Here is a new notation I am thinking about. I think i already reached w^w in FGH, which was my initial goal; now my goal is to hit epsilon-zero. When I saw PlantStar's Notation and how easy it was to create your own notation, I got the inspiration to create my own. However, the first time I tried, I utterly and completely fucking failed. I extended upon Graham's function, which has growth w+1, but I did a bunch of complicated stuff and only got to w+9 :( For my second attempt, I will try to make my own version of BEAF, based off of the factorial function. Right now, I have not developed very much and my notation is not very strong yet, but it will be! I promise.
Here is the basics:
- {a} = a!!!...!!! with a factorial signs. Pretty basic for 1-entry arrays. Similar to Aarex's warp notation.
- {0} = 0
- {1} = 1! = 1
- {2} = 2!! = 2! = 2
- {3} = 3!!! = 6!! = 720! = 10^1,746.42
- {4} = 4!!!! = 24!!! = 10^10^10^24
- In general, {a} is about 10^^a
With one-entry arrrays, we already have created a strong function, but we're only in between \(f_2(n)\) and \(f_3(n)\) in the fast-growing hierarchy, so we need to keep going. Let's see what two-entry arrays look like:
- {a, b} = {{{{...{{{a}}}...}}}, b-1} where there are a brackets. We've created a pretty strong recursion so far.
- {a, 1} = {a}. Just like BEAF, if there are any 1's at the end, they can be removed.
- {1, 2} = {{1}, 1} = 1
- {2, 2} = { {2}, 1} = 2
- {2, megafugafzgarboogagoogolplexian} = 2. If the first entry is 1 or 2, the whole array simplifies to just 1 or 2.
- {3, 2} = {{{{3}}}, 1} = {{{{3}}}} = {{{10^1,746.42}}} = { {10^^10^^10^^10^^(10^1,750)}}
- {4, 2} = {{{{{4}}}}, 1} = {{{{{4}}}}} = {{{{10^10^10^24}}}} = 10^^10^^10^^10^^(10^10^10^24)
- In general, {a, 2} is about 10^^^a
- {3, 3} = {{{{3}}}, 2} = {10^^10^^10^^(10^1,750), 2} = {{{{...}}}, 1} = about 10^^^10^^^(10^1,750)
- In general, {a, 3} is about 10^^^^a
- In general, {a, b} = 10^^^...^^^a with b+1 arrows
Wow! With just two entries, we have created something that grows as fast as Knuth's arrows do. Thus, we have hit growth rate w in the fast-growing hierarchy, starting from scratch! This is as powerful as the Ackermann function and three-entry BEAF. But the three-entry arrays in my notation are way crazier.
- {a, b, c} = {{{{...{{{a}}}...}}}, b-1, c} with a bracket sets
- {a, 1, c} = {a, a, c-1}
So {a, b, 2} has growth rate w*2, {a, b, 3} has growth rate w*3, etc. so the limit of 3-entry arrays is w^2. Leaving my previous record of w+9 IN THE DUST!
With 4+ entries, it's very similar to normal BEAF (thanks for the suggestion Syst3ms):
- signifies the rest of the array.
- {a,1,...,1,c+1,#} = {a,1,...,a,c,#}
So 4-entry arrays have growth rate w^3, 5-entry arrays have growth rate w^4, etc., and a-entry arrays have growth rate w^(a-1). So we can say that the limit of this is w^w. But beyond this, it's a little different.
- {a | b} = {a, a, a, ..., a, a} with b a's
- {a, 2 | b} = the following:
- Stage 1 = {a | b} = {a, a, a, ..., a, a}
- Stage 2 = {{a | b} | b} = {a, a, a, ..., a, a} with Stage 1 a's
- Stage 3 = {{{a | b} | b} | b} = {a, a, a, ..., a, a} with Stage 2 a's
- ...
- {a, 2 | b} = Stage a
Insane, right??
- {a, 3 | b} = the following:
- Stage 1 = {a, 2 | b}
- Stage 2 = {{a, 2 | b}, 2 | b}
- Stage 3 = {{{a, 2 | b}, 2 | b}, 2 | b}
- ...
- {a, 3 | b} = Stage a
{a, 4 | b}, {a, 5 | b}, etc. should be obvious.
Well, that's it for now. I promise, I will extend this further in the future!
Please, anyone who has experience with fast-growing hierarchies and stuff, tell me the maximum growth rate of my notation. I suck at fast-growing hierarchies and probably a lot of what I have written in here is complete and utter nonsense. Someone pls help me