User blog:ArtismScrub/Re: New array notation idea? (looking back, revising)

(original blog post)

A few days ago, I cranked out some ideas for a linear array notation based off a slight modification to Conway chained arrows. I presumed it to be about as powerful as ωω at its limits, but it may have only been slightly more powerful than CCAN (something like ω2×2) from others' analyses.

I gave  μ(3,3,3,3) as an example.

Rule 3: Another "1" bites the dust
This one got on my nerves once I realized it.

I originally wrote:

μ(a,b,☆,1,c,☆) = μ(a,b,☆,c,c-1,☆) (if there is a chain of multiple "1"s, solve from the right, turning for example (1,1,1,n) into (n,n-1,n-1,n-1))

The major error here is how it prioritizes over Rule 4. Take for example something like:

μ(3,3,1,1,3)

μ(3,3,1,3,2)

μ(3,3,3,2,2)

μ(3,3,3, μ(3,3,3,1,2),1)

μ(3,3,3, μ(3,3,3,2,1),1)

μ(3,3,3, μ(3,3,3,2))

μ(3,3,3, μ(3,3, μ(3,3,2,2),1)

μ(3,3,3, μ(3,3, μ(3,3, μ(3,3,1,2),1),1)

μ(3,3,3, μ(3,3, μ(3,3, μ(3,3,2,1),1),1)

μ(3,3,3, μ(3,3, μ(3,3, μ(3,3,2)))

μ(3,3,3, μ(3,3, μ(3,3, 7625597484987))

μ(3,3,3, μ(3,3, 3{7625597484987}3))

μ(3,3,3,3{ 3{7625597484987}3}3)

...which does solve to an extremely large number using the original rules, BUT it can be made better.

μ(a,b,☆,1,c) = μ(a,b,☆,c,c-1)

In other words, DON'T evaluate Rule 3 until the second-to-last entry is 1.

Then:

μ(3,3,1,1,3)

μ(3,3,1,3,2)

μ(3,3,1, μ(3,3,1,2,2),1)

μ(3,3,1, μ(3,3,1, μ(3,3,1,1,2),1),1)

μ(3,3,1, μ(3,3,1, μ(3,3,1,2,1),1),1)

μ(3,3,1, μ(3,3,1, μ(3,3,1,2),1),1)

μ(3,3,1, μ(3,3,1, μ(3,3,2,1),1),1)

μ(3,3,1, μ(3,3,1, μ(3,3,2),1),1)

μ(3,3,1, μ(3,3,1, 7625597484987 ,1),1)

μ(3,3,1, μ(3,3,1, 7625597484987 ))

μ(3,3,1, μ(3,3, 7625597484987, 7625597484986 ))

See the difference?

I don't believe a string of 1s like that would appear in evaluation unless it was there to start off with, but still, it's a flaw.

(i must use a 5-entry example because the first 2 entries are preserved, so  μ(a,1, ☆) will always solve to a.)

Rule 4: Left or right?
I've been told that it's better to iterate from the left rather than the right.

Rule 4 in my original notation was:

μ(a,b,☆,c,d) = μ(a,b,☆,μ(a,b,☆,c-1,d),d-1)

whereas normal array notation uses:

(a,b,c,☆) = (a,(a,b-1,c,☆),c-1,☆)

as pointed out by PsiCubed2.

Apparently, the latter is much stronger. I didn't want to do exactly that because I wanted to preserve the first 2 entries instead of just the first, so what I could have done was:

(a,b,c,d,☆) = (a,b,(a,b,c-1,d,☆),d-1,☆)

This doesn't make a difference for 4 entries, so I must again use a 5-entry example.

So how this compares... I can only presume that since there are more layers of arrays to evaluate, the result on the righthand side will indeed be much, much larger.

Another thing involving "1"s
μ(3,3,1,1,1,1,1,1,1,1,2)

μ(3,3,1,1,1,1,1,1,1,2,1)

μ(3,3,1,1,1,1,1,1,1,2)

μ(3,3,1,1,1,1,1,1,2,1)

μ(3,3,1,1,1,1,1,1,2)

μ(3,3,1,1,1,1,1,2,1)

μ(3,3,1,1,1,1,1,2)

μ(3,3,1,1,1,1,2,1)

μ(3,3,1,1,1,1,2)

μ(3,3,1,1,1,2,1)

μ(3,3,1,1,1,2)

μ(3,3,1,1,2,1)

μ(3,3,1,1,2)

μ(3,3,1,2,1)

μ(3,3,1,2)

μ(3,3,2,1)

μ(3,3,2)

7625597484987

lol ok

Basically...
This definitely wasn't as good of an idea as I thought it was initially.

‾\_(ツ)_/‾