Mixed arrow notation

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Mixed arrow notation
Growth rate	\(f_{\omega}(n)\) (limit)
Notations	\(a \uparrow^{b_1}\downarrow^{b_2}... c\)
Author(s)	hyp cos and IsTakenIsTaken
Year	2013 and 2020

Mixed arrow notation is an arrow notation independently made by hyp cos and IsTakenIsTaken.

Historical background

In 2013, hyp cos created a notation called mixed arrow notation in one of his user pages in this wiki. Since this wiki does not allow to create an article which refers to a page in this wiki as the first source unless it passes the voting system, there had not been an article on the mixed arrow notation. Later in 2020, IsTakenIsTaken created another notation called mixed arrow notation. Since IsTakenIsTaken prepared an external source of the own mixed arrow notation, this article can refer to it as the first source of the version.

Since there had not been an article on the mixed arrow notation by hyp cos, IsTakenIsTaken perhaps does not know hyp cos's preceding work. As a user pointed out at the talk page of this article, the mixed arrow notation by IsTakenIsTaken seems to accidentally coincide with the mixed arrow notation by hyp cos. Therefore we credit both of the independent creators. This article explains the convention by IsTakenIsTaken, although the resulting function is equivalent.

Definition

Expressions in mixed arrow notation are of the form a@b, where @ is any sequence of arrows. Although it was not clarified in the original definition, it is reasonable to guess that a and b are assumed to be positive integers. Later, the creator clarified that a and b are positive integers. Mixed arrow notation in the original definition has four rules to compute a@b:

1. a↑b = a↓b = a^b
2. a↑...↑1 = a↓...↓1 = a
3. a@↑b = a@(a@(...a)...) with b copies of a
4. a@↓b = (...(a...)@a)@a with b copies of a

It is right-associative by default so a↑↓↑b↑↓c is a↑↓↑(b↑↓c). Although it was not clear from the original description, the second rule applies when @ is any sequence of arrows, even though it looks like it only applies when there is only one type of arrow. This is later clarified in the updated definition.

Later, the creator updated the definition in the following way:

1. a↑b = a↓b = a^b
2. a@1 = a
3. a@↑b = a@(a@(...a)...) with b copies of a
4. a@↓b = (...(a...)@a)@a with b copies of a

It is also right-associative by default so a↑↓↑b↑↓c is a↑↓↑(b↑↓c). This does not change the definition, but it makes it clearer.

Functions defined using mixed arrow notation

a↑↓↑b (equal to a↓↓↑b) is called "mixation."

Mixation has an approximate growth rate of \(f_3\) in the fast-growing hierarchy.

a↓↑↓b (equal to a↑↑↓b) is called "ixmation."

Ixmation has an approximate growth rate of \(f_4\) in the fast-growing hierarchy.

Examples

• a↑@b=a↓@b, as a↑b=a↓b.
• 2↑↓↑3=2↑↓(2↑↓2)=2↑↓(2↑2)=2↑↓4=((2↑2)↑2)↑2=(4↑2)↑2=16↑2=256.
• 2↑↓↑4=2↑↓256=2↓↓256=2↑2↑255.