My favorite part of BEAF is the array-of operator, specifically its ability to nest as in 3&3&3. This allows BEAF to define its own structures and describe itself, adding immense power to the notation. It also makes the notation far more complicated to define, because now we have to define the notation over both natural numbers and countable ordinals.
ExE has no such operator, and it lacks both the power and the complexity that arise from it. In this post I will attempt to define something analogous to an array-of operator, perhaps allowing us to develop ExE legion space.
Background[]
ExE forms power towers with a base of 10. If we replace 10 with the symbol "#" and pretend that this substitution is perfectly valid and does not require any clarification, we can define some neat arrays such as this:
\[E100(E_\#100\#100)100 = E100\#^{\#^{.^{.^{.^{\#^{\#^{100}}}}}}}100\]
To make things more visually distinguishable, we'll use \(H\) instead of \(E_\#\).
Okay, let's try to formalize the basics of this F notation. Recall the rules of Extensible-E:
- BASE RULE : If there is a single argument: Ea = 10^a
- DECOMPOSITION RULE : If the last cascader of the last hyper product is not of the form ###...###, then apply a transformation on the hyperion expression: @a&b --> @a&[b]a (where &[b] is the bth member of the fundamental sequence for &)
- TERMINATION RULE : If Rules 1 and 2 don't apply, the last argument = 1: @a&1 --> @a
- EXPANSION RULE : If Rules 1, 2, and 3 don't apply and the last delimiter =! #: @a&*#b --> @a&a&*#(b-1)
- RECURSIVE RULE : If Rules 1, 2, 3, and 4 don't apply: @a#b --> @(@a#(b-1))
[WIP]