User blog:Boboris02/MBOT

I have something new today!

And it's big.....It's a whole new theory that could construct very,very large numbers,possibly larger than the current record holder BIG FOOT!

It's MBOT (Mush9-Boris Output Theory).

Background
Ok,a while ago Mush9 wrote a blog post about the NOOP FUNCTION,which got me really interasted.

This could possibly beat the largest of numbers ever defined and change the way we think about the creation of numbers.There was no information for the function itself,how it works or what it actualy does.

So I decided to make one myself! On Mush's talk page I wrote a few messages about my idea.It would be very helpfull if you read them before reading this blog post.

Introduction
MBOT is based of ZF or ZFC.

The thing that is different than in set theory or any other particular Mathematical theory that I can think of right now is that in MBOT we can add systems which can recreate any existing function,given enough space.Infact I believe (although I can't proove it) that MBOT can recreate EVERY possible function - computable or not,even when there are infinitly many of them!

So the first thing we need is to know what an operation type is.

In MBOT there are 6 types of operations: Statments,Theorems,Calculations,Functions,Systems and Notators.

For now I will focus on systems.

A system is the way that a function works:what gives it it's growth.

In MBOT we use \(\Phi\) to denote this is a system-type.

\(\Phi\)(Phi) basics
First,\(\Phi\) means the operation is a system type.

Here we use the logical connectivities from set theory.However,non of them have anything to do with their traditional meanings! This should not confuse people.

\(\Rightarrow\) here means next step/character/move

\(\cup\) means a long chain of repetitions.

\(\Leftrightarrow\) means that the system returns _ if _.

\({_n}\),where n is a subscript means an n-level system.For those sequences to be infinite usualy they collapse in on themselves.

\(\neg\) on itself has no meaning and can be set to anything,even more than the others!

Now,you might be noticing a problem with those:There definition is very weak and ill-defined.

So we can have an equasion-like explanation of them in the actual system.

Since their definitions aren't well-defind that leaves them open for any definition that you can set,respecting the conditions above.This makes Phi systems particularly strong,since we do not need a new symbol for every type of recursion possible,but rather set the meaning of them in the system.

Next we can define more than one systems,that operate differently,but have one be based on the other.Then we can say one of them is inside the other.

Examples
These are some examples of Phi systems in action:

The Ackerman function(\(Ack(a,b)\)) would be  a \(\Phi(a)\Rightarrow b=\Phi(a-1)\Rightarrow (\Phi(a)\Rightarrow (b-1))\) system,inside of a \(\Phi(a)\Rightarrow 0 = \Phi(a-1)\Rightarrow 1\) -system,inside of a \(\Phi(0)\Rightarrow b = b+1\) system.

What would four-entry arrays in BEAF(\(\{a,b,c,d\}\)) be?

It would be a...

\(\Phi(a) \Rightarrow b \Rightarrow c \Rightarrow d = \Phi(a) \Rightarrow (\Phi(a) \Rightarrow (b-1) \Rightarrow c \Rightarrow d) \Rightarrow (c-1) \Rightarrow d\) -system,inside of a \(\Phi(a)\Rightarrow 1 \ldots = a\) system inside of a \(\Phi(a)\Rightarrow b\Rightarrow 1\Rghtarrow d = \Phi(a)\Rightarrow b\Rightarrow (\Phi(a)\Rightarrow (b-1) \Rightarrow 1\Rightarrow d)\Rightarrow (d-1)\) system inside a \(\Phi(a)\Rightarrow b = a^b\) system.

If at some point the system changes,it will have to be pointed out that the system has changed and the primary system will be sent back inside the secondary system.

Example:What would \(f_3(n)\) in the fast-growing hierarchy be?

It would be \(\Phi_3(n)\) inside a \(\Phi_x(n) = \Phi^{n}_{(x-1)}(n)\) -system inside a \(\Phi_0(n) = n+1\) -system

Pretty easy....right?

Well at this level,yes!...

Now we can start using other symbols for stronger definitions:

Let's now get the same definition for the BEAF system and put it all inside of a

\(\Phi(a)\cup b = \Phi(a)\underbrace{\Rightarrow a\Rightarrow a\Rightarrow \ldots \Rightarrow a}_b\) - system.

 This way \(\Phi(n)\cup n\) has the growth rate of \(f_{\omega^{\omega}}(n)\)!

 Now some stronger definitions:

 \(\Rightarrow n\) means repeating everything previously set in the operation \(n\) times.

 \(\Rightarrow n\Rightarrow m\) means repeating the system n,m times.

 \(\Rightarrow n\Rightarrow m\Rightarrow k\) means repeating the system \(n\Rightarrow m\), k times.

<p style="font-weight:normal;"> \(a \cup 1\) means a long chain of repetitions

<p style="font-weight:normal;"> \(\Phi(a) \cup 1 = \Phi(a) \Rightarrow (\Phi(a)\Rightarrow a \Rightarrow \ldots \Rightarrow a)\Rightarrow a\)

<p style="font-weight:normal;"> \(\Phi(a) \cup 2 = \Phi(a) \cup 1 \Rightarrow \Phi(a-1) \cup 2) \Rightarrow a\)

<p style="font-weight:normal;">With just these operators we can set any system we want:

<p style="font-weight:normal;">\(\Phi(a) \Rightarrow b = \Phi^b(a) \Rightarrow (\Phi(a) \Rightarrow (b-1)) \cup a \Rightarrow a\) -system for example is a system whith a growth rate of something like \(f_{\psi(\Omega^{\Omega^{\varepsilon_0}+b})}(a)\)

<p style="font-weight:normal;">Another good system is

<p style="font-weight:normal;">\(\Phi(a) \Rightarrow b \cup c \Rightarrow d = \Phi^b(a) \Rightarrow (\Phi(a) \cup b) \Rightarrow c \cup \Phi_c(b-1)\)

<p style="font-weight:normal;">where \(\Phi_a(b) = \Phi_a(b-1) \Rightarrow a \cup b\),where \(\Phi_a(1) = \Phi_{a-1}(Phi(a\cup a))\Rightarrow a\)

<p style="font-weight:normal;">This has a growth rate of \(f_{\psi(\psi_I(0))}(a)\)

<p style="font-weight:normal;">Now,It's time for uncomputable functions!

Uncomputable systems!
The problem with what we were doing previously is that no matter how long we make it,it will allways be computable and thus will be surpassed by the Busy beaver function,the Xi function,Rayo's function,FOOT function,......

So we now percieve the logical conectivity \(\Leftrightarrow\) and the meaningless by itself \(\neg\).

We also use \(\mathbb{C}\) to denote if a system is computable,\(\mathbb{U}\) if it's uncomputable,\(\lor\) for minimum,\(^{>}\) for overgrowing,\(^{>^*}\) for eventual domination and \(^{>}+(@) F(n)\) for overgrowing another function/system F(n) by @ at a time.

With this we can make uncomputable systems,stronger than all computable systems and now,just like we could make computable systems as strong as we like,we can now make uncomputable systems as strong as we like.

Now there is no limit to how far they go! Infact Phi systems are like word descriptions,but with symbols and thus any function that can be described on paper can be described in a Phi system.

That means that there exists a system using a finite amount of charecters that is larger than FOOT function.

I don't know how many symbols this requires but it should not be too much since the Busy beaver can be described in about 20 symbols:

\(\Phi(n) = n\neg n \Leftrightarrow k \Leftrightarrow m=\ldots 00\underbrace{111\ldots 111}_m00\ldots \Leftrightarrow 0=\infty\)

This represents \(\Sigma(n)\),where the tape is infinite and set completely to zeros.

n represents the input,k represents the number of states and because of the \(\neg n\) part that becomes n and m is the longest possible line of ones that could be written using the other conditions and is the output.

My guess is that FOOT function should be able to be described in less than 1000 symbols,but I can't prove any of that without actualy showing the solution and I don't have it.

Conclusion
The whole aim of this theory was to show that there are ways of writing a (sort of) turing machine for all functions and that there are obvious outputs for most functions.However it also shows that sometimes there is no obvious output!

This is where NOOP function comes in!

This whole thing was inspired by many things: We define \(\Delta_x\) as the strongest possible system describable in \(x\) symbols or less(describing every number as one symbol and excluding and Phi).
 * Mush9's idea for NOOP function and the whole No Obvious OutPut thingy.
 * ZFC
 * Turing machines
 * Computation
 * FOOT
 * Unprovability and Uncomputability Theory
 * Bowers' K-system

Now we define \(\Delta_x[n]\) as the largest number describable in n symbols or less in that system!

This is exactly K-systems,exept this time it's well-defined! The reason why Bower's function was ill-defind is because it did not use any fixed languige for it's definition.Mine has.....and it uses MBOT as it's fixed languige!

Now we define our final function \(\text{NOOP(n)}\) as \(\Delta_n[n]\) !

My final number will be \(\text{NOOP}^{10}(10^{100})\) and I will call it BIG NOOB!!!