Basic Ampersand Notation

Basic Ampersand Notation (B&N) is the first part of Ampersand Notation.

Definition

Let @ represent any sequence of ampersand symbols (&).

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

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

Examples

2[1]
= 2[0][0]
= 2[][]
= 2[]^2[]
= (2²)⁽²²⁾
= 4⁴
= 256

2[&1]
= 2[&0][&0]
= 2[&][&]
= 2[2][&]
= 2[1][1][&]
= 2[0][0][1][&]
= 2[][][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[] ≥ f₂(a)
• a[1] ≥ f₃(a)
• a[2] ≥ f₄(a)
• a[3] ≥ f₅(a)
• a[4] ≥ f₆(a)
• a[&] ≥ f_ω(a)
• a[&1] ≥ f_ω+1(a)
• a[&2] ≥ f_ω+2(a)
• a[&3] ≥ f_ω+3(a)
• a[&&] ≥ f_ω×2(a)
• a[&&1] ≥ f_ω×2+1(a)
• a[&&&] ≥ f_ω×3(a)
• a[&&&&] ≥ f_ω×4(a)