optimise its search procedure in a way that's largely orthogonal to the predicted rewards).

Schmidhuber later defined "PowerPlay" as a framework for building up capabilities in a more practical way, which is more adaptive than just measuring the score on a fixed benchmark. A PowerPlay system searches for (problem, replacement) pairs, where it switches to the replacement if (a) the current system cannot solve that problem, (b) the replacement can solve that problem, and (c) the replacement can also solve all the problems that caused previous replacements (maintained in a list).

I formalised that in Coq many years ago ( http://www.chriswarbo.net/projects/powerplay ), and the general idea can be extended to (a) include these genetic-programming approaches, rather than using a single instance; and (b) could be seeded with desirable benchmarks, etc. to guide the system in a useful direction (so it's "self-invented" problems can include things like "achieves X% on benchmark Y")