User:Vel!/Uncomp

What is the fastest known function in googology? No, it is not BEAF. It is not the R function. It is not loader's function. There is something faster.

The Busy Beaver function, $$\Sigma(n)$$, and the frantic frog function, $$FF(n)$$, are faster than all of those. They are faster than any function a Turing machine can compute. Thus, a Turing machine cannot compute either of them. Now, Turing machines can compute "almost anything". And any device buildable in the real world cannot be any more powerful than a Turing machine. So what this realistically means is that only a few values can be known of these functions.

But there are faster ones, like Rayo's function, $$R(n)$$, and the FOOT function. So what do we do?

How to define an uncomputable
To define an uncomputable, you need to diagonalize over something that is Turing-complete - that is, can compute anything a Turing machine can. Doing this with a Turing machine can give you the busy beaver function.

First, let's explain how a Turing machine works. A Turing machine is an imaginary machine that would edit an infinitely long strip, like ...,0,0,1,1,0,1,1,0,0,1,0,0,.... A Turing machine code consists of commands, which consist of these parts:


 * On which state to execute this command
 * On which color [we will use 2 "colors" for this, '0' and '1'] to execute this command
 * Whether to change the color of the piece it is on
 * Whether to go to the left or to the right. [not moving is also allowed but no busy beavers have it.]
 * What state to go into after completing the command

There should be a command for each combination of state and color. For example, Turing machine code can be like this:


 * State 1, color 0: set color to 1, then go left
 * State 1, color 1: go to state 2
 * State 2, color 0: go to state 1
 * State 2, color 1: set color to 0, then go right

One special state is the "halt" state (write this as H) - if the Turing machine goes into the halt state, it halts. Let's assume the Turing machine will always start in state 1.

An N-state Turing machine is one with N states other than the halt state. A busy beaver is a Turing machine, that when given a completely 0-color strip, will turn the most pieces of the strip color 1 before halting, than any other Turing machine of the state (non-halting machines will not count).

The busy beaver function, $$\Sigma(n)$$, is defined as the number of pieces an n-state busy beaver can color.

Here are the known busy beavers:

1-state:


 * State 1, color 0: set color to 1, go right, then halt
 * State 1, color 1: do nothing [this combination is not used]

Colors 1 piece after 1 step.

2-state:


 * State 1, color 0: set color to 1, go right, then go to state 2
 * State 1, color 1: go to the left, then go to state 2
 * State 2, color 0: set color to 1, go left, then go to state 1
 * State 2, color 1: go to the right, then halt

Colors 4 pieces after 6 steps.

3-state:


 * State 1, color 0: set color to 1, go right, then go to state 2
 * State 1, color 1: go to the right, then halt
 * State 2, color 0: go to the right, then go to state 3
 * State 2, color 1: go to the right
 * State 3, color 0: set color to 1, then go left
 * State 3, color 1: go to the left, then go to state 1

Colors 6 pieces after 14 steps.

4-state:


 * State 1, color 0: set color to 1, go right, then go to state 2
 * State 1, color 1: go to the left, then go to state 2
 * State 2, color 0: set color to 1, go left, then go to state 1
 * State 2, color 1: set color to 0, go left, then go to state 3
 * State 3, color 0: set color to 1, go right, then halt
 * State 3, color 1: go to the left, then go to state 4
 * State 4, color 0: set color to 1, then go to the right
 * State 4, color 1: set color to 0, go right, then go to state 1

Colors 13 pieces after 107 steps

[5- and 6-state record holders coming soon]

So we have: