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Compact FGH Notation

The Fast-growing hierarchy (FGH) function is very useful to understand and compare fast growing functions in general. The visual notation however can become difficult to read and compare once the ordinals become nested in complex permutations. This makes it difficult to evaluate and investigate the patterns and properties of individual FGH functions.


Proposed Notation

This blog introduces a proposed notation to simply the presentation of complex FGH functions. This Compact FGH Notation is then used in my other blogs on a fast growing J Function that I am designing.

The notation is meant to be easy to read, but also, easy to translate back into LaTeX math notation so that the relevant FGH function can be presented visually in its usual form.


Notation Example

Here is the first example, which shows the key notational change. The use of [ square ] brackets implicitly notates the \(\omega\) ordinal and makes it possible to compactly notate various powers of \(\omega\).

f[3.1]^2(4) \(= f_{\omega^{3}}^2(4)\)

In this example we have many more complex powers of omega:

f[6.2+3.1]^2(7) \(= f_{\omega^{6}.2+\omega^{3}}^2(7)\)

The advantage of this notation is that it uses less characters overall, in particular the removal of all instances of \omega which normally would appear many times.

Another advantage is the ability to do side by side comparisons of two functions and identify the differences faster, as in this pair of examples:

f[6.1+1.2+0.4].6+[6.1+1.2+0.3].5+[6.1+1.2+0.3].6+3.4+0.1^7(7) \(= f_{\omega^{\omega^{6}+\omega.2+4}.6+\omega^{\omega^{6}+\omega.2+3}.5+\omega^{\omega^{6}+\omega.2+3}.6+\omega^{3}.4+1}^7(7)\)

f[6.4+2.4+0.5].6+[6.1+0.5].1+[6.1+0.2].3+[3.2+1.2+0.3].4+0.2^2(7) \(= f_{\omega^{\omega^{6}.4+\omega^{2}.4+5}.6+\omega^{\omega^{6}+5}+\omega^{\omega^{6}+2}.3+\omega^{\omega^{3}.2+\omega.2+3}.4+2}^2(7)\)

f[[6.1+1.2+0.4].6].4+[6.1+1.2+0.3].5+0.1^7(7) \(= f_{\omega^{\omega^{\omega^{6}+\omega.2+4}.6}.4+\omega^{\omega^{6}+\omega.2+3}.5+1}^7(7)\)

f[[[6.4]+2.4+0.5].6].4+[6.1+0.5].1+6.1+0.2^2(7) \(= f_{\omega^{\omega^{\omega^{\omega^{6}.4}+\omega^{2}.4+5}.6}.4+\omega^{\omega^{6}+5}+\omega^{6}+2}^2(7)\)


Notation Definition

The notation is not dissimilar to normal FGH notation. The use of "[" square "]" brackets implicitly notates the \(\omega\) ordinal. The start of the function "f" is a shortened version of "f_{\omega^{" and the end of the function is the usual nested powers of the FGH function and the usual single input parameter.

Note that any number in the form "p.m" equates to \(\omega^p.m\) such as in this example:

f0.2^2(4) \(= f_2^2(4)\)

and ordinal towers can be constructed using one or more sets of square brackets. In this example there are two ways to represent \(\omega\):

f[0.2]^2(4) = f1.2^2.4 \(= f_{\omega.2}^2(4)\)

Otherwise the notation uses unique sequences to reference various FGH functions.

A complex tower or ordinals is shown here:

f[[1.1].1].1^1(3) \(= f_{\omega^{\omega^{\omega^1.1}.1}.1}^1(3) = f_{\omega^{\omega^{\omega}}}(3) = f_{\epsilon_0}(3)\)

The following Compact FGH Notations are equivalent:

f[1.1].1^1(2) \(= f_{\omega^{\omega}}(2) = f_{\epsilon_0}(2)\)

f1.1+0.1^2(2) \(= f_{\omega+1}^2(2) = f_{\omega^{\omega}}(2) = f_{\epsilon_0}(2)\)

Some Observations Constructing FGH functions using this notation can be intuitive once you become comfortable with the definition of the "p.m" numbers and behavior of the square brackets. Here are some observations:

Any sequence of "p.m" numbers should be presented in monotonically decreasing order of "p". Duplicate instances of "p" are redundant because they can simply be combined and the "m" number adjusted accordingly. Therefore:

f0.1+2.1^2(5) this notation is incorrect

f2.1+0.1^2(5) this notation is the correct version of the above (and easier to read)

f2.2+2.1^2(5) this notation is redundant and should be simplified

f2.3^2(5) this notation simplifies the redundant previous example (and is easier to read)

Another observation is that all instances of square brackets can be repeated to create complex but still valid FGH functions. Any sequence of square brackets should be presented in monotonically decreasing order. The order is defined by the "p" numbers contained in the square brackets or by the associated "m" numbers if the "p" numbers are the same. Therefore:

f[2.2+1.2].1+2.1^3(4) is a valid FGH function and so are the following:

f[2.2+1.2].1+[2.1+1.2].2+2.1^3(4)

f[2.2+1.2].1+[2.1+1.2].2+[2.1+1.1+0.3].2+2.1^3(4)

f[2.2+1.2].1+[2.1+1.2].2+[2.1+1.1+0.3].2+[1.1+0.2].3+2.1^3(4)

and

f[2.2+1.2].1+[2.1+1.2].2+[2.1+1.1+0.3].2+[1.1+0.2].3+[1.3].2+2.1^3(4)

This is not a surprise. This is only the start of how complex FGH functions can become. Nested square brackets can be repeated in a similar way. The following is only a small example of the complexity of some FGH functions:

f[[3.2].1].2+[2.2+0.2].3+3.1^3(5) is the first example of FGH with nested square brackets. Here are more:

f[[3.2].1+[3.1].2].2+[2.2+0.2].3+[2.1+1.4].4+3.1^3(5)

f[[3.2].1+[3.1].2].2+[[3.2].1+[3.1].1+2.4].4+[2.2+0.2].3+[2.1+1.4].4+3.1+2.2^3(5)

f[[3.2].1+[3.1].2].2+[[3.2].1+[3.1].1+2.4].4+[3.3+2.1+1.4+0.3].2+0.2].4+[2.2+0.2].3+[2.1+1.4].4+3.1+2.2^3(5)

The general rule is that any instance of square brackets can be repeated to form another valid FGH function. This can occur within nested brackets or the repetition of the entire nested bracket, provided each instance is monotonically smaller than the last. For example:

f[2.3+0.5].2+[2.1+0.6].1+[1.1+0.5].2+1.1+0.6^1(7) \(= f_{\omega^{\omega^{2}.3+5}.2+\omega^{\omega^{2}+6}+\omega^{\omega+5}.2+\omega+6}(7)\)


Notation Translation

Compact FGH Notation can be translated back into LaTeX math notation using the following string replacements:

Replace "^" with "}^"

Replace "]." with "."

Replace "." with "}."

Replace "f" with "f_{\omega^{"

Replace "+" with "+\omega^{"

Replace "[" with "\omega^{"

Replace "]" with "}"

The following translations will remove redundant powers of zero and multiplication by 1:

Replace "\omega^{0}." with ""

Replace "^{1}" with ""

Replace ".1" with ""

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