User blog:Primussupremus/ Alpha notation array notation.

Since I have been away from this wiki for what seems like years and forgotten almost completely about my old notation I decided to start anew with a brand new notation which I call alpha array notation.

Since I've "literally" just thought about this notation a few minutes before writing these words I won't develop a rule set until later on since I don't want to rush into things.

First of all we need to start at the bottom and work up.

I have decided to call this regiment the unary regiment and with it comes the unary vocabulary.

The unary vocabulary is listed as so: The first stage in developing the notation would be to describe what happens with 1 variable.
 * { } are used to hold the array in place.
 * Variables {a,b,c...z ,a,c,b,....z, b,a,c,....z, b,c,a,....z and so on) are used to represent values in which to input into the array.
 * numbers such as: 0,1,2, 56, 723 and 859 are used to represent symbols such as ^ , + , * and so on.

Since a is the first letter in the English alphabet and unary is related to single variable logic that's what we'll use.

{a} =  a or b iff b is in the brackets or the array protectors.

Now we can move on to using 2 variables.

{a,b} = a,b since their is no symbol to connect the two variables together.

So the array {1,2} would simply equal 1,2 since arrays in unary alpha notation aren't valid unless they have a number connecting them together.

Now with a number:

{a,(0),b} = a or b iff b is on the left hand side.

{1 (0) 2} = 1.

{a,(1) ,b} = a*b or b*a iff b is on the left hand side.

{a,(2),b} = a^b = a(1)a(1).....a(1)a  with b-1 (1)'s.

{a,(3),b} = a↑↑b = a(2)a(2)....a(2)a with b-1 (2)'s.

It seems like we have come across an unexpected surprise because a beautiful pattern is beginning to emerge, for every number x>=2 the value of {a,(x),b} is a{x-1 up arrows}b.

We can say that once you've got any arbitrary value of x under your belt you can safely say that you've completed the 1st stage of the notation.

The growth rate at this stage would be fω(n).

The 2nd stage of the notation would let us develop such puny things as Grahams number and all of that nonsense.

I am calling it nonsense because Grahams number is a circus trick made by a circus acrobat who liked to play with coloured pencils and graphs.

Any way enough of that let us continue on with the tour.

The 2nd stage (or the first stage if you count 0 is the first ordinal which I do but for the sake of argument let's not) is called the binary regiment.

Like with the unary regiment we need to develop an alphabet of symbols to work with. The symbols are essentially the same as the unary regiments except for the fact that a few new symbols are being implemented. Let's continue with on with some examples: The first would be working out what {2(2)4} is then inputing that into x then using the array with that value to find {2(2,2)4}.
 * (x:y) = {a(x)b} recursed into "x" y times.
 * {2(2,2)4}

{2(2)4} = 2(1}2(1)2(1)2 = 2*2*2*2 = 16.

That gives us our first value of x which we plug into {2(x)4} to get {2(16)4} = 2{14 up arrows}4.

Now by this into x we can find out what {2(2,2)4} equals.

{2(2,2)4} = {2({2(16)4})4}.

The next stage the trinary regiment is when things really start to kick off.

Once again an alphabet of symbols is required. Here's an example of a number written in this format:
 * (x,y,z) = {a,(x,y)b}R{a,(x,y)b}.
 * R stands for recurse by the way.

{2,(4,4,4),8} = {2,(4,4),8}R{2,(4,4)8}.

The name for the 4th regiment should be called something: strange, unseen , unnoticed by the eyes of watchful men , perhaps it is to do with number 4 perhaps not? , who knows? , ah ha that's it I'll call it the Bakers regiment since Tom Baker was the 4th doctor and he said "who knows".

The Bakers regiment also involves an alphabet of symbols.

(x,y,z,w) = {a,(x,y,z)b}R{a(x,y,z)b}.

Of course you can probably work out some examples in your minds so I won't give any examples.

The 5th regiment also known as the witches symbol regiment also uses an alphabet of symbols:

(x,y,z,w,v) = {a(x,y,z,w)b}R{a(x,y,z,w)b}.

The 6th regiment is known as the venus regiment since 6 stands for venus in symbology.

Like all of the other regiments this also involves an alphabet of symbols.

(x,y,z,w,v,u) = {a(x,y,z,w,v)b}R{a(x,y,z,w,v)b}.

A peculiar thing to note out is that each time you go to a new regiment the symbol you produce in the alphabet is 1 less than the symbol of the previous regiment. So by continuing on eventually you'll get back to a, I calculate the regiment number to be 26 since their are 26 letters in the alphabet and 1 cycle would be a succession of 26 letters beginning at a and ending at a. If we continue on we can imagine the symbols from the alphabet as being on an infinite conveyer belt that loops aroud continuously cycling the letters for all of eternity.