User blog:Syst3ms/Chained Array Notation (Revised)

Chained Array Notation
Okay, this is my second attempt at Chained Array Notation, and this one should be much better.

So we're starting at LCAN, i.e these rules :

1) \(a[0] = a^a\)

2) \(a[\#,0] = a[\#]\)

3) \(a[b+1,\#] = a\underbrace{[b,\#]\cdots[b,\#]}_{a}\)

4) \(a[0,\ldots,0,c+1,\#] = a[a,\ldots,a,c,\#]\)

This reaches \(\omega^\omega\).

But then we're going to make some changes regarding NCAN (Nested CAN) :

Stage 3 : Nested (NCAN) :
First off, let's change rule 4 :

4) \(a[0,\ldots,0,c+1,\#] = a[0,\ldots,[0,\ldots,0,0,\#],c,\#]\)

This acts like HAN, and this makes it much, MUCH more powerful.

But we first need to add our process :

If none of the rules apply, begin this process, starting at the first entry : Now what's the difference with my first attempt at this notation ? Well, take a look at the expansion of \(2[0,1,1]\), first with the original definition, and then with the revised one :
 * 1) If the entry is a 0, jump to the next one
 * 2) Otherwise (the entry has to be a nested array) :
 * 3) If the entry is \([0]\), replace it with the base.
 * 4) If the entry is of the form \([b+1,\#]\), then decrease b by 1, and duplicate the main array \(a\) times, where \(a\) is the base.
 * 5) If the entry is of the form \([0,\ldots,0,b+1,\#]\), then replace it with \([0,\ldots,[0,\ldots,0,0,\#],b,\#]\)
 * 6) Otherwise, start this process inside of the entry.

\(\begin{align*} 2[0,1,1] &= 2[2,0,1] \\ &= 2[0,0,1][0,0,1][1,0,1] \\ &= 2[2,2][0,0,1][1,0,1] \\ &= \ldots \end{align*}}\)

\(\begin{align*} 2[0,1,1] &= 2[[0,0,1],0,1] \\ &= 2[[0,[0,0,0],0],0,1] \\ &= 2[[0,2],0,1] \\ &= 2[[2,1],0,1] \\ &= 2[[0,1],0,1][[0,1],0,1][[1,1],0,1] \\ &= 2[[2],0,1][[0,1],0,1][[1,1],0,1] \\ &= 2[[0],0,1][[0],0,1][[1],0,1][[0,1],0,1][[1,1],0,1] \\ &= 2[2,0,1][[0],0,1][[1],0,1][[0,1],0,1][[1,1],0,1] \\ &= \ldots \end{align*}\)

So yeah, as you can see, this makes NCAN much more powerful. But how much more powerful ?

Well, previously, NCAN reached somewhere around \(\varepsilon_0\). Now, it reaches all the way up to \(\varphi(\omega,0)\). Not only that, but the limit of NCAN is no longer \(n\cdots[0]\cdots\), but rather \(n[0,\ldots,0,1]\).

Stage 4 : Dimensional (DCAN) :
Now we're taking a 180° turn from what I had previously done, and we're jumping straight to dimensional.

Tthe comma is a shorthand for \(\{0\}\).

The next separator, as obvious as it seems, is \(\{1\}\). How does this expand, you might ask ? To answer that, we just need to add change rule 2 to "If the entry is an array" and add another rule after that.

3) Otherwise (the entry is a number preceded by a dimensional separator and a zero entry), replace \([\ldots 0 \{A+1\} b+1 \ldots]\) by \([\ldots \underbrace{0\{A\}\ldots\{A\}0}_a 1 \{A+1\} b]\)

Now, \([0\{1\}1]\) is the supremum of linear arrays, giving it a strength of \(\varphi(\omega,0)\).

The limit of this is \(n[0\{n\}1]\), which has a mere strength of \(\varphi(\omega^\omega,0)\).

Stage 5 : Hyperdimensional (HDCAN) :
Now we're going to take a big step and allow for dimensional arrays to be used inside of separators. If you can't deal with a separator because it is an array, apply the regular rules to it.

This means that \([0\{0,1\}1]\) has strength \(\varphi(\omega^\omega,0)\).

But now we can the full power of dimensional arrays to our advantage, which gets us to \(0\{0\{\cdots\{0\}\cdots\}1\}1\), which I believe has strength \(\Gamma_0\).

Stage 6 : Chain-dimensional (CDCAN) :
This is related to the name of this notation.

Now is the time where I could introduce brand new, more powerful separators like the double comma or the like. But I chose to go a different route. Now that we've exhausted what we could put inside of separators, I decided to chain separators together.

Formal definition : [WIP]

The first chained separator is \(\{0\}\{0\}\). \(0\{0\}\{0\}1\) is the fixed point of \(a \mapsto 0\{a\}1\), aka \(0\{0\{\cdots\{0\}\cdots\}1\}1\). The next separator is not \(\{0\}\{0\}\{0\}\), which is very powerful, but rather \(\{0\}\{1\}\).

Now what's \(\{0\}\{1\}\) ? Well, this is simple enough, \(0\{0\}\{1\}1\) expands to \(0\{0\}\{0\}0\{0\}\{0\}\ldots\{0\}\{0\}1\).

What's the limit of \(\{0\}\{n\}\) ? Simply \(\{0\}\{0,1\}\).

Then we have \(\{1\}\{0\}\), and this is what I like to call the "{1} fixed point" of \(a \mapsto \{a\}\). \(0\{1\}\{0\}1\) expands to \(0\{0\{\cdots\{1\}\cdots\}1\}1\) rather than \(0\{0\{\cdots\{0\}\cdots\}1\}1\).

Then we have \(\{0\}\{0\}\{0\}\), which is the fixed point of \(a \mapsto \{0\}\{a\}\).

\(0{0}{1}{0}1\) is the "{1} fixed point" of \(a \mapsto 0{0}{a}1\)

\(0{2}{1}{0}1\) is the "{1} fixed point of \(a\mapsto 0{2}{a}1\), and so on.

The limit of this system is \(0\{0\}\cdots\{0\}1\).

Stage 7 : Powerdimensional (PDCAN) :
You know what time it is, it's time to diagonalize over this.

Now we introduce a new kind of separator, which I will call the chain separator. The first chain separator is \(\langle 0 \rangle\). This is equal to \(\{0\}\cdots\{0\}\).

We can use the full power of chain-dimensional arrays inside all of the separators, including chain ones.

Now we need to formalize some details. What happens when we do \(0\langle 0 \rangle\{0\}1\) ? This is the fixed point of \(a \mapsto 0\langle a \rangle1\), but it's the "{0} fixed point" of it. Meaning that it expands to \(0\langle0\langle\cdots\{0\}\cdots\rangle1\rangle1\). And \(0\{0\}\langle 0 \rangle 1\) is the opposite, it expands to \(0\{0\{\cdots\langle 0\rangle\cdots\}1\}1\).

But this is only the beginning of powerdimensional separators. We can let \([] = []_0\), \(\langle\rangle = []_1\), and so on. In general, \([0]_a\cdots[0]_a = [0]_{a+1}\).

These subscripts don't have to be numbers, they can be linear arrays, nested arrays, dimensional arrays, up to powerdimensional arrays !

This means the limit of powerdimensional arrays is the fixed point of \(a \mapsto 0[0]_a1\)

Formal definition : [WIP]

This is the end of CAN... for now. I plan on :
 * Comparing it to the FGH
 * Writing out the formal definitions
 * Coining numbers
 * Introducing more expansions' to the notation