User blog:Hyp cos/Non-Classic Growth Rate

The post is a continuation of this one.

A better definition of "comparable" functions
There are many definitions of eventual domination. Some are listed below from tighter to looser. And \(f\approx g\) (f is comparable to g) if \(f\ge^*g\land g\ge^*f\). So \(f\approx f\) holds for all f.
 * 1) \(f\ge^*g\) (f eventually dominates or is comparable to g) if \(\exists N\forall n>N(f(n)\ge g(n))\)
 * 2) \(f\ge^*g\) if \(\lim\limits_{n\to\infty}\frac{g(n)}{f(n)}\le1\)
 * 3) \(f\ge^*g\) if \(\exists a,b,N\forall n>N(f(n+a)+b\ge g(n))\)
 * 4) \(f\ge^*g\) if \(\exists a,b,N\forall n>N(f(an+b)\ge g(n))\)

Definition #1 is very strict. There are few common functions can be classified into the same comparable class. And so is Definition #2 for fast-growing functions (googological functions). As PsiCubed2 said, Definition #3 is still too strict, if we want to eliminate the effect of "base numbers" of multivariable notations. For instance, \(\lambda n.n\uparrow\uparrow n\) (using lambda abstraction to denote functions) is not comparable to \(\lambda n.2\uparrow\uparrow n\) according to Definition #3, but they are comparable according to Definition #4. However, Definition #4 is too loose to distinguish \(H_\alpha\) and \(H_{\alpha+\beta}\) where \(\beta<\omega^2\) in Hardy hierarchy. The "resolution" of Definition #4 in the HH scale is \(\omega^2\), defined as the least \(\beta\) that \(H_\alpha\) is not comparable to \(H_{\alpha+\beta}\). Definition #3 has resolution \(\omega\). A definition of "comparable" with resolution \(<\omega\) would be too strict.

Here is the definition used in this post, with resolution \(\omega\) but can eliminate the effect of "base numbers":
 * \(f\ge^*g\) if \(\forall a>1\exists N\forall n>N(f(\lfloor an\rfloor)>g(n))\) (here a is a real number)

By the way, this definition, #3 and #4 are very loose in the SGH scale.

Growth rate (HH) addition and subtraction
If the FS system fits \((\alpha+\beta)[n]=\alpha+(\beta[n])\), then \(H_{\alpha+beta}(n)=H_\alpha(H_\beta(n))\). The composition of functions results addition of growth rates (in HH scale). Since inverse function is the inverse of composition, we can define growth rate (in HH scale) addition and subtraction as follows.

A growth rate expression is built by the following rules: Growth rate expressions work in Hardy hierarchy as follows: Function f has growth rate \(a\) if \(f\approx H_a\).
 * 1) An ordinal \(\alpha\) is a growth rate expression
 * 2) If \(a\) is a growth rate expression, then \((-a)\) is a growth rate expression
 * 3) If \(a\) and \(b\) are growth rate expressions, then \((a+b)\) is a growth rate expression
 * 1) For ordinal \(\alpha\), \(H_\alpha\) works as usual
 * 2) \(H_{(-a)}=H_a^{-1}\) (inverse function)
 * 3) \(H_{(a+b)}=H_a\circ H_b\) (composition, i.e. \(H_{(a+b)}(n)=H_a(H_b(n))\))

Growth rate expressions have these properties: Further, functions with costs (e.g. Goodstein function), or even functions with costs, which again have costs, which again have costs, and so forth, can get their growth rate in the form of \(\alpha_1+(-\beta_1)+\alpha_2+(-\beta_2)+\cdots+\alpha_k+(-\beta_k)\), where \(\alpha_i\) and \(\beta_i\) are ordinals (only have addition).
 * 1) \(((a+b)+c)=(a+(b+c))\) (so we can omit the brackets among addition)
 * 2) \((-(-a))=a\)
 * 3) \((-(a+b))=((-b)+(-a))\)

Growth rate (HH) multiplication and division by ordinal
If the FS system fits \((\alpha\beta)[n]=\alpha(\beta[n])\), then \(H_{\alpha\beta}\) can be build in the following way: From now on, it is hard to find an inverse of the operation. So it is hard to define how growth rate expressions work in Hardy hierarchy.
 * 1) \(h_0(n)=n\)
 * 2) \(h_{\gamma+1}(n)=h_\gamma(H_\alpha(n))\)
 * 3) \(h_\gamma(n)=h_{\gamma[n]}(n)\) for limit \(\gamma\)
 * 4) \(H_{\alpha\beta}=h_\beta\)

A growth rate expression is built by the following rules: Growth rates work as follows: Approximately, the "division by ordinal" eliminates the rightmost ordinal factor of a growth rate expression.
 * 1) An ordinal \(\alpha\) is a growth rate expression
 * 2) If \(a\) is a growth rate expression, then \((-a)\) is a growth rate expression
 * 3) If \(a\) and \(b\) are growth rate expressions, then \((a+b)\) is a growth rate expression
 * 4) If \(a\) is a growth rate expression, \(\alpha\) is an ordinal, then \((a\alpha)\) is a growth rate expression
 * 5) If \(a\) is a growth rate expression, \(\alpha\) is an ordinal, then \((a/\alpha)\) is a growth rate expression
 * 1) Function f has growth rate \(\alpha\) if \(f\approx H_\alpha\)
 * 2) Function f has growth rate \((-a)\) if \(f=g^{-1}\), where g has growth rate \(a\)
 * 3) Function f has growth rate \((a+b)\) if \(f=g\circ h\), where g has growth rate \(a\) and h has growth rate \(b\)
 * 4) Function f has growth rate \((a\alpha)\) if
 * 5) \(f=h_\alpha\)
 * 6) \(h_0(n)=n\)
 * 7) \(h_{\beta+1}(n)=h_\beta(g(n))\)
 * 8) \(h_\beta(n)=h_{\beta[n]}(n)\) for limit ordinal \(\beta\)
 * 9) g has growth rate \(a\)
 * 10) Function f has growth rate \((a/\alpha)\) if
 * 11) \(g=h_\alpha\)
 * 12) \(h_0(n)=n\)
 * 13) \(h_{\beta+1}(n)=h_\beta(f(n))\)
 * 14) \(h_\beta(n)=h_{\beta[n]}(n)\) for limit ordinal \(\beta\)
 * 15) g has growth rate \(a\)