Our novel Gödel machine^{1}derives its name and its power
from exploiting provably useful changes of
*any* part of its own code in self-referential fashion.
Its theoretical advantages over previous approaches
can be traced back to the fact
that its notion of optimality is less restricted and
that it has *no* unmodifiable software *at all*.

Its bootstrap mechanism is based on a simple idea.
We provide it with an axiomatic description of
(possibly stochastic) environmental properties and of the
Gödel machine's goals and means.
The latter not only include some initial
problem soving strategy but also a systematic proof searcher seeking an
algorithm for modifying the current Gödel machine together with a formal proof that
the execution of this algorithm will improve the Gödel machine,
according to some utility function or optimality criterion
represented as part of the goals.
*Any* utility function (such as expected future reward in the remaining
lifetime) can be plugged in as an axiom stored in initial program
. In particular, utility may take into account
expected computational costs of proof searching and other actions,
to be derived from the axioms.
During runtime, systematically makes pairs
*(switchprog, proof)*
until it finds a *proof* of:
*`the immediate rewrite of p through current program
switchprog implies higher utility than leaving p as is'.*
Then it executes

Unlike AIXI and HSEARCH, the Gödel machine
can improve the proof searcher itself.
Unlike HSEARCH it does not waste
time on finding programs that provably compute for *all*
when the *current*
is the only object of interest.
It is neither limited to AIXI's assumption of an enumerable environmental
distribution nor to its particular *expected reward* utility function,
since we may plug in *worst-case* or other types of utility functions
as well.
While the hardwired brute force theorem provers
of AIXI and HSEARCH systematically search in *raw* proof
space--they can hide
the proof search cost (exponential in proof size)
in their asymptotic optimality notation
[16,15]--the
*initial*, not yet self-improved Gödel machine
already produces many proofs much faster
as it searches among *online
proof techniques:* proof-generating programs that
may read the Gödel machine's current state.
Hence, unlike AIXI, the Gödel machine can profit
from *online* proof search which may exploit
information obtained through interaction with the environment.

This approach raises
several unconventional issues concerning the
connection between syntax and semantics though.
Proofs are just symbol strings produced
from other symbol strings according to certain
syntactic rules.
Such a symbol string, however, may be interpreted in online fashion as
a statement about the computational costs of
the program that computes it (a semantic issue),
and may suggest a Gödel machine-modifying algorithm
whose execution would be semantically useful *right now* as the
proof is being created.
The proof searcher must deal with the fact
that the utility of certain self-modifications may depend on the
remaining lifetime, and with the problem of producing
the right proof at the right time.

Section 2 will formally describe details of a particular Gödel machine, focusing on its novel aspects, skipping over well-known standard issues treated by any proof theory textbook. Section 2.2.1 will introduce the essential instructions invoked by proof techniques to compute axioms and theorems and to relate syntax to semantics. Section 2.3 will describe one possible initialization of the Gödel machine's proof searcher: Bias-Optimal Proof Search (BIOPS) uses a variant of the already mentioned Optimal Ordered Problem Solver OOPS [38,40] to efficiently search the space of proof techniques. Section 3 will discuss the Gödel machine's limitations, possible types of self-improvements, and additional differences from previous work.

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