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REPLACED

This blog has been replaced. Simpler and more rigorous calculations of the Strong D Function can be found at another blog that can be accessed by this link.


Strong D Function Overview

The Strong D Function will be calculated in this blog hopefully up to a growth rate of \(f_{\epsilon_0}\). This blog will also refer to the omega to epsilon nought ordinal hierarchy index of f().

From the Strong D Function blog, we can start with these formulas:

\(D(4,1) = D(3,D(4,0)) = D(3,D(3,D(3,4))) = d(4,1)\)

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

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


D(4,n) Calculations

\(D(4,2) = f_{3}(4)\)

Using formula 15 \(f_n(a+1) >> f_{n-1}(f_{n}(a))\) we get

\(D(4,2) = f_{3}(4) >> f_{2}(f_{3}(3)) = f_{2}(f_{\omega}(3))\)

similarly

\(D(4,2) = D(3,D(4,1)) >> D(3,f_{\omega}(3)) >> f_{3-1}(f_{\omega}(3)+2) = f_2(f_{\omega}(3)+2) >> f_2(f_{\omega}(3))\)

\(D(4,3) = D(3,D(4,2)) >> D(3,f_2(f_{\omega}(3))) >> f_2^2(f_{\omega}(3))\)

\(D(4,n) >> f_2^{n-1}(f_{\omega}(3))\)

\(D(4,D(4,1)) >> D(4,f_{\omega}(3)+1)) >> f_2^{f_{\omega}(3))+1-1}(f_{\omega}(3)) = f_3(f_{\omega}(3))\)


D(5,n) Calculations

\(D(5,0) = D(4,D(4,5)) >> f_3(D(4,5)) >> f_3(f_2^{5-1}(f_{\omega}(3))) >>\) very low bound \(>> f_3(f_{\omega}(3))\)

\(D(5,1) = D(4,D(5,0)) >> f_3(f_3(f_{\omega}(3))) = f_3^2(f_{\omega}(3))\)

\(D(5,2) = D(4,D(5,1)) >> f_3(f_3^2(f_{\omega}(3))) = f_3^3(f_{\omega}(3))\)

\(D(5,n) = D(4,D(5,n-1)) >> f_3(f_3^n(f_{\omega}(3))) = f_3^{n+1}(f_{\omega}(3))\)

\(D(5,D(4,1)) = D(4,f_{\omega}(3)) >> f_3^{f_{\omega}(3)}(f_{\omega}(3)) = f_4(f_{\omega}(3))\)


D(m,n) Calculations

\(D(m,0) >> f_{m-2}(f_{\omega}(3))\)

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

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


D(1,0,n) Calculations

\(D(1,0,0) = D(0,D(0,1,1),D(0,1,1)) = D(D(1,1),D(1,1)) = D(4,4) >> f_2^3(f_{\omega}(3))\)

\(D(1,0,1) = D(0,D(1,0,0),D(1,0,0)) = D(D(4,4),D(4,4)) >> D(f_2^3(f_{\omega}(3)),f_2^3(f_{\omega}(3)))\)

\(>> f_{f_2^3(f_{\omega}(3))}(f_2^3(f_{\omega}(3))) = f_{\omega}(f_2^3(f_{\omega}(3))) >>\) very low bound \(>> f_{\omega}^2(3)\)

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

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

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

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

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


D(1,m,n) Calculations

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

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

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

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

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

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

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

then

\(D(1,2,2) = D(1,0,2-1+4.3/2) = f_{\omega}^{2-3+6}(f_{\omega+1}(3)) = f_{\omega}^5(f_{\omega+1}(3))\)

and

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


D(2,0,n) Calculations

\(D(2,0,0) = D(1,D(1,2,2),D(1,2,2)) >>\) very low bound \(>> D(1,0,D(1,2,2))\)

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

and

\(D(2,0,0) = D(1,D(1,2,2),D(1,2,2)) >> f_{\omega}^{m-3+(m+2).(m+1)/2}(f_{\omega+1}(3))\) where \(m >> f_{\omega}^5(f_{\omega+1}(3))\)

\(= f_{\omega}^{m-3+m^2/2+m.3/2+2/2}(a)\) where \(m >> f_{\omega}^5(a)\) and \(a = f_{\omega+1}(3)\)

\(>> f_{\omega}^{m^2/2+2m}(a) >> f_{\omega+1}(a)\) need to check this

then

\(>> f_{\omega}^{f_{\omega}^5(a)^2/2+f_{\omega}^5(a).2}(a) >> f_{\omega+1}(a)\) need to check this

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

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

Using formula 24 \(f_{\omega}^{f_{\omega+1}^n(3).2}(3) >> f_{\omega+1}^{n+1}(3)\)

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

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

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

\(>> f_{\omega}^{f_{\omega+1}^2(a).2}(a) >> f_{\omega+1}^3(a) = f_{\omega+1}^4(3) = f_{\omega+1}(f_{\omega+2}(3))\)

and

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


D(l,0,n) Calculations

Let \(D(l-1,0,n) = f_{\phi}^{n-1}(f_{\phi+1}(3))\) where \(\phi\) is any ordinal up to \(\epsilon_0\)

then

\(D(l,0,0) = D(l-1,D(l-1,l,l),D(l-1,1,1)) >> D(l-1,D(l-1,l,l),0)\)

Using Rule: NL from above then

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

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

\(= f_{\phi}^{f_{\phi+1}(3)}(f_{\phi+1}(3)) = f_{\phi+1}(f_{\phi+1}(3))\)

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

and

\(D(l,0,1) = D(l-1,D(l,0,0),D(l,0,0)) >> D(l-1,D(l,0,0),0)\)

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

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

and

\(D(l,0,2) >> D(l-1,D(l,0,1),D(l,0,1)) >> D(l-1,D(l,0,1),0)\)

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

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

\(= f_{\phi+1}(f_{\phi+2}(3))\)

and when

\(D(l,0,n-1) = f_{\phi+1}^{n-2}(f_{\phi+2}(3))\)

then

\(D(l,0,n) >> D(l-1,D(l,0,n-1),D(l,0,n-1)) >> D(l-1,D(l,0,n-1),0)\)

\(>> D(l,0,D(l,0,n-1)+D(l,0,n-2)+D(l,0,n-3)+ ... +D(l,0,1)+D(l-1,0,1)+1)\)

\(= f_{\phi}^{f_{\phi+1}^{n-2}(f_{\phi+2}(3))+f_{\phi+1}^{n-3}(f_{\phi+2}(3))+ ... +f_{\phi+1}(f_{\phi+2}(3))+f_{\phi+2}(3)+f_{\phi+1}^2(3)+f_{\phi+1}(3)}(f_{\phi+1}(3))\)

\(= f_{\phi}^{f_{\phi+1}^{n-2}(f_{\phi+2}(3))+f_{\phi+1}^{n-3}(f_{\phi+2}(3))+ ... +f_{\phi+1}(f_{\phi+2}(3))+f_{\phi+2}(3)+f_{\phi+1}^2(3)}(f_{\phi+1}^2(3))\)

\(= f_{\phi}^{f_{\phi+1}^{n-2}(f_{\phi+2}(3))+f_{\phi+1}^{n-3}(f_{\phi+2}(3))+ ... +f_{\phi+1}(f_{\phi+2}(3))+f_{\phi+2}(3)}(f_{\phi+2}(3))\)

\(= f_{\phi}^{f_{\phi+1}^{n-2}(f_{\phi+2}(3))+f_{\phi+1}^{n-3}(f_{\phi+2}(3))+ ... +f_{\phi+1}(f_{\phi+2}(3))}(f_{\phi+1}(f_{\phi+2}(3)))\)

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

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

and

Rule: LN \(D(l,0,n) >> f_{\phi+1}((D(l,0,n-1)))\) where \(D(l,0,n-1) = f_{\phi+1}^{n-2}(f_{\phi+2}(3))\)

and

Rule: LG \(D(l,0,n) >> f_{\mu}^{n-1}(f_{\mu+1}(3))\)  where \(\mu=\phi+1\) and

\(D(l-1,0,n) = f_{\phi}^{n-1}(f_{\phi+1}(3))\)

and

Rule: L1 \(D(l,0,1) >> f_{\phi+1}(3)\) where \(D(l-1,0,1) = f_{\phi}(3)\)


General Proof of Strong D Function Growth Rate

A general proof of the Strong D Function Growth Rate is available on another blog post.

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