Nested Ampersand Notation

Nested Ampersand Notation (N&N) is the third part of Ampersand Notation.

Definition

Define an ampersand-structure as an expression of the form &_@x, where @ is any sequence of ampersand-structures, and x is a non-negative integer. x is optional and may be removed.

A valid expression in N&N is of the form a[@b], where @ is any sequence of ampersand structures, and a and b are non-negative integers. b is optional and may be removed. N&N outputs a non-negative integer for every valid expression, determined using this set of rules:

a[] = a^a
a[@0]= a[@]
b > 0: a[@b]= a[@b - 1][@b - 1]...[@b - 1][@b - 1], with a copies of "[@b - 1]"

In the above definition, the string S(x) for a non-negative integer x is defined as follows:

S(0) = ø (empty string)
x > 0: S(x) = &_{S(x - 1)}

For example, S(1) = &, S(2) = &_&, and S(3) = &_{&_&}.

Note: ø_@ where @ is an ampersand-structure sequence is simply @.

Examples

2[&_&1]
= 2[&_&][&_&]
= 2[&₂][&_&]
= 2[&₁][&₁][&_&]
= 2[&₀][&₀][&₁][&_&]
= 2[&][&][&₁][&_&]
= 2[2][&][&₁][&_&]
= 2[1][1][&][&₁][&_&]
= ...

Comparison

Below is a comparison of expressions in this notation with the fast-growing hierarchy, using the Wainer hierarchy for fundamental sequences.

Here, a is a non-negative integer. All inequalities presented only hold true for sufficiently large a, rather than any a.

a[&_&] ≥ f_ω^ω(a)
a[&_&1] ≥ f_ω^ω+1(a)
a[&_&&] ≥ f_ω^ω+ω(a)
a[&_&1] ≥ f_ω^ω+1(a)
a[&_&&] ≥ f_ω^2ω(a)
a[&_&&&] ≥ f_ω^3ω(a)
a[&_&₁] ≥ f_ω^ω²(a)
a[&_{&_&}] ≥ f_ω^{ω^ω}(a)
a[&_{&_{&_&}}] ≥ f_{ω^{ω^{ω^ω}}}(a)