# Sbiis Cascading-E notation

“Seven planets in the Universe:
Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon.
Seven days in the Year: the seven days of the week.
Seven gates in the Soul, male and female:
Two eyes, two ears, two nostrils and the mouth.”
– Sefer Yetzirah 4:7

First read my review of Hyper-E notation.

# Hyper-E into the hyper-dimension

When we translate expressions of cascading-E and superchained Arrows
to a function format (E and A) in the style of Bowers for comparison,
the difference of the rules in both algorithms as shown, will manifest
as a small difference (think 1 is significant?) in the number of dimensions,
while we keep evaluating expressions past #{s2} and →↑{s} from the right,
as we jump into hyper-dimensions ,[s,t] and nested separators ,[T] soon.

A(X,[T]q1,2) = A(X,[T]A(X,[T]q,2))

            == A(X,[T]..A(X).) :q:



E(X,[T]1,q1) = E(X,[T]E(X,[T]1,q))

            == E(X,[T]..E(X).) :q:



E(X,[T]p,q1) = E(X,[T]E(X,[T]p,q))  (cascading-E motor)

            == E(X,[T]..E(X,[T]p).) :q:



A(X,[T]q1,p1) = A(X,[T]A(X,[T]q,p1),p)  (superchains motor)

             == A(X,[T]..A(X).,p) :q:

Because E is 1 dimension slower than A it seems doable to fine-tune function speed
on a dimensional scale, by adapting the motor rule of your Cascading-F, so that:
for a large size increase you let part of the final entry q (now dropped) remain
in the outer expression; to decrease dimensionally insert the new word F(X,[T]p,q)
in an earlier entry (left of the penultimate p) and drop (or reduce) the outer right;
but to grow faster than a dimension of E (while maintaining the same structure)
your F must employ a partial upload rule (or waitor mechanism) as in Beaf.

Reviewed by Giga Gerard, North-Holland, February 2013.