User blog:B1mb0w/The Alpha Function

The Alpha Function
The Alpha Function has one parameter: \(\alpha(r)\) where r is any real number. It is derived from the The J Function with a variable number of input parameters.

This blog replaces two other previous attempts at this function. Links to the original blogs are available in the References section at the end of this blog. However, they will provide links back to here, and this blog will not use any of the material from those older attempts.

What is the Alpha Function
My motivation to create this function was to develop a finely grained number notation system for really big numbers. \(\alpha(1)\) for example can be used to reference the number 2. Therefore 1 is the Alpha Index for the number 2. Alpha needs to reference big numbers very quickly to be useful, therefore it uses the Strong D Function for this purpose. Alpha should also be strictly hierarchical and every number a > b, must reference larger numbers, so that \(\alpha(a) >> \alpha(b)\) in all cases. The function is finely grained. It accepts a real number input and offers some finesse to locate large big numbers.

The Alpha Function is defined recursively based on a series of binary decisions. The logic can be explained by referring to the following notation.

Some Notation
WORK IN PROGRESS

e Rule: M4'''

The final rule is used for D functions of the form:

\(D(n+1,0_{[y]}) = D(n,D(n,n_{[y]}+1)_{[y]}\)

therefore

\(D(n,p_{[y]}) < D(n,D(n,n_{[y]}+1)_{[y]})\) \(<= D(n,D(n,n_{[y]}+1)_{[y]}-1)\)  or alternatively

\(<= D(n,D(n,n_{[y]}+1)_{[y-1]},D(n,n_{[y-1]}+1,n))\)  Rule M4a 

and

\(p_i <= D(n,n_{[y]}+1)\)  for all   \(1 <= i < y\)   Rule M4b

and

\(p_y <= D(n,n_{[y-1]}+1,n)\)  Rule M4c

Some Calculations
Program Code was used to calculate these examples.

\(\alpha(0.00) = D = 0\)

\(\alpha(0.50) = D(0) = 1\)

\(\alpha(1.00) = D(1) = 2\)

\(\alpha(2.00) = D(1,0) = 3\)

\(\alpha(2.08) = D(1,1) = 4\)

\(\alpha(2.16) = D(1,2) = 5\)

\(\alpha(2.56) = D(2,1) = 11\)

\(\alpha(e) = D(2,6) = 26\)

\(\alpha(3.00) = D(3,0) = 59\)

\(\alpha(3.04) = D(3,2) = 563\)

\(\alpha(\pi) = D(3,14) = 301327043\)

\(\alpha(3.52) = D(4,0) >>\) Googol

\(\alpha(3.6) = D(4,1) >> f_{\omega}(3)\)

\(\alpha(3.64) = D(4,2) >>\) Googolplex

\(\alpha(4.00) = D(1,0,0)\)

\(\alpha(4.0005) = D(1,0,1) >> g_2\) where \(g_{64}\) is Graham's number

\(\alpha(4.008304) = D(1,11,0) >> D(1,9,9) >>\) Graham's number

\(\alpha(4.1250) = D(2,0,0) >> f_{\omega+1}^2(3)\)

\(\alpha(4.1255) = D(2,0,1) >> f_{\omega+2}(3)\)

\(\alpha(4.2501) = D(3,0,1) >> f_{\omega.2}(3)\)

\(\alpha(6) = D(D(1,0,0),0,0) = D(\alpha(4),0,0)\)

\(\alpha(6.5) = D(D(1,0,1),0,0) = D(D(D(4,4),D(4,4)),0,0)\)

\(\alpha(8) = D(1,0,0,0) = D(D(1,1,1),D(1,1,1),D(1,1,1))\)

\(\alpha(8.tba) = D(D(3,0,1),0,0) >> f_{\omega.2+1}(3)\) ''This result is being checked. Work in progress''

\(\alpha(9) = D(D(59,0,0),0,0,0,0,0,0)\)

Program Code and Description
Program code will be provided here.

WORK IN PROGRESS

Comments and Questions
Look forward to comments and questions. I am learning heaps by writing these blogs and correcting all the mistakes the community finds in them !

Cheers B1mb0w.