User blog comment:Hyp cos/R function v2.0.1/@comment-27516045-20160711200118/@comment-5529393-20160712002524

I don't quite have a clear picture of how the notation works in full, but I can work out an example with separators.

Let's start with nR.

We scan to the right until we reach the {}, and look to the left; it is preceded by, which is a separator since it has a +1 before the }.

So we are in case A3, which means we (temporarily) replace {} with {}, set s to 1 and set A1 to, and repeat the main process starting inside the first. We hit the end with +1}, so we are in case D.

Step D.1 says to set S1 to our current brace, which is, and set t = s = 1.

In Step D.2, we reduce t to 0, and set S0 equal to the brace surrounding S1, which will be the original array. Step D.2.3 says to break repeating when t = 1, but I'm pretty sure we also break out of it when t = 0.

In Step D.3, This says to set k = m = 1, s(1) = s = 1, B(1) = =  = {}, X(1) = { and Y(1) = {}}.

In Step D.4, we reduce k to 0, set s(0) = s(1) = 1. Then have a problem, since in Step D.4.3 we reduce s(0) to 0, and then compare A0 to A1 - except we haven't defined A0. I believe in this case we skip D.4.3.2 and D.4.4, and proceed to D.4.5, which says to break out of the loop.

In Step D.5, we have Ss(0) = S0 = and Ss(1) = S1 =, and P and Q are set to the part of S0 to the left and right of S1 respectively; that is, P = { and Q = {}}. The final step is to replace S0 (which is the entire array) with P P ... P P Q Q ... Q Q with n P's and Q's, giving us, and this is our final array. We recognize that as n goes to infinity this goes to the limit of the notation without separators, so is at the level of $$\varepsilon_0$$.

I haven't studied R version 2.0.1 to the extent that I have the original R function, but I believe there is a rough equivalence between the two. If that is the case, then notations of the form {A B}, where A and B are either braces without separators or also of this form, are roughly equivalent to Buchholz hydras, where we can think of B as the label of node A. So we can think of as an equivalent of $$\Omega$$,  as an equivalent of $$\Omega_2$$, { – } as an equivalent of $$\Omega_\omega$$, and so on. So the R function at this level works similarly to an ordinal collapsing function.

Note that Hyp Cos has updated his notation, which is listed here: https://stepstowardinfinity.wordpress.com/array/

The R function supposedly has the same strength as the "dropping array notation" in his new notation.