User blog:B1mb0w/Alpha Spreadsheet q3.0

Alpha function code (Spreadsheet) version 3.0
Javascript code for my Alpha Function based on my Quantum Function is available in Google Sheets.

Javascript code in Google Sheets
You are free to use and copy the code as you like. Version 2.0 of the Javascript code in Google Sheets is available here:

Alpha Function Javascript

DRAFT Version 3.0 Examples
I have started working on a new version. Here is some work in progress:

\(\alpha(10) = Q(Q^{2}(Q^{t_0(0)}(t_0(0))_*,t_0(0)),3)\)

\(\alpha(11) = Q(Q(Q(Q^{2}(t_0(0)_*,t_0(0)),t_0(1)),t_0(1)),3)\)

\(\alpha(12) = Q(Q^{Q(Q(2,t_0(0)),t_0(0))}(t_0(1)_*,t_0(1)),3)\)

\(\alpha(13) = Q(Q(Q^{t_0(0)}(1,t_0(2)_*),t_0(2)),3)\)

\(\alpha(14) = Q(Q^{Q(2,t_0(0))}(t_0(2)_*,t_0(2)),3)\)

\(\alpha(15) = Q(Q^{Q^{t_0(0)}(t_0(1)_*,t_0(1))}(t_0(2)_*,t_0(2)),3)\)

Code changes (in this version)
The javascript code is not complete. I plan to add some improvements and fix a number of errors.

Improvement #1: The output code should use base 1 instead of 2. I think this is more interesting and a little easier to make comparisons between the Quantum output function and a similar FGH function. Refer to my Alpha Function blog for examples. This is easy to change but using base 2 makes the next version of the code easier to debug as it is being developed.

Improvement #2: The algorithm will be calibrated to approximately:

\(\alpha(2^{512}) = Q(1,0,1) = Q(t_{Q(1,1)}(0),1) = Q(t_3(0),1)\)

where

\(t_3(0) = t_2(1,0_{[t_2(0)]})\)

The calibration algorithm will be adjusted when the full code has been completed.

Error #1: The code is not complete. The algorithm does not generate t function values of \(t_1(0)\) or greater. This will be the focus of the next version.

Recursion & Summation Functions
This version introduces further development of my Notation. I use notation that is not in common use, and I have extended my Recursion notation and added a 'Summation' function to extend the flexibility of the notation to reference highly nested functions.

Here is an updated definition of the Quantum function rules using the new notation functions:

Rule 0: \(Q(n) = n + 1\)

Rule 1: \(Q^{c + 1}(n) = Q(C) = Q(Q^c(n))\)

Rule 2: \(Q(1,n) = Q^{Q(n)}(n) = R^{Q(n)}(n,Q(r))\)

Rule 3: \(Q^{c + 1}(1,n_*) = Q(1,C) = Q(1,Q^{c}(1,n_*))\)

Rule 4: \(Q^{c + 1}(1_*,n) = Q(C,n) = Q(Q^{c}(1_*,n),n)\)

Rule 5: \(Q(c + 1,n) = Q^{C}(c,n_*)\)

Rule 6: \(Q(t_0(0),n) = Q(n,n)\)

Rule 7: \(t_0(c + 1) = Q^{C}(C_*,C)\)

Rule 8: \(t_0^{c + 1}(x) = t_0(C) = D\)

then

\(Q^{D}(D_*,D) = t_0^{c + 1}(Q(t_0^{[c]}(x)))\)

here is a worked example of how Indexed Recursion notation is calculated using rules 7 and 8:

Iterate \(E = t_0^{c + 1}(Q(t_0^{[c]}(x))) = t_0^{c + 1}(Q(F_c))\) where \(t_0^{[c]}(x) = F_c\)

\(Q^{E}(E_*,E) = t_0^{c + 1}(Q^{2}(F_c))\)

Iterate until \(E = t_0^{c + 1}(Q^{C}(F_c))\)

\(Q^{E}(E_*,E) = t_0^{c + 1}(Q(1,F_c))\)

Iterate until \(E = t_0^{c + 1}(Q^{F_c}(F_c*,F_c)) = t_0^{c + 1}(Q(t_0^{[c - 1]}(x))) = t_0^{c + 1}(Q(F_{c - 1}))\)

continue to Iterate until \(E = t_0^{c + 1}(Q(t_0^{[1]}(x))) = t_0^{c + 1}(Q(F_1))\)

Iterate until \(E = t_0^{c + 1}(Q^{F_1}(F_1*,F_1)) = t_0^{c + 1}(x + 1)\)

continue to Iterate until \(E = t_0^{c + 1}(t_0(0)) = t_0^{c + 2}(0)\)

continue to Iterate until \(E = t_0^{c + 2}(x) = t_0(D)\)

Rule 9: \(t_0(1,0_{[c + 1]}) = t_0^{C}(1_*,0_{[c]})\)

e.g.

\(t_0(1,0) = t_0^{C}(1_*,0_{[0]}) = t_0^{t_0(1,0_{[c]})}(1) = t_0^{t_0(1,0_{[0]})}(1) = t_0^{t_0(1)}(1)\)

\(t_0(1,0) = t_0^{t_0(1)}(1) = D\)

then

\(D = t_0^{t_0(s)(D)}(1)\)

\(Q^{D}(D_*,D) = t_0^{t_0(s)(D)}(Q(t_0^{[t_0(s)(D) - 1]}(1)))\)

\(t_0(D) = t_0^{t_0(s)(D) + 1}(1)\)

\(t_0^{D}(D) = t_0^{t_0(s)(D) + D}(1) = t_0^{t_0(s)(t_0(1,1))}(1) = t_0(1,1)\)

Rule 10: \(t_0(1,c + 1) = t_0^{C}(C) = D\)

then

\(D = t_0^{t_0(s)(D)}(1)\)

\(Q^{D}(D_*,D) = t_0^{t_0(s)(D)}(Q(t_0^{[t_0(s)(D) - 1]}(1)))\)

\(t_0(D) = t_0^{t_0(s)(D) + 1}(1)\)

\(t_0^{D}(D) = t_0^{t_0(s)(D) + D}(1) = t_0^{t_0(s)(t_0(1,c + 2))}(1) = t_0(1,c + 2)\)

Rule 11: \(t_0^{c + 1}(1,0_*) = t_0(1,C) = D\)

then

\(D = t_0^{t_0(s)(D)}(1)\)

\(Q^{D}(D_*,D) = t_0^{t_0(s)(D)}(Q(t_0^{[t_0(s)(D) - 1]}(1)))\)

\(t_0(D) = t_0^{t_0(s)(D) + 1}(1)\)

\(t_0^{D}(D) = t_0^{t_0(s)(D) + D}(1) = t_0^{t_0(s)(t_0(1,C + 1))}(1) = t_0(1,C + 1)\)

\(= t_0(1,Q(C)) = t_0^{c + 1}(1,Q(t_0^{[c]}(1,0_*))_*)\)

Rule 12: \(t_0(c + 1,0) = t_0^{C}(c,0_*) = D\)

e.g.

\(t_0(2,0) = t_0^{C}(1,0_*) = t_0^{t_0(1,0)}(1,0_*) = D = t_0^{t_0(1,s)(D)}(1,0_*)\)

using

\(t_0(1,s)(t_0(2,0)) = t_0(1,0)\) by Rule s3

then

\(D = t_0^{t_0(s)(D)}(1)\)

\(D = t_0^{t_0(1,s)(D)}(1,0_*) = t_0^{ t_0(1,0)}(1,0_*) = t_0(2,0)\)

\(D = t_0^{t_0(2,s)(D)}(2,0_*) = t_0^{1}(2,0_*) = t_0(2,0)\)

\(Q^{D}(D_*,D) = t_0^{t_0(s)(D)}(Q(t_0^{[t_0(s)(D) - 1]}(1)))\)

\(t_0(D) = t_0^{t_0(s)(D) + 1}(1)\)

\(t_0^{D}(D) = t_0^{t_0(s)(D) + D}(1) = t_0^{t_0(1,s)(D)}(1,Q(t_0^{[t_0(1,s)(D) - 1]}(1,0_*))_*)\)

\(t_0(1,D) = t_0^{t_0(1,s)(D) + 1}(1,0_*)\)

\(t_0^{D}(1,D_*) = t_0^{t_0(1,s)(D) + D}(1,0_*) = t_0^{t_0(1,s)(t_0(2,1))}(1,0_*) = t_0(2,1)\)

because

\(t_0(1,s)(t_0(2,c + 1)) = t_0(2,c) + C = t_0(2,c) + t_0(1,s)(t_0(2,c))\) by Rule s4 where \(x = 1\)

and the general case

\(t_0(c + 1,x) = D = t_0^{t_0(c,s)(D)}(c,0_*)\)

\(t_0(c,D) = t_0^{t_0(c,s)(D) + 1}(c,0_*)\)

\(t_0(c + 1,x + 1) = t_0^{t_0(c,s)(t_0(c + 1,x + 1))}(c,0_*) = t_0^{t_0(c,s)(t_0(c + 1,x)) + t_0(c + 1,x)}(c,0_*) = t_0^{t_0(c,s)(D) + D}(c,0_*)\)

Rule 13: \(t_0(b + 1,c + 1) = t_0^{C}(b,C_*)\)

e.g.

\(t_0(2,c + 1) = t_0^{C}(1,C_*) = t_0^{t_0(2,c)}(1,t_0(2,c)_*) = D\)

then

\(D = t_0^{t_0(s)(D)}(1)\)

\(D = t_0^{t_0(1,s)(D)}(1,0_*)\)

\(D = t_0^{t_0(2,s)(D)}(2,0_*)\)

\(Q^{D}(D_*,D) = t_0^{t_0(s)(D)}(Q(t_0^{[t_0(s)(D) - 1]}(1)))\)

\(t_0(D) = t_0^{t_0(s)(D) + 1}(1)\)

\(t_0^{D}(D) = t_0^{t_0(s)(D) + D}(1) = t_0^{t_0(1,s)(D)}(1,Q(t_0^{[t_0(1,s)(D) - 1]}(1,0_*))_*)\)

\(t_0(1,D) = t_0^{t_0(1,s)(D) + 1}(1,0_*)\)

\(t_0^{D}(1,D_*) = t_0^{t_0(1,s)(D) + D}(1,0_*) = t_0^{t_0(1,s)(t_0(2,c + 2))}(1,0_*) = t_0(2,c + 2)\)

\(t_0(2,D) = t_0(2,t_0(2,c + 1)) = t_0^{2}(2,c + 1_*)\)

and the general case

\(t_0(b + 1,c) = D\)

then

\(t_0(x,D) = t_0(x,t_0^{t_0(x,s)(D)}(x,0_*)) = t_0^{1 + t_0(x,s)(D)}(x,0_*)\)

\(t_0^{2}(x,D_*) = t_0^{2 + t_0(x,s)(D)}(x,0_*)\)

\(t_0^{D}(x,D_*) = t_0^{D + t_0(x,s)(D)}(x,0_*)\)

when \(x < b\)

\(t_0^{D}(x,D_*) = t_0^{t_0(x + 1,s)(D)}(x + 1,Q(t_0^{[t_0(x + 1,s)(D) - 1]}(x + 1,0_*))_*)\)

when \(x = b\)

\(t_0^{D}(b,D_*) = t_0(b + 1,c + 1)\)

because

\(t_0^{t_0(b,s)(D) + 1}(c,0_*) = t_0^{t_0(b,s)(D)}(b,t_0(b,0)_*) = t_0^{t_0(b,s)(D)}(b,t_0^{1}(b,0_*)_*)\)

\(t_0^{t_0(b,s)(D) + D}(b,0_*) = t_0^{D}(b,t_0^{t_0(b,s)(D)}(b,0_*)_*) = t_0^{D}(b,D_*)\)

Rule 14: \(t_0^{c + 1}(1_*,0) = t_0(C,0) = D\)

then

\(D = t_0^{t_0(s)(D)}(1)\)

\(D = t_0^{t_0(1,s)(D)}(1,0_*)\)

\(D = t_0^{t_0(2,s)(D)}(2,0_*)\)

\(D = t_0^{t_0(b,s)(D)}(b,0_*)\) where \(b < C = t_0^{c}(1_*,0)\)

\(Q^{D}(D_*,D) = t_0^{t_0(s)(D)}(Q(t_0^{[t_0(s)(D) - 1]}(1)))\)

\(t_0(D) = t_0^{t_0(s)(D) + 1}(1)\)

\(t_0^{D}(D) = t_0^{t_0(s)(D) + D}(1) = t_0^{t_0(1,s)(D)}(1,Q(t_0^{[t_0(1,s)(D) - 1]}(1,0_*))_*)\)

\(t_0(1,D) = t_0^{t_0(1,s)(D) + 1}(1,0_*)\)

\(t_0^{D}(b,D_*) = t_0^{t_0(b,s)(D) + D}(b,0_*) = t_0^{t_0(b + 1,s)(D)}(b + 1,Q(t_0^{[t_0(b + 1,s)(D) - 1]}(b + 1,0_*))_*)\)

\(t_0(b + 1,D) = t_0^{t_0(b + 1,s)(D) + 1}(b + 1,0_*)\)

\(t_0^{D}(C - 1,D_*) = t_0^{t_0(C - 1,s)(D) + D}(C - 1,0_*) = t_0^{t_0(C - 1,s)(t_0(C,1))}(C - 1,0_*) = t_0(C,1)\)

\(t_0(C,1) = t_0^{c + 1}(1_*,0[1])\) refer to notation Rule i6

Iterate \(E = t_0^{c + 1}(1_*,0[1])\)

\(t_0^{E}(C - 1,E_*) = t_0(C,2) = t_0^{c + 1}(1_*,0[2])\)

Iterate until \(E = t_0^{c + 1}(1_*,0[t_0^{t_0(C,0) - 1}(C,0_*)])\)

\(t_0^{c + 1}(1_*,0[t_0^{t_0(C,0) - 1}(C,0_*)]) = t_0(C,t_0^{t_0(C,0) - 1}(C,0_*))\)

\(= t_0^{t_0(C,0)}(C,0_*) = t_0(Q(C),0) = t_0^{c + 1}(Q(t_0^{[c]}(1_*,0))_*,0)\)

continue to Iterate \(E = t_0^{c + 1}(Q(t_0^{[c - 1]}(1_*,0))_*,0)\)

continue to Iterate \(E = t_0^{c + 1}(Q(t_0^{[1]}(1_*,0))_*,0)\)

continue to Iterate \(E = t_0^{c + 1}(t_0^{[1]}(1_*,0[1])_*,0)\)

\(t_0^{c + 1}(t_0^{[1]}(1_*,0[1])_*,0) = t_0^{c + 1}(t_0^{[2]}(1_*,0[+0])_*,0)\) refer to notation Rule i7

continue to Iterate \(E = t_0^{c + 1}(t_0^{[c]}(1_*,0[+0])_*,0)\)

continue to Iterate \(E = t_0^{c + 1}(1_*,0[+0])\)

continue to Iterate \(E = t_0^{c + 1}(1_*,0[+1]) = t_0^{c + 1}(1_*,1)\)

Further References
Further references to relevant blogs can be found here: User:B1mb0w