Gödel Machine — Jürgen Schmidhuber / Serious Science

Summary

The 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

00:00

🤖 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.

05:02

🔄 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.

10:04

🧠 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.

Keywords

💡Girdle Machines

Girdle Machines are self-referential, universal problem solvers that are designed to make provably optimal self-improvements. They formalize the concept of an intelligence explosion as proposed by I.J. Goode in 1965, where super intelligences improve themselves to achieve greater intelligence. The concept is inspired by Kurt Gödel's self-referential formulas and his foundational work in theoretical computer science.

💡Self-Referential

Self-referential refers to the ability of a system or formula to refer to itself, often used in the context of logic, mathematics, and computer science. In the video, Girdle Machines are described as self-referential because they can analyze and modify their own code, which is a direct reference to their own structure and operation.

💡Universal Problem Solvers

Universal Problem Solvers are theoretical constructs that can solve any problem that is computable or solvable within a given framework. In the context of the video, Girdle Machines are described as such solvers because they can operate on any formal axiomatic system or digital computer, formalizing operations through numbers and manipulating data and programs.

💡Optimal Self-Improvement

Optimal self-improvement refers to the process of enhancing one's capabilities in the most efficient and effective way possible. In the context of the video, Girdle Machines are designed to undergo self-improvement in a manner that is provably optimal, meaning that any changes they make to themselves are guaranteed to be beneficial and cannot be improved upon.

💡Proof Searcher

A proof searcher is a piece of software that systematically tests computable proof techniques to generate new theorems and lemmas from a set of axioms. In the context of Girdle Machines, the proof searcher is an integral part of the initial code, tasked with finding provably useful computable self-rewriting strategies.

💡Axiomatic Systems

Axiomatic systems are mathematical or logical frameworks based on a set of axioms, or basic assumptions, from which all other truths are derived through formal reasoning. In the video, Girdle Machines operate within an axiomatic system where the initial code, utility function, and hardware properties are all encoded as axioms.

💡Computational Limits

Computational limits refer to the inherent boundaries or constraints on what can be computed or solved by a machine or algorithm. These limits are often defined by mathematical and logical principles, such as Gödel's incompleteness theorems. In the video, the concept of computational limits is central to understanding the design and operation of Girdle Machines.

💡Meta Learning

Meta learning, also known as learning to learn, is the concept of designing algorithms or machines that can improve their own learning process. It involves creating systems that can adapt and become more efficient at learning from data over time. In the context of the video, Girdle Machines implement a form of meta learning by learning how to optimize their own self-improvement strategies.

💡Global Optimality

Global optimality refers to the state where a system has achieved the best possible outcome or solution across the entire range of its possible states. In the video, the concept is used to describe the self-improvement process of Girdle Machines, ensuring that any self-rewrite is not just locally optimal but globally optimal, meaning no better solution exists.

💡Uncertainty and Probabilistic Settings

Uncertainty and probabilistic settings refer to the handling of unknowns or variables in a system or model, often using statistical methods to make predictions or decisions. In the context of the video, Girdle Machines can be adapted to operate in uncertain environments by incorporating standard axioms for representing uncertainty and dealing with probabilistic settings.

💡Recursive Fashion

Recursive fashion refers to a method of operation where a process or function calls itself as part of its execution, often used in programming and algorithms. In the context of the video, the self-referential setup of Girdle Machines operates in a recursive fashion, where each level of self-modification is based on proofs that apply to all future levels of self-modification.

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