This blog presents a summary of the growth rates for the Strong D Function growth rate. Refer to the blog with the general proof for all the detail.

Growth Rates of the Strong D Functions

Growth rate of the 2 parameter Strong D Function

\(D(n+1,n+1) >> f_n(n+3) >> f_n(n) = f_{\omega}(n)\)

Growth rate of the 3 parameter Strong D Function

\(D(1,0,n) >> f_{\omega}^4(n)\) Refer to First Proof

\(D(1,m,0) >> f_{\omega}^m(f_{\omega+1}(m))\) Refer to Second Proof

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

\(D(2,0,n) >> f_{\omega+1}^2(n)\) Refer to Third Proof

\(D(2,m,0) >> f_{\omega+1}^m(f_{\omega+2}(m))\) Refer to Fourth Proof

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

The next two proofs assume the following:

  • \(D(l-1,0,n) >> f_{\phi-1}^2(n)\)
  • \(D(l-1,m,n) >> f_{\phi-1}^{m+n}(f_{\phi}(m))\)

\(D(l,0,n) >> f_{\phi}^2(n)\) Refer to Fifth Proof

\(D(l,m,n) >> f_{\phi}^{m+n}(f_{\phi+1}(m))\) Refer to Sixth Proof

Therefore the growth rates for the following values of l are:

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

\(D(4,m,n) >> f_{\omega+3}^{m+n}(f_{\omega+4}(m))\)

\(D(5,m,n) >> f_{\omega+4}^{m+n}(f_{\omega+5}(m))\)

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