There is another mathematical framework for AGI: the Gödel machine, which was proposed by Jürgen Schmidhuber (who also worked with Marcus Hutter). In the paper Gödel Machines: Self-Referential Universal Problem Solvers Making Provably Optimal Self-Improvements (2003), Schmidhuber describes the Gödel machine as follows
They are universal problem solving systems that interact with some (partially observable) environment and can in principle modify themselves without essential limits apart from the limits of computability. Their initial algorithm is not hardwired; it can completely rewrite itself, but only if a proof searcher embedded within the initial algorithm can first prove that the rewrite is useful, given a formalized utility function reflecting computation time and expected future success (e.g., rewards). We will see that self-rewrites due to this approach are actually globally optimal (Theorem 4.1, Section 4), relative to Gödel's well-known fundamental restrictions of provability. These restrictions should not worry us; if there is no proof of some self-rewrite's utility, then humans cannot do much either.
The initial proof searcher is $O$()-optimal (has an optimal order of complexity) in the sense of Theorem 5.1, Section 5. Unlike hardwired systems such as Hutter's and Levin's (Section 6.4), however, a Gödel machine can in principle speed up any part of its initial software, including its proof searcher, to meet arbitrary formalizable notions of optimality beyond those expressible in the $O$()-notation. Our approach yields the first theoretically sound, fully self-referential, optimal, general problem solvers.
This work combines Russell's bounded rationality with Legg's and Hutter's definition of universal intelligence to provide a framework for AGI that takes into account resource (space and time) constraints and other issues (such as that the environment can modify or even destroy the agent).
