Extended Ampersand Notation

Extended Ampersand Notation (X&N) is the second part of Ampersand Notation.

Definition

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

Let @ represent any sequence of ampersand-structures.

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

  1. a[] = aa
  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]
  5. a[@&0]= a[@&]
  6. x > 0: a[@&x]= a[@&x - 1&x - 1...&x - 1&x - 1] with a copies of "&x - 1"

Examples

2[&10]
= 2[&1]
= 2[&0&0]
= 2[&&]
= 2[&2]
= 2[&1][&1]
= 2[&0][&0][&1]
= 2[&][&][&1]
= ...

2[&2&1]
= 2[&2&0][&2&0]
= 2[&2&][&2&]
= 2[&22][&2&]
= 2[&21][&21][&2&]
= 2[&20][&20][&21][&2&]
= 2[&2][&2][&21][&2&]
= 2[&1&1][&2][&21][&2&]
= ...

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.