User blog:Ecl1psed276/Introducing a simple yet very powerful notation - Depth Notation!

In this blog post I will introduce the rules for my new notation called Depth Notation (which can be abbreviated as EDN - Ecl1psed's depth notation). I will also analyze it to some extent.

Linear Depth Notation (LDN)
The notation looks like this: D{a,b;c,d,e...} where there are exactly 2 entries before the semicolon and a nonnegative number of entries after the semicolon. Every entry must be a nonnegative integer. The first entry in the array is called the base, and the second entry is called the key. The ruleset for linear array notation is as follows:


 * 1) Trailing zeros can be removed. For example, D{3,4;2,1,0} = D{3,4;2,1}.
 * 2) If there are no entries after the semicolon, then the expression equals the sum of the base and the key. So D{9,13;} = 22.
 * 3) If the key is 1, then the whole expression equals the base. So D{5,1;6,7} = 5.
 * 4) If the third entry is not 0, then do this:
 * 5) Let X be the entire array, except with the key decremented by 1.
 * 6) Then, subtract 1 from the third entry, and replace the key with the value of X. For example, D{3,3;3,3,3} = D{3,D{3,2;3,3,3};2,3,3}.
 * 7) If the third entry is 0, then find the first nonzero entry after the third entry. Subtract 1 from it, and replace the previous entry with the value of the key. For example, D{5,6;0,0,0,3,4} = D{5,6;0,0,6,2,4}

Linear Depth Notation (LDN) acts very similarly to other linear array notations like SAN, BEAF, and BAN, so I will skim through the analysis pretty fast. The limit of LDN is \(\omega^\omega\).

Dimensional Depth Notation (DDN)
With Dimensional Depth Notation, we add new types of seperators. The comma is a shorthand for (0), but now we can also have (1), (2), and we can have (n) for all natural numbers n. For example, D{3,3;3(5)8(123)4} is a valid array. The ruleset is as follows:


 * 1) Trailing zeros can be removed.
 * 2) If there are no entries after the semicolon, then the expression equals the sum of the base and the key.
 * 3) If the key is 1, then the whole expression equals the base.
 * 4) If the third entry is not 0, then do this:
 * 5) Let X be the entire array, except with the key decremented by 1.
 * 6) Then, subtract 1 from the third entry, and replace the key with the value of X.
 * 7) If the third entry is 0, then find the first nonzero entry after the third entry.
 * 8) Look at the separator immediately before said entry. If it's (0), do this:
 * 9) Subtract 1 from the entry, and replace the previous entry with the key.
 * 10) If the separator is (n) and n>0, then do this:
 * 11) Replace the (n) with (n-1)0(n-1)0(n-1)...(n-1)0(n-1) where the number of (n-1)'s equals the key. For example, D{4,6;0(7)0(3)5,2} becomes D{4,6;0(7)0(2)0(2)0(2)0(2)0(2)0(2)5,2}.

Analysis
In this analysis, I will simplify things a little bit. Instead of writing out the FGH expression in full, I will just be giving the ordinal. Also, instead of writing the array out in full, like D{a,a;5,6,7}, I will only be giving the array part, which in this case is 5,6,7. Just like with the previous section, I will go decently fast through the analysis, because it is very similar to dimensional arrays in BEAF and BAN.

So the limit of Dimensional Depth Notation is \(\omega^{\omega^\omega}\).

MORE COMING SOON!
I have an idea to extend this notation really far, but the extension is pretty unusual. However, I don't think it's too complicated to understand.