User blog:B1mb0w/Strong D Function

Strong D Function
The strong D function is based on a weaker deeply nested Ackermann function called d. The rules are similar with the significant change being that the D function:

\(D(x_1,x_2,x_3,x_4,...,x_n)\)

expands to this function:

\(D( x_1-1, D(x_1,x_2,x_3,x_4,...,x_n-1), ..., D(x_1,x_2,x_3,x_4,...,x_n-1))\)

The same expansion is used to replace each input parameter \(x_2\) to \(x_n\). Also refer to the Leading and Trailing Zero rules: L1 and T1, for more information.

Definition
For 2 parameters, the D function is equivalent to the d function:

\(d(a,b)=d(a-1,d(a,b-1))=D(a,b)=D(a-1,D(a,b-1))\)

For 3 parameters, the D function quickly dominates the weaker d function:

\(d(a,b,c)=d(a-1,d(a,b-1),d(a,b,c-1))\)

\(D(a,b,c)=D(a-1,D(a,b,c-1),D(a,b,c-1))\)

Calculated Examples up to D(1,D(1,0,2),0)
\(D = 0\) This is a null function that always returns zero.

\(D(3) = 4\) This is the successor function

\(D(1,2) = 5\) This is the same as d(1,2)

\(D(2,3) = 17\) This is the same as d(2,3)

Using the Comparison Rule C1 \(d(m,n) >> f_{m-1}(n+2)\) we get

\(D(2,3) >> f_1(5) = 10\)

\(D(3,4) = 5099 >> f_2(6) = 6.2^6 = 384\)

\(D(3, 9) = 1,240,025 >> 1,000,000\)

\(D(3, 206) = 122*10^{98} >>\) Googol

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

\(D(4,2) >>\) Googolplex

\(D(6,1) >> g_1\) where \(g_64 = G\) Graham's number

Other calculations give these results

\(D(D(4,1)+1,D(4,1)-1) >> f_{\omega}^2(3)\)

\(D(1,0,0) = D(0,D(0,1,1),D(0,1,1)) = D(4,4) >> f_3(6) >> f_{\omega}(3)\)

\(D(1,0,1) = D(0,D(1,0,0),D(1,0,0)) = D(D(4,4),D(4,4)) >>\) very low bound \(>> f_{\omega}^2(3)\)

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

Using Rule: N1 \(D(1,0,n) >> f_{\omega}^{n-2}(f_{\omega+1}(3))\) when n>2

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

\(= f_{\omega}^{f_{\omega+1}(3)+2-2}(f_{\omega+1}(3)) = f_{\omega}^{f_{\omega+1}(3)}(f_{\omega+1}(3))\)

\(= f_{\omega+1}(f_{\omega+1}(3)) = f_{\omega+1}^2(3)\)

Using Rule: NL \(D(l,m,n) = D(l,0,n-1+(m+2).(m+1)/2)\) or

\(D(1,m,n) = D(1,0,n-1+(m+2).(m+1)/2)\)

\(D(1,9,9) = D(1,6,36) = D(1,3,54) = D(1,0,63) >> g_{64} = G\) Graham's number

More Calculated Examples
\(D(2,0,0) = D(1,D(1,2,2),D(1,2,2)) >> D(1,0,D(1,2,2)) >> D(1,0,D(1,0,2)+2)\)

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

\(D(2,0,2) >> D(1,0,D(2,0,1)+2) >> \) work in progress

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

\(>> D(2,f_{\omega+1}^{11}(f_{\omega+2}(3)),f_{\omega+1}^{11}(f_{\omega+2}(3))) >> f_{\omega+2}^2(3)\)

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

Speculating Ahead
\(D(1,0,2) >> f_{\omega}^3(3) >> f_{\omega+1}(3)\)

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

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

work in progress

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

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

\(D(3,m,n) = D(3,0,n-1+(m+2).(m+1)/2)\)

\(D(4,0,1) = D(3,D(4,0,0),D(4,0,0)) >> f_{\omega.2+1}(3)\) need to check this

\(D(5,0,1) = D(4,D(5,0,0),D(5,0,0)) >> f_{\omega.2+2}(3)\) need to check this

\(D(6,0,1) = D(5,D(6,0,0),D(6,0,0)) >> f_{\omega.2+3}(3) = f_{\omega.2+\omega}(3)\) need to check this

\(D(l,m,n) = D(l,0,n-1+(m+2).(m+1)/2)\)

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

work in progress

Next - the Alpha Function
My next blog post will introduce a new Alpha function that I have been thinking about.