User blog:B1mb0w/Strong D Function

Strong D Function

The strong D function is based on the weaker d function defined in User blog:B1mb0w/Deeply Nested Ackermann. 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

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

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

Next

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

Similarly

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

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

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

and \(D(1,m,n)\) is comparable to \(f_{\omega}^{n+(m+2).(m+1)/2+1}(3) >> f_{\omega}^{n+(m+2).(m+1)/2-2}(f_{\omega+1}(3))\) where m or n > 0

D function examples with 3 parameters - continues

\(D(2,0,0)\) grows significantly faster \(= D(1,D(1,2,2),D(1,2,2)) >> D(1,f_{\omega}^{6}(f_{\omega+1}(3)),f_{\omega}^{6}(f_{\omega+1}(3))\)

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

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

\(D(2,0,1) >> f_{\omega}^{m.2+m^2/2}(3)\) where \(m = f_{\omega+2}(7) >> f_{\omega}^{m.12+8}(3) >> f_{\omega}^{m.6+f_{\omega+1}(7).6}(f_{\omega+1}(7))\)

and \(= f_{\omega}^{f_{\omega+2}(7).6}(f_{\omega+2}(7)) = f_{\omega+3}(7)\)

\(D(2,0,2) >> f_{\omega}^{m.2+m^2/2}(3)\) where \(m = f_{\omega+3}(7)\) or \(>> f_{\omega+4}(7)\)

\(D(2,0,3) >> f_{\omega+5}(7)\) and

\(D(2,0,n) >> f_{\omega+n+2}(7)\)

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

\(D(2,1,0) = D(1,D(2,0,1),D(2,0,1)) >> D(1,f_{\omega+3}(7),f_{\omega+3}(7))\)

and \(>> f_{\omega}^{m.18+8}(3)\) where \(m = f_{\omega+3}(7)\)  and \(>> f_{\omega+4}(7)\)

\(D(2,1,1) = D(1,D(2,1,0),D(2,1,0)) >> f_{\omega+5}(7)\)

\(D(2,1,n) >> f_{\omega+n+4}(7)\)

\(D(2,2,0) = D(1,D(2,1,2),D(2,1,2)) >> f_{\omega+7}(7)\)

\(D(2,2,n) >> f_{\omega+n+7}(7)\)

\(D(2,m,n) >> f_{\omega+n+(m+2).(m+1)/2+1}(7) >> f_{\omega.2}(3)\) when m or n > 1

then

\(D(3,0,0) = D(2,D(2,3,3),D(2,3,3)) = f_{\omega+m+(m+2).(m+1)/2+1}(7)\) where \(m = f_{\omega.2}(3)\)

and \(>> f_{\omega+m+m}^6(3) >> f_{\omega+m.169+169}^5(f_{\omega+13}(13)) >> f_{\omega+m.169+169}^5(m)\)

Speculating ahead a little

\(D(3,m,n) >> f_{\omega.(n+(m+2).(m+1)/2+1)}(7)\)  need to check this

\(D(4,0,0) >> f_{\omega^2}(7)\)  need to check this

\(D(4,m,n) >> f_{\omega^{n+(m+2).(m+1)/2+1}}(7)\)  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. You can find it here: User blog:B1mb0w/Alpha Function