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## 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:

$$d(a,b)=d(a-1,d(a,b-1))=D(m,n)=D(m-1,D(m,n-1))$$

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

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

$$D(k,m,n)=D(k-1,D(k,m,n-1),D(k,m,n-1))$$

## Calculated Examples up to D(3,4)

$$D() = 0$$ 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)

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

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

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

## Calculated Examples up to D(1,0,n)

Using the Comparison Rule C1 $$d(m,n-2+\delta) >> f_{m-1}(n)$$ where $$\delta << n$$

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

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

$$D(4,4) >> f_{3}^3(f_{\omega}(3))$$

$$D(6,1) >> g_1$$ where $$g_{64} = G$$ Graham's number

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

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

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

$$D(1,0,2) >> f_{\omega}^2(f_3(6)) >> f_{\omega}^2(f_3(3)) >> f_{\omega}^2(f_{\omega}(3)) >> f_{\omega}^3(3) >> f_{\omega+1}(3)$$

$$D(1,0,3) >> f_{\omega}^3(f_3(6))$$

$$D(1,0,4) >> f_{\omega}^4(f_3(6)) >> f_{\omega}^4(4) >> f_{\omega+1}(4)$$

and

$$D(1,0,n+\delta) >> f_{\omega}^n(n) = f_{\omega+1}(n)$$ where $$\delta << n$$

## Calculated Example for Graham's number

$$D(1,1,0) = D(0,D(1,0,1),D(1,0,1)) = D(1,0,2)$$

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

$$D(1,1,n) = D(1,0,n+2)$$

$$D(1,2,0) = D(0,D(1,1,2),D(1,1,2)) = D(1,1,3) = D(1,0,5)$$

$$D(1,2,n) = D(1,1,n+3) = D(1,0,n+5)$$

$$D(1,3,n) = D(1,2,n+4) = D(1,1,n+7) = D(1,0,n+9)$$

then

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

and

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

## Calculated Examples up to D(2,0,n)

Using $$D(1,0,n-1+(m+2).(m+1)/2)$$ we get

$$D(1,2,2) = D(1,0,1+4.3/2) = D(1,0,7) >> f_{\omega+1}(7)$$

then

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

and

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

$$D(2,0,2) >> f_{\omega+1}^4(7) >> f_{\omega+1}^4(4) = f_{\omega+2}(4)$$

$$D(2,0,3) >> f_{\omega+1}^5(7) >> f_{\omega+1}^5(5) = f_{\omega+2}(5)$$

and

$$D(2,0,n-2) >> f_{\omega+2}(n)$$ when $$n < 7$$

or

$$D(2,0,n-2+delta) >> f_{\omega+2}(n)$$ when $$\delta << n$$

## Calculated Examples up to D(n,0,n)

$$D(2,3,3) = D(2,0,2+5.4/2) = D(2,0,12) >> f_{\omega+2}(13)$$

then

$$D(3,0,0) = D(2,D(2,3,3),D(2,3,3)) >> D(2,0,D(2,3,3)) >> f_{\omega+2}^2(13)$$

and

$$D(3,0,1) >> f_{\omega+2}^3(13) >> f_{\omega+3}(3) = f_{\omega.2}(3)$$

$$D(3,0,2) >> f_{\omega+2}^4(13) >> f_{\omega+3}(4)$$

and

$$D(3,0,n-2) >> f_{\omega+3}(n)$$ when $$n < 13$$

or

$$D(3,0,n-2+\delta) >> f_{\omega+3}(n)$$ when $$\delta << n$$

then

$$D(4,0,n-2+\delta) >> f_{\omega+4}(n)$$

and

$$D(n,0,n-2+\delta) >> f_{\omega+n}(n) = f_{\omega.2}(n)$$ where $$\delta << n$$

## Calculated Examples up to D(n,0,0,n)

$$D(1,0,0,0) = D(0,D(0,1,1,1),D(0,1,1,1),D(0,1,1,1))$$

$$= D(D(1,1,1),D(1,1,1),D(1,1,1)) >> f_{\omega.2}(D(1,1,1)) = f_{\omega.2}(D(1,0,3)) >> f_{\omega.2}(f_{\omega}^3(f_3(6)))$$

$$D(1,0,0,1) >> f_{\omega.2}^2(n)$$ where $$n <= f_{\omega}^3(f_3(6))$$

then

$$D(1,0,0,n-1+\delta) >> f_{\omega.2}^n(n) >> f_{\omega.2+1}(n)$$ where $$\delta << n$$

$$D(2,0,0,n-2+\delta) >> f_{\omega.2+2}(n)$$ where $$\delta << n$$

$$D(3,0,0,n-2+\delta) >> f_{\omega.2+3}(n)$$ where $$\delta << n$$ or $$f_{\omega^2}(3)$$ when $$n = 3$$

and

$$D(n,0,0,n) >> f_{\omega.2+n}(n) >> f_{\omega.3}(n)$$ where $$\delta << n$$

## Strong D Function Growth Rate

Using subscript notation $$0_{[n-1]}$$ to represent n-1 input parameters with value 0:

$$D(1,0,0,0,n) = D(1,0_{[3]},n) >> f_{\omega.3+1}(n)$$

$$D(n,0_{[3]},n) >> f_{\omega.4}(n)$$ or $$f_{\omega^2}(4)$$ when $$n = 4$$

then

$$D(n,0_{[4]},n) >> f_{\omega.5}(n)$$ or $$f_{\omega^2}(5)$$ when $$n = 5$$

and

$$D(n,0_{[n-1]},n) >> f_{\omega.n}(n) = f_{\omega^2}(n)$$

## Some calculations for n=2

$$D(2,0) = 8 = f_2(2) = f_{\omega}(2)$$

$$D(2,0,2) >> D(2,0,0) >> f_{\omega^2}(2) = f_{\omega^{\omega}}(2) = f_{\epsilon_0}(2)$$

## Some calculations for n=3

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

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

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

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

$$D(3,0_{[2]},1) = D(3,0,0,1) >> f_{\omega^2}(3)$$

$$D(D(3,0_{[2]},1),0_{[D(3,0_{[2]},1)]},1) >> f_{\omega^2}^2(3)$$

## Next - the Alpha Function

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

## References

Strong D Function

The following references are outdated because the above proofs and examples are more reliable. Please keep this in mind if you refer to any of these blogs.

• (Outdated) References