User:Vel!/pu/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:

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

2-state:

Colors 4 pieces after 6 steps.
 * 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

3-state:

Colors 6 pieces after 14 steps.
 * 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

4-state:

Colors 13 pieces after 107 steps
 * 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

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

So we have:

The frantic frog function is like the busy beaver function but enumerates state changes instead of colored pieces. The Turing machines start differing between the two at n=3. The frantic frog function grows faster than the busy beaver function. and are both probably larger than anything ever defined with computable functions.

Rayo's number
Rayo's number works using FOST, a theory of set theory. It is the smallest integer greater than or equal to the largest number expressible in first order set theory with a googol (10100) or less symbols. This can be generalized to n symbols, and the function is called the rayo function. R(n) outgrows the busy beaver and frantic frog functions by a lot. Another number, BIG FOOT, is based on an extension of FOST called oodle theory - 'FOOT'. FOOT(n) is like R(n) but with FOOT instead of FOST. BIG FOOT is FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(10100)))))))))). Later, it was found that FOOT is equivalent to FOST with a truth predicate (something that can evaluate diagonalization over FOST).

Yet another is little biggedon, which is defined using further extensions.