User blog comment:MachineGunSuper/The Final HTN notation/@comment-30754445-20181118004650/@comment-30754445-20181119092330

Yeah.

You basically need to make sure that all the following things happen:

(1) All the dimensions are infinite in scope (a symbol that can be repeated an arbitrary number of times, or a index which is a number that can go to infinity).

(2) The numbers along any dimension can change independently from one another (for example, in your notation up to the !'s, you can have the 123rd symbol repeated 456 times with the number after the symbols being 789.

(3) One of your dimensions must keep track of actual "recursion levels". A "recursion level" is whenever you repeat the previous function n times. As Alemagno stated, repeating a function twice or even a million times does not count. The number of repetitions must grow with n.

(4) Every dimension (except the one mentioned in #3) can be reduced, at a single point, to going (at least) n steps in the previous dimension (for the exact meaning of this strange statement, see below where I analyze your own notation).

Now let's see why the 3 dimensions of your !'s obey these 4 rules:

(1) The three dimensions are: the type of symbol, the number of times the symbol is repeated, and the final index. Clearly all these can go to infinity, so we're all good here.

(2) You can have any symbol repeated any number of times, followed by any number as an index Checked.

(3) The final index counts recursion-levels perfectly. Tr####7(n) repeats Tr####6(n)... which repeats Tr####5(n)... which repeats Tr####4(n)... and so on. The number of repetitions in every case grows with n. So again, we're good here.

(4) (a) In your notation, only the final symbol (#,@,&,...) is transformed into a number. For example, Tr####(n) b Tr###n(n) where k>n. This only happens for the final symbol, which is what I mean by "it can be reduced at a single point".

(b) Similarly, !k+1 reduces to !k→n only when there's a single !. In other word: "@" becomes "####....", and "&" becomes "@@@@...." and so on. Again, this happens at  a single point (where there's a single symbol and no numbers after it).

So your notation obeys the 4 criteria, which is why it is - indeed - an ω3-level notation.

BTW here is an interesting challange you might want to take upon yourself:

Can you create an ω3-level notation using only the symbols # and @ along with final index?

Let us call it STr (for "supertriangle"). It could work precisely the same as your Tr up to Tr@, but from now on you'll use the number of @'s as your 3rd dimension

Note that by criteria (2), such a notation must allow for having an arbitrary number of @'s and an arbitrary number of #'s. So you'll have to deal with stuff like this:

STr@@@@@@####7(n)

Have any idea how to go about creating such a function?