Zillion Notation

Zillion Notation (ZN) is a BEAF-like notation useful for coining -illions similar to the ones in Jonathan Bowers' system.

Definition

Let @ represent any sequence of non-negative integers separated by commas. A valid expression in ZN is of the form z(@). ZN outputs a non-negative integer for every valid expression, determined using this set of rules:

1. z() = 0
2. z(a) = 1000a
3. z(@, 0) = z(@)
4. b > 0: z(a, b, @) = z(10^3a, (b-1), @)

If none of these rules apply, start the process shown below, starting from the second entry:

• If the entry is 0, then jump to the next entry.
• If the entry is not 0, then:
  • Let n equal to this entry, and a equal to the first entry of the input.
  • Decrease n by 1.
  • Change the entry before n to a.
  • Terminate the process.

Examples

z(x, y) = yth Tier x -illion in Jonathan Bowers' -illion system

z(5, 2)
= z(1000^5, 1)
= z(10^15, 1)
= z(1000^(10^15), 0)
= z(10^(3*10^15), 0)
= 10^(3*10^15 + 3)

z(5, 1, 1, 2)
= z(5, 1, 5, 1)
= z(5, 1, 5)
= z(5, 5, 4)
= z(1000^5, 4, 4)
...