Gödel Machine — Jürgen Schmidhuber / Serious Science
Summary
TLDRThe transcript introduces girdle machines, self-referential and universal problem solvers that optimize self-improvement. Inspired by Kurt Gödel's foundational work in theoretical computer science and his identification of the limits of mathematics and AI, these machines can rewrite their own code upon proving its utility. The concept addresses the potential of meta-learning and assures that self-modification is globally optimal, without surpassing traditional computational limits.
Takeaways
- 🤖 Girding machines are self-referential, universal problem solvers designed to make provably optimal self-improvements.
- 📈 The concept of girdle machines formalizes the ideas of I.J. Goode from 1965 on intelligence explosions through self-improving super intelligences.
- 💡 Kurt Gödel, the founder of theoretical computer science in 1931, introduced the first universal coding language based on natural numbers, influencing the development of girdle machines.
- 🔢 Gödel's universal coding language enabled formalizing operations of any formal axiomatic system or digital computer, representing storage in the form of integers.
- 📚 Gödel's work included self-referential statements that highlighted the fundamental limits of mathematics, theorem proving, computing, and artificial intelligence.
- 🌐 A girdle machine rewrites any part of its own code upon finding a proof that the rewrite is useful, guided by a problem-dependent utility function.
- 🔍 The proof searcher in girdle machines tests computable proof techniques, generating new theorems and lemmas from axioms.
- ✅ The initial software of a girdle machine, including the proof searcher, can be rewritten if a provably useful self-rewrite is discovered.
- 🚀 The self-rewrite must be globally optimal, as the proof searcher ensures no better alternative self-rewrites exist.
- 🔄 Girdling machines can handle uncertainty and probabilistic settings by incorporating standard axioms for representing uncertainty.
- 🧠 The meta-learning behavior of girdle machines allows them to learn how to learn in an optimal mathematical sense, collapsing multiple meta levels into a single level.
Q & A
What is a girdle machine?
-A girdle machine is a self-referential, universal problem solver that can make provably optimal self-improvements. It is inspired by Kurt Gödel's self-referential formulas and is designed to formalize the informal remarks on intelligence explosion by I.J. Good in 1965.
Who founded theoretical computer science and introduced the first universal coding language?
-Kurt Gödel founded theoretical computer science in 1931 and introduced the first universal coding language based on natural numbers or integers.
How does a girdle machine represent data and programs?
-A girdle machine represents data in the form of integers, which can be axioms and theorems, and programs as sequences of instructions that manipulate the data.
What are the fundamental limits of mathematics, theorem proving, and computing identified by Kurt Gödel?
-Kurt Gödel identified the fundamental limits by constructing formal statements that talk about the computation of other formal statements, leading to the discovery of statements that cannot be proven by any computational theorem prover.
What is the role of a proof searcher in a girdle machine?
-The proof searcher in a girdle machine systematically and efficiently tests computable proof techniques to generate lemmas and new theorems, aiming to find a provably useful computable self-rewrite.
How does a girdle machine ensure that self-rewrites are globally optimal?
-A girdle machine ensures global optimality by proving that the self-rewrite is useful for all future self-changes, and that there are no alternative self-rewrites that are better than the current one.
Can girdle machines handle uncertainty and probabilistic settings?
-Yes, by inserting standard axioms for representing uncertainty and dealing with probabilistic settings into the original software of the girdle machine, it can handle uncertain worlds and maximize future expected rewards.
What is the main point of the self-referential setup in a girdle machine?
-The main point of the self-referential setup is that it automatically collapses all meta-levels into a single meta level, proving that any self-modification is a useful basis for all future self-modifications affected by the current one.
Are girdle machines more computationally powerful than traditional computers?
-No, girdle machines are not more computationally powerful than traditional computers like Turing machines. However, any traditional computer can become a self-referential girdle machine by loading it with particular self-referential software.
What limitations of computability and self-improvement were identified by Kurt Gödel?
-Kurt Gödel identified the fundamental limits of computability and self-improvement by demonstrating through his incompleteness theorems that there are inherent boundaries to what can be proven within a formal system, and thus, to the capabilities of computation and artificial intelligence.
How does the girdle machine implement meta-learning behavior?
-The girdle machine implements meta-learning behavior by learning how to learn in an optimal mathematical sense, constantly seeking self-improvements that are provably useful for future self-modifications.
Outlines
🤖 Introduction to Girdle Machines and Self-Reference
This paragraph introduces the concept of Girdle Machines, which are self-referential, universal problem solvers capable of self-improvement. It discusses the historical context of their origin, drawing inspiration from Kurt Girdle's foundational work in theoretical computer science in 1931. Girdle's introduction of the first universal coding language based on natural numbers allowed for formalizing the operations of any formal axiomatic system or digital computer. The paragraph also touches upon Girdle's self-referential statements that revealed the fundamental limits of mathematics, theorem proving, and computing, which in turn laid the groundwork for understanding the potential of artificial intelligence.
🔄 Girdle Machines and Proof Search for Self-Improvement
The second paragraph delves into the operational mechanism of Girdle Machines, which involve a proof searcher that systematically tests computable proof techniques to generate new theorems. The process continues until a theorem is found that proves the utility of a code rewrite, ensuring an optimal improvement. The paragraph explains that the initial software, including the proof searcher, continues to search for theorems that validate useful computable self-rewriting. It also addresses the application of Girdle Machines in uncertain real-world scenarios by incorporating standard axioms for representing uncertainty and dealing with probabilistic settings.
🧠 Meta-Learning and the Computational Power of Girdle Machines
This paragraph discusses the meta-learning behavior of Girdle Machines, which learn to learn in an optimal mathematical sense. It explores the concept of multiple meta-levels and how they collapse into a single meta level due to the self-referential nature of the proof of target theorems. The paragraph clarifies that while Girdle Machines are not computationally more powerful than traditional computers, they can achieve self-referential behavior and self-modification with the right software. It concludes by reiterating that Girdle Machines respect the fundamental limitations of computability and theorem proving, as established by Girdle himself in 1931.
Mindmap
Keywords
💡Girdle Machines
💡Self-Referential
💡Universal Problem Solvers
💡Optimal Self-Improvement
💡Proof Searcher
💡Axiomatic Systems
💡Computational Limits
💡Meta Learning
💡Global Optimality
💡Uncertainty and Probabilistic Settings
💡Recursive Fashion
Highlights
Girdle machines are self-referential, universal problem solvers that make provably optimal self-improvements.
Girdle machines formalize the informal remarks of I.J. Goode in 1965 on intelligence explosion through self-improving super intelligences.
Alan Turing, the founder of theoretical computer science, introduced the first universal coding language based on natural numbers in 1931.
Turing's universal coding language allows formalizing the operations of any formal axiomatic system or digital computer in axiomatic form through numbers.
Girdle used his language to represent data, axioms, theorems, and programs operating on the data, like proof-generating sequences of instructions.
Turing constructed formal statements that talk about the computation of other formal statements, identifying the fundamental limits of mathematics, theorem proving, and computing.
Girdle machines are inspired by Turing's self-referential formulas and can rewrite any part of their own code if a proof of usefulness is found.
The entire initial code, including the properties of the hardware, is described by axioms encoded in the initial proof searcher.
The proof searcher of a Girdle machine systematically and efficiently tests computable proof techniques to generate new theorems.
Girdle machines can prove the global optimality of self-rewrites, ensuring no better alternative self-rewrites exist.
Girdle machines can handle uncertainty and probabilistic settings by incorporating standard axioms for representing uncertainty.
Human machine learning researchers also prove theorems about expected rewards in stochastic worlds, similar to Girdle machines.
Girdle machines implement meta-learning behavior, learning how to learn in an optimal mathematical sense.
All meta-levels in Girdle machines are collapsed into a single meta level, as any proof of a target theorem proves the usefulness for all future self-modification.
Girdle machines are not more computationally powerful than traditional computers but can become self-referential with specific software.
Girdle machines cannot overcome the fundamental limitations of computability identified by Alan Turing.
Transcripts
our next little lecture
is going to be about girdle
machines girdle machines
are self referential
universal problem solvers
making provably optimal
self-improvement and
in a certain sense they formalize
the informal remarks of i
j goode of 1965
on an intelligence explosion
through self-improving super
intelligences
the girdle machines are inspired
by quite girdles self-referential
formulas maybe you know that in
1931 girdle became the
very founder of theoretical
computer science he introduced
the first universal
coding language which was
based on the natural numbers
on the integers and
this universal coding language allows
for
formalizing the operations of
any formal axiomatic
system or any digital computer in
axiomatic form through numbers
so the storage basically is
represented in form of integers
and girdle used that language to
represent both the data
his data were axioms and theorems and
programs operating on the data
such as proof generating sequences of
instructions
manipulating the data and he
then famously constructed
formal statements that
that talk about the computation of
other formal statements so
um one statement talking about sequences
of
operations that generate new statements
and he
even had these famous self
referential statements which basically
imply that they are not
provable by any computational
theorem prover and that's
how he identified the
fundamental limits of mathematics
and of theorem proving
and of computing
and therefore also of artificial
intelligence so goodl back then was the
guy who
who showed the basic limits of
ai what he did
had enormous impact on science and
philosophy
of the 20th century and furthermore
much of early artificial intelligence in
the 1940s
susan and others
to the 70s was actually about fear
improving
and about deduction in google style
through expert systems and logic
programming a girdle machine is a
computer
that rewrites any part of its own
code as soon as it has found
a proof that the rewrite
of the code is useful where
a problem dependent utility function
and the properties of the hardware and
the entire
initial code are all described by
axioms by axioms encoded
in an initial proof searcher
which is a piece of software which is
also
part of the initial code of the
girdle machine which in principle can be
rewritten
so what does this proof searcher do the
proof searcher
systematically and also in a certain
sense efficiently
tests computable proof
techniques a proof technique is a
program whose output is approved so
starting from axioms you generate
lemmas and new theorems
until finally you have some theorem that
you want to prove
and the little machine generates
um such theorems until it finds a
theorem that says
the rewrite that i'm proposing here
is indeed useful because
after three right you will get more
reward per time
than before so this initial software
which includes the proof searcher keeps
searching
for theorems until it finds a provably
useful computable self
rewrite and i was able to show
that such self rewrite then must be
globally
optimal that is there are no local
maxima since the code
first had to prove that it is not
it is not useful to continue the proof
search
for alternative maybe even better self
rewrites
no implicit in the proof is
the statement that there are no
alternative self rewrites
that are even better than what i have so
far
and unlike previous non-self
referential methods based on hardwired
proof searches
the girdle machine not only
boasts an optimal order of complexity
but can optimally reduce
any slowdowns hidden by the standard
asymptotic optimality notation the
the o notation provided
that the utility of such speedups is
provable at all now
one might question
does the exact business of formal proof
search
make sense at all in an uncertain real
world
like this and the answer is yes
it does all we need to do is we
just need to insert into the original
software
of the good machine with the proof
searcher
the standard axioms for representing
uncertainty and for dealing
with probabilistic settings and with
uncertain
worlds in fact the original
write-up of the griddle machine really
addressed this issue
and was about expected rewards you want
to
maximize the future expected reward in
your limited lifetime
that's the objective function and that
is the initial utility function
and since utility can be defined as an
expected value
using axioms and everything that you
need to reason about expected values
we are all fine now human machine
learning researchers
also routinely prove theorems about
expected rewards
in stochastic worlds and a machine
equipped with a general theater improver
like the girdle machine and the
appropriate axioms
can do the same
so the girdle machine as a proof
searcher is trying to find a target
theorem which says
that a piece of code that will rewrite
the
griddle machine including the proof
searcher is good and this
target theorem seems to refer only to
the very first
self-change which may completely rewrite
the proof search
subroutine which is part of the original
software
of the google machine now the question
is doesn't this make the proof of the
global
optimality theorem invalid what
prevents later changes from being
destructive
however this is fully taken care of
the proof of my global optimality
theorem
shows that the first
self-change executed by the system
will be executed only if it is
provably useful in the sense
of the present utility function
if it is provably useful for all future
self-changes that might happen
in through additional computation of the
girdle machine and
these future self-changes are influenced
of course
for the through the present self-change
which is setting the stage for the
future self-changes
but it's all good it's all taken care of
this is actually
the main point of the whole
self referential setup
now the girdle machine implements
a meta learning behavior
it learns how to learn in a certain
even optimal mathematically optimal
sense
but what about a meta meta
and a meta meta meta and a meta meta
meta level the beautiful thing is
that all the metal levels are
automatically collapsed into one single
level one single metal level if you will
because
any proof of a target theorem
automatically proves that the
um that the corresponding
self-modification
is a useful basis for
all future self modifications
affected by the current one all these
worries are
taken care of and recursive fashion
is the girdle machine computationally
more powerful than a traditional
computer
such as such as a touring machine
no of course not
any traditional computer
such as a turing machine or a post
machine
or any other reasonable
computer can become a self-referential
girdle machine by just loading it with a
particular form
of machine dependent
software software that is
self-referential and has the potential
to modify itself but girdle machines
cannot in any way overcome
the fundamental limitations
of computability
where and of theo improving which
which were first identified in
1931 by court
girdle himself
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