User blog:Ecl1psed276/Star Notation Revamped! - Introduction and Analysis

Well, I made a blog post a few months back about my new notation "Star notation". I've decided to overhaul this notation and extend it to be really powerful. I know how the notation will work up until at least part 5, all I have to do is write this blog post.

Part 1 - Bracket notation
This part is really simple, so it will be brief. Bracket notation consists of a[b]c where a, b, and c are numbers. \(a\) is called the base, and \(b\) is called the key. The rules are as follows: With these rules, a[0]b is equivalent to multiplying a and b, a[1]b equals a^b, a[2]b equals a tetrated to b, and so on. In general, a[b]c equals \(a\uparrow\uparrow...\uparrow c\) with b arrows. Thus, a[1]b is approximately \(f_2(b)\), a[2]b is approximately \(f_3(b)\), and so on. The ordinal limit of this notation is therefore \(\omega\).
 * 1) \(a[b]1 = a\)
 * 2) \(a[0]b = a*b\)
 * 3) \(a[b]c = a[b-1](a[b]c-1)\)

Part 2 - Linear array notation
In this section, the notation changes to allow arrays of numbers in the square brackets. It looks like \(b[n_1,n_2...n_m]k\). The rules for evaluating arrays are as follows: (note that you can remove trailing zeros) According to rule 3, a[0,1]b will expand to a[b]b, which has a growth rate of \(f_\omega\), as per the previous section. a[1,1]b will expand to a[0,1](a[1,1]b-1), which will eventually become a[0,1]a[0,1]a...a[0,1]a with b a's. This has a growth rate of \(f_{\omega+1}\). Thus, \(\3[1,1]64 \approx G_64\) where \(G_64\) is Graham's number. Contining, we find that a[2,1]b has a growth rate of \(f_{\omega+2}\), a[3,1]b has a growth rate of \(f_{\omega+3}\), and a[0,2]b = a[b,1]b has a growth rate of \(f_{\omega2}\). We also have the following comparisons: [1,2] corresponds to \(\omega2+1\). [2,2] corresponds to \(\omega2+2\). [0,3] corresponds to \(\omega3\). [0,0,1] corresponds to \(\omega^2\). [1,0,1] corresponds to \(\omega^2+1\). [0,1,1] corresponds to \(\omega^2+\omega\). [0,0,2] corresponds to \(\omega^22\). [0,0,0,1] corresponds to \(\omega^3\). [0,0,0,0,1] corresponds to \(\omega^4\). Thus, the ordinal limit of linear array notation is \(\omega^\omega\).
 * 1) Key rule: \(b[n_1,n_2...n_m]1 = b\)
 * 2) Base rule: \(b[0]k = bk\)
 * 3) Successor rule: \(b[n_1+1,n_2...n_m]k = b[n_1,n_2...n_m](b[n_1+1,n_2...n_m]k-1)\)
 * 4) Expansion rule: \(b[0,0...0,n_1+1,n_2...n_m]k = b[0,0...k,n_1,n_2...n_m]k\)

Part 3 - Nested array notation
In Nested array notation, we will introduce the new concept of a separator. A separator looks like \({n_1 S_1 n_2 S_2 ... n_{m-1} S_{m-1} n_m}\) where \(n_i\) are numbers, and \(S_i\) are also separators. The comma that we've seen in linear array notation is just a shorthand for {0}. We can also have {1}, {2}, {0,1}, {0,0,1}, {0{1}1}, etc... A valid array in Nested array notation could look something like this: [0{2{1,4}5,1}3{4}1]. The next separator after the comma is {1}. [0{1}1] expands to [0,0,0...0,1] with k commas. Then, we have [1{1}1], [0,1{1}1], [0,0,1{1}1]... etc. And then [0{1}2] expands to [0,0,0...0,1{1}1] with k commas. We can also have [0{1}3], [0{1}0,1], [0{1}0{1}1], [0{1}0{1}0{1}1]... etc. Then, [0{2}1] expands to [0{1}0{1}0{1}1] with k {1}'s. Further, we can have [0{3}1], [0{0,1}1], [0{1,1}1], [0{0,2}1], [0{0,0,1}1], [0{0{1}1}1],, [0{0{0{1}1}1}1], and so on... which forms the limit of nested array notation.

Analysis
The following is a table of arrays with their ordinal correspondences. Thus, the ordinal limit of nested array notation is \(\varepsilon_0\).

Part 4 - Hyper-nested array notation
This part will reach the ordinal level of \(\psi(\Omega_\omega)\). WIP.

Part 5 - Primary star notation
This part will reach the ordinal level of Hyp cos's primary dropping array notation (pDAN). This is the last part that I'm sure how it works. WIP.

Part 6 - Hyper star notation
This part will reach past dropping array notation, and it will be upper bounded by (0,0,0)(1,1,1)(2,2,2) in BMS. For this part, I'm not entirely sure how it will work, but I have a pretty good idea. WIP.