User blog:B1mb0w/new Alpha Function

(new) Alpha Function
The Alpha Function has one parameter: \(\alpha(r)\) where r is any real number. It is derived from the Strong D Function with a variable number of input parameters. This blog replaces the old description of the Alpha Function.

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 for every number a > b: a 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
Following notation helps to explain the behaviour of Strong D Functions and the logic of the Alpha Function.

\(D(m_{[x]}) = D(m_1,m_2,...,m_x)\)

\(D(m_{[x]},n_{[y]}) = D(m_1,m_2,...,m_x,n_1,n_2,...,n_y)\)

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

\(D(m,n_{[y]}) = D(m-1,D(m,n_{[y-1]},n_y-1)_{[y]})\)

\(D(m_{[x]},n,0_{[y]}) = D(m_1-1,D(m_{[x]},n-1,n_{[y]})_{[x+y]})\) when x>0 and which is equal to

\(= D(m_{[x]},n-1,n_{[y-1]},n_y+1)\)

Alpha Function Logic
The Alpha Function is defined using the following logic.

\(\alpha(r) = D(D(m_{[x]})_{[x]})\) where \(2^{x-1} <= r < 2^x\)   and

\(\alpha(2^x) = D(1, 0_{[x]})\)

The values of \(m_{[x]}\) are calculated based on the value of r but only legal values can be used which follow these restrictions: The second constraint is important to force \(\alpha(a) >> \alpha(b)\) whenever a > b.  Additional logic that is used is derived from the following rules:
 * duplicate values should be avoided
 * out of sequence results must be avoided

Maximum Value Rule: M1

\(D(1,0_{[x]}) = D(D(1_{[x]})_{[x]})\)

therefore

\(D(m_{[x]}) < D(D(1_{[x]})_{[x]})\)

\(<= D(D(1_{[x]})_{[x]})-1\) or alternatively

\(<= D(D(1_{[x]})_{[x-1]},D(1_{[x-1]},0))\) Rule M1a

and

\(m_1 >= 1\)   Rule M1b

Maximum Value Rule: M2

\(D(m_{[x]},n+1,0_{[y]}) = D(m_{[x]},n,n_{[y-1]}+1,n_y+2)\)

therefore

\(D(m_{[x]},n,p_{[y]}) < D(m_{[x]},n,n_{[y-1]}+1,n_y+2)\)

\(<= D(m_{[x]},n,n_{[y-1]}+1,n_y+2)-1\) or alternatively

\(<= D(m_{[x]},n,n_{[y]}+1)\) Rule M2a

or

\(p_{[y]} <= n_{[y]}+1\) Rule M2b

Maximum Value Rule: M3

It can also be shown that the only legal values for D functions in the form:

\(D(m_{[x]})\)

are when

\(m_i <= m_{i-1}+1\)   for all   \(1 <= i <= x\)

Maximum Value 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
\(\alpha(0) = D = 0\)

\(\alpha(1) = D(1) = 2\) \(\alpha(2) = D(1,0) = 3\)

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

Program Code and Description
Version 1 of the program code for the Alpha Function will be available here (work in progress). The code is not complete and various errors will be corrected in Version 2 (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.