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 \(f_{\omega}^{n+(m+2).(m+1)/2}(3) >>  f_{\omega}^{n+m^2/2+m}(3)\)

D function examples with 3 parameters - continues

\(D(2,0,0)\) grows significantly faster \(= D(1,D(1,2,2),D(1,2,2)) >> D(1,f_{\omega}^8(3),f_{\omega}^8(3))\)

which is comparable to \(f_{\omega}^{m.2+m^2/2}(3)\)   where   \(m = f_{\omega}^8(3)  >>  f_{\omega+1}(7)\)

and \(>> f_{\omega}^{m.2+m.2+m+8}(3)  >>  f_{\omega}^{m.2+m.2+f_{\omega+1}(7)}(f_{\omega+1}(7))\)

\(= f_{\omega}^{m.3 + f_{\omega+1}(7)}(f_{\omega+1}^2(7))  =  f_{\omega}^{m.3}(f_{\omega+1}^3(7)) =  f_{\omega+1}^6(7)  >>  f_{\omega+2}(6)\)

\(D(2,0,1) >>  f_{\omega}^{m.2+m^2/2}(3)\)   where   \(m = f_{\omega+2}(6)  >>  f_{\omega}^{m.2+m.2+m+7}(3)  >>  f_{\omega}^{m.4+f_{\omega+2}(6)}(f_{\omega+1}(6))\)

and \(>> f_{\omega}^{f_{\omega+2}(6).4+f_{\omega+1}(6)}(f_{\omega+1}(6))  >>  f_{\omega}^{f_{\omega+2}(6).4}(f_{\omega+2}(6))\)

\(D(2,0,2) >>  f_{\omega}^{m.2+m^2/2}(3)\)   where   \(m = f_{\omega}^{f_{\omega+2}(6).4}(f_{\omega+2}(6))\)

\(D(2,0,3) >>  \)  Need to check this result.

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