The Gödel machine [56]
explicitly addresses the `Grand Problem of Artificial Intelligence'
[58]
by optimally dealing with limited resources in general reinforcement
learning settings,
and with the possibly huge (but constant) slowdowns buried
by AIXI
[22]
in the somewhat misleading 
-notation.
It is designed to solve arbitrary computational problems beyond those
solvable by plain OOPS,
such as maximizing the expected future reward of a robot in a possibly
stochastic and reactive environment
(note that the total utility of some robot behavior may be hard
to verify--its evaluation may consume the robot's entire lifetime).
How does it work? While executing some arbitrary initial problem solving strategy, the Gödel machine simultaneously runs a proof searcher which systematically and repeatedly tests proof techniques. Proof techniques are programs that may read any part of the Gödel machine's state, and write on a reserved part which may be reset for each new proof technique test. In an example Gödel machine [56] this writable storage includes the variables proof and switchprog, where switchprog holds a potentially unrestricted program whose execution could completely rewrite any part of the Gödel machine's current software. Normally the current switchprog is not executed. However, proof techniques may invoke a special subroutine check() which tests whether proof currently holds a proof showing that the utility of stopping the systematic proof searcher and transferring control to the current switchprog at a particular point in the near future exceeds the utility of continuing the search until some alternative switchprog is found. Such proofs are derivable from the proof searcher's axiom scheme which formally describes the utility function to be maximized (typically the expected future reward in the expected remaining lifetime of the Gödel machine), the computational costs of hardware instructions (from which all programs are composed), and the effects of hardware instructions on the Gödel machine's state. The axiom scheme also formalizes known probabilistic properties of the possibly reactive environment, and also the initial Gödel machine state and software, which includes the axiom scheme itself (no circular argument here). Thus proof techniques can reason about expected costs and results of all programs including the proof searcher.
Once check() has identified a provably good switchprog, the latter is executed (some care has to be taken here because the proof verification itself and the transfer of control to switchprog also consume part of the typically limited lifetime). The discovered switchprog represents a globally optimal self-change in the following sense: provably none of all the alternative switchprogs and proofs (that could be found in the future by continuing the proof search) is worth waiting for.
There are many ways of initializing the proof searcher. Although identical proof techniques may yield different proofs depending on the time of their invocation (due to the continually changing Gödel machine state), there is a bias-optimal and asymptotically optimal proof searcher initialization based on a variant of OOPS [56] (Section 9). It exploits the fact that proof verification is a simple and fast business where the particular optimality notion of OOPS is appropriate. The Gödel machine itself, however, may have an arbitrary, typically different and more powerful sense of optimality embodied by its given utility function.