Introduction and Outline

All traditional algorithms for problem solving / machine learning / reinforcement learning [20] are hardwired. Some are designed to improve some limited type of policy through experience, but are not part of the modifiable policy, and cannot improve themselves in a theoretically sound way. Humans are needed to create new / better problem solving algorithms and to prove their usefulness under appropriate assumptions.

Here we will eliminate the restrictive need for human
effort in the most general way possible,
leaving all the work
including the proof search to a system
that can rewrite and improve itself in arbitrary
computable ways and in a most efficient fashion.
To attack this *``Grand Problem of Artificial Intelligence,''*
we introduce a novel class of optimal,
fully self-referential [11]
general problem solvers called
*Gödel machines* [45,46].^{1}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 [11].
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 -optimal
(has an optimal order of complexity)
in the sense of Theorem 5.1, Section 5.
Unlike hardwired systems such as Hutter's
[16,17]
and Levin's [24,26]
(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 -notation. Our approach yields the first
theoretically sound, fully self-referential,
optimal, general problem solvers.

**Outline.** Section 2 presents basic concepts
and fundamental limitations, Section 3
the essential details of a self-referential axiomatic system,
Section 4 the Global Optimality Theorem 4.1,
and Section 5 the -optimal
(Theorem 5.1) initial proof searcher.
Section 6 provides examples
and relations to previous work,
briefly discusses issues such as
a *technical* justification of consciousness,
and lists answers to several frequently asked questions
about Gödel machines.