Hex function

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Hex function
Growth rate	\(f_{\varepsilon_0}(n)\) (limit)
Notations	H(n)
Author(s)	Googology wiki user DeepLineMadom
Year	2022

Hex function is a notation made by googology wiki user DeepLineMadom. It is based on the hexadecimal (base 16) numeral system. It comes in multiple parts. An example of a valid expression is H(5,1,2).

An expression consists of different kinds of entries and separators at lower levels, and, in later levels, it also introduces some new separators, which diagonalizes over many groups consisting of one or more items, which may be either nested separators (according to the creator, the limit of the level which diagonalizes some fundamental sequences over \(\varepsilon_\alpha\)), star (*), ampersand (&), caret (^), or an hyper-explodion array (#), starting from nested separators first.

Unfortunately, the creator abandoned this notation on June 5, 2022.

Basic rules

1. For empty expressions, H() = 16.
2. For H(n), H(n) = 16*n.
3. If the first entry is 1, H(1,x) = H(1) = 16 (x is the latter of the expression.)
4. If there are 1’s at the end of the input, get rid of 1.
5. For two-entry functions, H(a,b) = H(H(a-1,b),b-1).
6. For those the first entry is 2, and the second entry is greater than 1, H(2,x,…) = H(16,x-1,…)
7. For three or more entries:
   • Reuse the previous rules.
   • The final … is the latter of the expression.
   • If there are 1’s in the intermediate entries, H(a,1,c,…) = H(a,(a-1,1,c,…),c-1,…), H(a,1,1,…,1,1,1,d,…) = H(a,1,1,…,1,1,H(a-1,1,1,…,1,1,1,d,…),d-1,…)

If the second entry is greater than 1, H(a,b,c,…) = H(H(a-1,b,c,…),b-1,c,…)

Note that all the arguments must be natural numbers (not including 0 in the entries, but including 0 in the separators), otherwise the expression will be ill-defined.

Extended rules

The extended notation introduces separators, which can be denoted using curly brackets ({}).

1. Commas indicate {0} separator.
2. H(a{1}2) = H(a,1,1,1,…,1,1,1,2) with a entries of 1’s.
3. H(a{1}n,…) = H(a,1,1,1,…,1,1,1,2{1}n-1,…) with a entries of 1’s.
4. H(a{1}1,n,…) = H(a{1}H(a-1{1}1,n,…),n-1,…), H(a{1}1,1,…,1,1,1,n,…) = H(a{1}1,1,…1,1,H(a-1{1}1,1,…,1,1,1,n,…),n-1,…)
5. H(a{1}1{1}2) = H(a{1}1,1,1,…,1,1,1,2) with a entries of 1’s after {1}.
6. H(a{2}n,…) = H(a{1}1{1}1{1}…{1}1{1}1{1}n-1,…) with a-1 entries of 1’s separated by {1}.
7. H(a{2}1{1}n,…) = H(a{2}1,1,1,…,1,1,1,2{1}n-1) with a entries of 1’s between {2} and {1}.
8. H(a{k}n,…) = H(a{k-1}1{k-1}…{k-1}1{k-1}2{k}n-1,…) with a-1 entries of 1’s separated by {k-1}.
9. H(a{0,1}2) = H(a{a}2).
10. H(a{0,1}n,…) = H(a{a}2{0,1}n-1,…).
11. H(a{k,1}n,…) = H(a{k-1,1}1{k-1,1}…{k-1,1}1{k-1,1}2{k,1}n-1,…) with a-1 entries of 1’s separated by {k-1,1}.
12. H(a{0,k}2) = H(a{a,k-1}2).
13. H(a{0,0,0,…,0,0,0,k}2) = H(a{0,0,0,…,0,0,a,k-1}2).
14. H(a{0{1}1}2) = H(a{0,0,0,…,0,0,0,1}2) with a entries of 0’s inside {}.
15. H(a{0{1}n}2) = H(a{0,0,…,0,0,1{1}n-1}2) with a entries of 0’s inside {}.
16. H(a{0{1}0{1}2}2) = H(a{0{1}0,0,…,0,0,1}2) with a entries of 0’s after {1} inside {}.
17. H(a{0{n}x}2) = H(a{0{n-1}0{n-1}…{n-1}0{n-1}x-1}2) with a entries of 0’s separated by {n-1}.

The previous rules remain unchanged.

It is easy to show that H(a,b) is equal to 16 ↑^b-1 a, using arrow notation.

According to the creator, the expressions like H(2,1,1,2) is comparable to \(\omega+1\) in the fast-growing hierarchy with respect to some unspecified system of fundamental sequences, H(2,1,1,1,2) is comparable to \(\omega 2+1\), H(2{1}1,2) is comparable to \(\omega^{\omega+1}+1\), and these do not match exactly over the terminal fundamental sequences of \(\omega\)’s without +1, under the totalities of each functions that are provably recursive within Peano arithmetic.

Also according to the creator, the pre-Alpha level of the notation is comparable to \(\varepsilon_0\) in the fast-growing hierarchy with respect to some unspecified system of fundamental sequences.

Low-level examples

H(3,2)

• = H(H(2,2))
• = H(H(H(1,2)))
• = H(H(16)))
• = H(16*16)
• = H(256)
• = 256*16
• = 4,096

4,096