So on Username5243's site, he described his array notation, which is sort of like BEAF but uses arrays as operators instead of self-contained expressions (the number on the right is called the iterator while the number on the left is called the base). He claims that it is equivalent to BEAF up until \(\omega^2\), but not for the full linear notation. However, I think there is an equivalence between it and BEAF for all of linear arrays, and this wiki seems to agree. So let's analyze it.
BEAF rules (note: for all rulesets in this article, in case of conflicting rules resolve using the highest rule):
- {a} = a
- {a,b} = ab
- {#,1} = {#}
- {a,1,#} = a
- If the third entry is 1, decrease the first non-one entry after the first two entries by 1, replace the entry directly before that entry with the original array with the second entry decreased by 1, and replace all other entries before that entry with the first entry.
- {a,b,c} = {a,{a,b-1,c},c-1}
- {a,b,c,#} = {a,{a,b-1,c,#},c-1,#}
UNAN rules:
- a[0]b = a*b
- a[#,0]b = a[#]b
- a[#]1 = a
- If the first entry is 0, decrease the first non-zero entry by 1, replace the entry before it with the iterator, and replace the iterator with the base.
- a[c]b = a[c-1](a[c](b-1))
- a[c,#]b = a[c-1,#](a[c,#](b-1))
UNAN arrays use nonnegative integers (although the numbers they operate on are positive integers) while BEAF only uses positive integers.
- a[0]b = a*b. BEAF cannot express this.
- a[1]b = {a,b} = ab
- a[2]b = {a,b,2} = a↑↑b
- a[3]b = {a,b,3} = a↑↑↑b
- a[n]b = {a,b,n}
- a[0,1]b = a[b,0]a = a[b]a = {a,a,b}
- a[1,1]b = a[0,1]a[0,1]...[0,1]a with b a's = {a,b,1,2}
- a[2,1]b = {a,b,2,2}
- a[n,1]b = {a,b,n,2}
- a[0,2]b = {a,a,b,2}
- a[1,2]b = {a,b,1,3}
- a[0,3]b = {a,a,b,3}
- a[0,0,1]b = a[0,0,b]a = {a,a,a,b}
- a[1,0,1]b = {a,b,1,1,2}. UN5423 claims that a[1,0,1]3 = a[0,0,1]a[0,0,1]a = {a,a,a,{a,a,a,a},{a,a,a,a}}. I calculate {a,a,a,{a,a,a,{a,a,a,a}}}.
- a[2,0,1]b = {a,b,2,1,2}
- a[0,1,1]b = {a,a,b,1,2}
- a[1,1,1]b = {a,b,1,2,2}
- a[c,d,1]b = {a,b,c,d+1,2}
- a[0,0,2]b = a[0,b,1]a = a[a,b-1,1]a = {a,a,a,b,2}
It seems that {a,b,c,d,e,f,...,z} can be expressed as a[c,d-1,e-1,...,z-1]b.
The array notation used to define PsiCubed2's letter notation is also equivalent to BEAF, but it is harder to see as it is designed for continuous numbers. Luckily, he defines the identity (n-1,...,c-1,b)|a = {10,a,b,c,.....,n}. Combining it with the UNAN-BEAF identity, we see that (n,...,c,b)|a = 10[b,c,...,n]a.
We could also reformat UNAN as a tiered arrow notation:
- a#1 = a
- a↑0b = ab
- a#↑0b = a#(a#↑0(b-1))
- a#↑nb = a#↑n-1ba (where the superscript denotes how many times the arrow is to be repeated)
I am sure something like this has been invented many times. The correspondence to UNAN is that each level of arrow corresponds to a position in the array and the number in that position is the number of that type of arrow. Note that we are only considering operators where higher-tiered arrows never come before lower-tiered arrows.
It is also equivalent to Denis Maksudov's extended arrow notation using ordinals. Each level of arrow corresponds to a power of omega. And finally there is X-Sequence Hyper-Exponential Notation, where the X's work like omegas.
Please excuse the informal tone of this post; it is just a quick overview, not a detailed analysis.
March 3, 2021 edit: I found out about the N-growing hierarchy, which is equivalent to Maksudov's arrows. In fact, the only real difference is notational. [m]α(n) = m↑αn. (also Maskudov's arrows don't support operator 0 which is multiplication). There is also Matthew's Function, which claims many of these same relations to BEAF but fails. Finally, it seems Bowers himself thought of the "ordinal hyper-operator" idea as can be seen in his tetrentrical and pententrical pages.