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 \(x_n\).

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

\(D = 0\) 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)

\(D(1,0,0)\) expands as follows:

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

\(D(1,0,1)\) expands as follows:

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

and is comparable to \(f_{\omega}(f_{\omega}(3)) = f_{\omega}^2(3)\)

My calculations show that \(D(1,0,n)\) is comparable to \(f_{\omega}^{n+1}(3)\)

More Examples with 3 parameters

\(D(1,1,0) = D(0,D(1,0,1),D(1,0,1))\) which is equal to \(D(1,0,2)\) and comparable to \(f_{\omega}^3(3)\)

Similarly

\(D(1,1,1) = D(0,D(1,0,2),D(1,0,2)) = D(1,0,3)\) and comparable to \(f_{\omega}^4(3)\)

My calculations show that \(D(1,1,n)\) is comparable to \(f_{\omega}^{n+3}(3)\)

Next

\(D(1,2,0) = D(0,D(1,1,2),D(1,1,2)) = D(1,1,3)\) and comparable to \(f_{\omega}^6(3)\)

Similarly

\(D(1,2,1) = D(0,D(1,1,3),D(1,1,3)) = D(1,1,4)\) and comparable to \(f_{\omega}^7(3)\)

My calculations show that:

\(D(1,2,n)\) is comparable to \(f_{\omega}^{n+6}(3)\)

\(D(1,3,n)\) is comparable to \(f_{\omega}^{n+10}(3)\)

and

\(D(1,m,n)\) is comparable to \(f_{\omega}^{n+(m+2).(m+1)/2}(3)\)

D function examples with 3 parameters - continues

\(D(2,0,0)\) grows significantly faster \(= D(1,D(1,2,2),D(1,2,2))\)

and \(>> f_{\omega}^{m+(m+2).(m+1)/2}(3)\) where \(m = f_{\omega}^8(3) >> f_{\omega+1}(3)\)

Using this formula 12 \(f_b^{f_{b+1}(a)+a}(a) = f_{b+1}^2(a)\) we get \(f_{\omega}^{f_{\omega+1}(3)+3}(3) >> f_{\omega+1}^2(3)\)

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

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

Using this formula 15 \(f_{b}^{f_{b+1}^{a-1}(a).2}(a) >> f_{b+2}(a)\) we get \(f_{\omega}^{f_{\omega+1}^2(3).2}(3) >> f_{\omega+2}(3)\)

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

\(D(2,0,2) >> f_{\omega}^{m.2}(3)\) where \(m >> f_{\omega+2}(3)\)

Using this formula 14 \(f_{b}^{f_{b+1}^n(a).2}(a) >> f_{b+1}^{n+1}(a)\) we get \(f_{\omega}^{f_{\omega+2}(3).2}(3) >> f_{\omega+2}^2(3)\)

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

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

\(D(2,0,n) >> f_{\omega+(n+3)/2}(3)\) if n is odd or \(>> f_{\omega+(n+2)/2}^2(3)\) if n is even

D function examples with 3 parameters - continues for \(D(2,m,n)\)

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

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

\(D(2,1,n) >> f_{\omega+(n+5)/2}(3)\) if n is odd or \(>> f_{\omega+(n+4)/2}^2(3)\) if n is even

\(D(2,2,0) = D(1,D(2,1,2),D(2,1,2)) = D(2,0,5) >> f_{\omega+4}(3)\)

\(D(2,2,n) >> f_{\omega+(n+8)/2}(3)\) if n is odd or \(>> f_{\omega+(n+7)/2}^2(3)\) if n is even

\(D(2,3,n) >> f_{\omega+(n+12)/2}(3)\) if n is odd or \(>> f_{\omega+(n+11)/2}^2(3)\) if n is even

\(D(2,m,n) >> f_{\omega+(n+2+(m+2).(m+1)/2)/2}(3)\) if n is odd or \(>> f_{\omega+(n+1+(m+2).(m+1)/2)/2}^2(3)\) if n is even

e.g.  \(D(2,3,3) >> f_{\omega+7}(3) >> f_{\omega+3}(3) = f_{\omega.2}(3)\)

then

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

Using this formula 25 \(f_{\omega}^{f_{\omega.n}^{m}(a).2}(a) >> f_{\omega.n}^{m+1}(a)\) we get \(f_{\omega}^{f_{\omega.2}^{1}(3).2}(3) >> f_{\omega.2}^{2}(3)\)

then \(D(3,0,0) >> f_{\omega.2}^2(3)\)

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

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

\(D(3,0,n) >> f_{\omega.2+(n+1)/2}(3)\) if n is odd or \(>> f_{\omega.2+n/2}^2(3)\) if n is even

e.g. \(D(3,0,5) >> f_{\omega.2+3}(3) = f_{\omega.3}(3) = f_{\omega^2}(3)\)

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

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

\(D(3,1,n) >> f_{\omega.2+(n+3)/2}(3)\) if n is odd or \(>> f_{\omega.2+(n+2)/2}^2(3)\) if n is even

\(D(3,2,n) >> f_{\omega.2+(n+6)/2}(3)\) if n is odd or \(>> f_{\omega.2+(n+5)/2}^2(3)\) if n is even

\(D(3,m,n) >> f_{\omega.2+(n+(m+2).(m+1)/2)/2}(3)\) if n is odd or \(>> f_{\omega.2+(n-1+(m+2).(m+1)/2)/2}^2(3)\) if n is even

Recap and Speculating ahead a little

\(D(3, 9) >> 1,000,000\)

\(D(3, 206) >> Googol\)

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

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

\(D(6,6) >> g1\) where g64 is Graham's number

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

\(D(1,4,47) = D(1,2,57) = D(1,0,63) >> g64 = G\) where g64 is Graham's number

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

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

\(D(3,0,5) >> f_{\omega^2}(3)\)

\(D(4,0,2) >> f_{\omega^{\omega}}(3)\)  need to check this

\(D(l,m,n)\) has a growth rate of \(f_{\epsilon_0}(3)\)  need to check this

Next

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