Wait, but isn't \(\lbrace a,b / c \rbrace = \lbrace a \& a \& a \cdots a \& a \& a / c-1\rbrace\) (b times)? Bowers defines it in "legion arrays" quite so. If we say that \(\lbrace a,b / c \rbrace = \lbrace a,\lbrace a,b-1 / c\rbrace / c-1\rbrace,\) it is no better than \(\lbrace a,b,c / 2\rbrace\). I shall fix that. Ikosarakt1 (talk) 21:47, January 30, 2013 (UTC)

\(c\) is indeed the pilot. The error is that \(b\) isn't the copilot and thus does not take on the sub-array. FB100Ztalkcontribs 22:47, January 30, 2013 (UTC)

Further extension

How do we extend this? FB100Ztalkcontribs 16:50, April 4, 2013 (UTC)

As far as I know, Bowers finished extending his array after defining meameamealokkapoowa oompa, i.e. on \(\lbrace LLL...LLL,n \rbrace_{b,p}\). What about using \(\lbrace X\& L, n \rbrace_{b,p}\), so that X&L collapses to LLL...LLL with p L's when calculating? What about L&L which collapses to b&b&b&...&L? Or was it already defined? LittlePeng9 (talk) 16:59, April 4, 2013 (UTC)

We used legions array of, n/&L = L&L...L&L (no comma, n L's). AarexTiao 16:47, July 12, 2013 (UTC)
Hyp cos uses [&].

Bowers doesn't used this notation, although it follows naturally in BEAF. In my previous comparisons of BEAF with FGH, I used that, but it has been deleted due to incorrect understanding of strength of \(\&\) operator in principle (LVO instead of \(\Gamma_0\)). Ikosarakt1 (talk ^ contribs) 18:27, April 4, 2013 (UTC)

Pentational arrays

Jiawhein, if you don't understand how it works, don't write about it. Ikosarakt1 (talk ^ contribs) 15:36, April 12, 2013 (UTC)

Triakulus is NOT {3, 3, 3, ..., 3, 3, 3} tritri times. That's a mere dupertri, which is {3, 3, 2 (1) 2}. FB100Ztalkcontribs 19:02, April 12, 2013 (UTC)

True, Triakulus is 3&3&3. 18:40, December 30, 2017 (UTC)

L structures

I have two issues with L structures and beyond.

First, look at these comparisons between Bowers' structures and separators:

\(L => / => 1 \&\& X\)

\(XL => (/1) => X \&\& X\)

\((X^2)L => (/2) => X^2 \&\& X\)

\((X^3)L => (/3) => X^3 \&\& X\)

\((X^X)L => (/0,1) => X^X \&\& X\)

Therefore, \(A \times L => A \&\& X\). When A changed to L, we get \(L^2 = \{L,2\} => \{n,n / 2\} \&\& X\), not \(X \&\& X \&\& X \cdots X \&\& X \&\& X\) (X X's) as expected. We need to take \(L^2 = L \times L \times L \cdots L \times L \times L\) (X L's) to get to expected structure.

Second, I'm in doubts what is the limit of sequence \(\{L,a\}_{b,c}, \{LL,a\}_{b,c}, \{LLL,a\}_{b,c}, \{LLLL,a\}_{b,c}\), etc. It can be both \(\{(1)X,a\}_{b,c}\) in reason that we have X L's or \(\{(1)L,a\}_{b,c}\), by analogy with \(\{b,p (1)/ 2\} = \{b,p ///\cdots /// 2\}\) (with p /'s).

Has anyone thoughts about it? Ikosarakt1 (talk ^ contribs) 15:59, April 23, 2013 (UTC)

The && operator isn't defined formally on Bower's page. We can only see some examples: 3&&3={A/A/A}, where A=3&3&3; 33&&3={A/A/A(/1)A/A/A(/1)A/A/A(/2)A/A/A(/1)A/A/A(/1)A/A/A(/2)A/A/A(/1)A/A/A(/1)A/A/A}, where A=3&3&3. Now think of this: what's 32&&5 ? Certainly, it must be something like {A/A/A(/1)A/A/A(/1)A/A/A}, but what's the A here? Is it 5&5&5, 5&5&5&5&5, or 5&5&5&5&5&5&5&5&5 ? {hyp<hyp··cos>cos} (talk) 15:55, July 13, 2013 (UTC)


As far as I know, BEAF is a function that has 3 informal rules and was defined (although implicitly) for all structures, not only for ones which Bowers gave. The limit ordinal \(\theta((\Omega^\Omega)\omega)\) is only for BEAN (notations like legions, lugions, lagions, ligions, L-arrays, etc...), but BEAF can actually work above it. Ikosarakt1 (talk ^ contribs) 16:16, June 10, 2013 (UTC)

I guess you are right. We can define H-space as structure defined just to be above all L-spaces, just like legion is above all spaces using &. With suffiscently complex arrays we can go wherever we want, like in ordinal BEAF. LittlePeng9 (talk) 18:15, June 10, 2013 (UTC)

Prime blocks

Is there exist an algorithm for computing prime blocks? I think for now that BEAF is uncomputable (there are many ways to define them, some of them can lead to ill-definition). Ikosarakt1 (talk ^ contribs) 17:29, June 12, 2013 (UTC)

Bowers have not gave one. This is main problem why BEAF is often considered ill-defined. Even for triakulus, one of simpliest pentational arrays, we have 3^^^3&3, and we can only speculate how this evaluates. Nuff said about something like {3,3,3,3,3}&3&3. LittlePeng9 (talk) 17:35, June 12, 2013 (UTC)

I see that Bowers sometimes doesn't use X structures, plugging numbers instead of X's. Once he said that Kungulus is X^^^100 & 10, but it another page it is defined as 10^^^100 & 10. Also, n^^(n+1) & m can be thought as either just n^n^n...n^n^n (with n+1 n's) & m (in other words, just a tetrational array) or X^^(X+1) & m (structurally another array, we know that X^^(X+1) ~ \(\epsilon_0^{\epsilon_0}). Ikosarakt1 (talk ^ contribs) 18:13, June 12, 2013 (UTC)


What does this LL(1)LL-attic array {LL(1)LL,LL(1)LL} means? Is it {LL(1)LL,2,2} , or a two-row LL(1)LL-attic array (the first row is "LL(1)LL,LL", and the second row is "LL")? {hyp<hyp··cos>cos} (talk)

Bowers didn't wrote it. I think the most questionable part in BEAN where we get arrays of L-marks. For example, what is the limit of {L,1}n,n, {LL,1}n,n, {LLL,1}n,n, etc...? It can be {L(1)2,1}n,n or {L(1)L,1}n,n as well, depending the way how it works. Ikosarakt1 (talk ^ contribs) 13:25, July 12, 2013 (UTC)
I mean, BEAN is ill-defined at this stage.In BEAN, the second "(1)" of {LL(1)LL,LL(1)LL} has two possible meaning. Bird's array notation is well-defined, because every expression has only one meaning.
By the way, L structure has a limit ordinal ψ(ΩΩΩ)(or LVO)in FGH, and L2 ~ ψ(ΩΩΩ2), Lm ~ ψ(ΩΩΩm), LX ~ ψ(ΩΩΩω), LL ~ ψ(ΩΩΩψ(ΩΩΩ)), and LL2 ~ ψ(ΩΩΩψ(ΩΩΩ2)), LLL ~ ψ(ΩΩΩψ(ΩΩΩψ(ΩΩΩ))), (1)L ~ ψ(ΩΩΩΩ) = θ(ΩΩ+1), (2)L ~ θ(ΩΩ+ω), (0,1)L ~ θ(ΩΩω), and L array of L's (no comma) ~ θ(ΩΩ*2). How can we get these? {hyp<hyp··cos>cos} (talk)
Yes, BEAN is problematic starting from arrays around L's. I can imagine that more intermediate notations must be defined to separate these ones.
I guess your comparisons are based on Chris Bird's ones, but I believe they're wrong somewhere. For example, Bird wrote that {X,X // 2} is structurally \([1 \backslash 2 [1 \neg 1 \neg 2] 2]\). From comparisons below \(\theta(\Omega^\Omega)\) we know that if A is BEAN structure, and its corresponding BAN separator is \([1 [B] 2]\), then A^^X will have separator \([1 \backslash 2 [B] 2\). If we have that:

A = {X,X / 2}

B = \([1 \neg 1 \neg 2]\)

Then {X,X / 2}^^X will have the separator \([1 \backslash 2 [1 \neg 1 \neg 2] 2]\), and {X,X / 2}^^X << {X,2X / 2} << {X,X // 2}. Ikosarakt1 (talk ^ contribs) 16:17, July 12, 2013 (UTC)

I wish Bowers explained his notations in more mathematical context. We have to interpret it ourselves! LittlePeng9 (talk) 16:47, July 12, 2013 (UTC)

Bowers last updated his array notation a few years ago. Probably he doesn't even interested in it currently. By the way, where I can contact him? Ikosarakt1 (talk ^ contribs) 16:58, July 12, 2013 (UTC)
From his site, here is his mail: hedrondude@suddenlink.net LittlePeng9 (talk) 18:44, July 12, 2013 (UTC)

Here I use square-bracket-array instead of no-comma-array to present L-array. Now LL(1)LL becomes [L,L(1)L,L]. If we use this notation, {LL(1)LL,LL(1)LL}b,p will becomes either {[L,L(1)L,L],[L,L(1)L,L]}b,p or {[L,L(1)L,L],[L,L](1)[L,L]}b,p. That would not cause problems above.

Square-bracket-arrays have the same Rule1 and Rule3 as normal arrays, but the Rule2 becomes:

{[L,A+1],1}b,p = b &@ b &@ b &@ ... b &@ b -- p times, where the "&@" represents "[L,A]-attic array of". Here A can be a number, X, X&X&X, L, L*2, L^2, L2, etc.

By the way, the sequence [L], [L,L], [L,[L,L]], [L,[L,[L,L]]], ... has limit [L,X,2], the sequence {[L],1}, {[L,L],1}, {[L,L,L],1}, {[L,L,L,L],1}, ...(also known as {L,1}, {LL,1}, {LLL,1}, {LLLL,1}, ...) has limit {[L,X(1)2],1},which is {LX(1)2,1} in no-comma-array. {hyp<hyp··cos>cos} (talk) 01:52, July 13, 2013 (UTC)

Question as always - how to resolve these types of arrays? I wonder if we'll ever be able to at least define & in well behaved way. LittlePeng9 (talk) 16:26, July 13, 2013 (UTC)


Let we define the rule for weak variant of BEAF as {# a,b,c,...,x,y,z #} = {# a+b+c+...+x+y+z #} (number of entries > 2) instead of iterating over some entry n times:

For example, {3,3,3,4,5,6} = {3,3+3+4+5+6} = {3,21} = 3+21 = 24 (let's admit that {a,b} = a+b for more naturalness.)

{3,3 (1) 3,3} = {3,3 (1) 3+3} = {3,3 (1) 6} = {3,3,3 (1) 5} = {3,3+3 (1) 5} = {3,6 (1) 5}

When would it catch normal BEAF? Ikosarakt1 (talk ^ contribs) 06:26, May 18, 2014 (UTC)

I'm guessing it would catch at X^^X structures. King2218 (talk) 06:37, May 18, 2014 (UTC)

Look to the following comparisons:
{a,b(1)2} = ab
{a,a(1)2} = a^2
{a,a,a(1)2} = {a,2a(1)2} = 2a^2
{a,a(1)3} = a^3
{a,a(1)b} = a^b
{a,a(1)a,a} = a^2a
{a,a(1)(1)2} = a^a^2
{a,a(2)2} = a^^(a+1)
{a,a,a(2)2} = a^^(2a+1)
{a,a(1)(2)2} ~ a^^(a^2)
{a,a(2)(2)2} ~ a^^(a^^a)
{a,a(3)2} ~ a^^^a
{a,a(0,1)2} ~ {a,a,a}
{a,2a(0,1)2} ~ {a,a,2a}
{a,a(1)(0,1)2} ~ {a,a,a^2}
{a,a(1,1)2} ~ {a,a,1,2}
{a,2a(1,1)2} ~ {a,2a,1,2}
{a,a(0,2)2} ~ {a,a,2,2}
{a,a(0,0,1)2} ~ {a,a,a,2}
{a,a(1,0,1)2} ~ {a,a,1,3}
{a,a(0,1,1)2} ~ {a,a,a,3}
{a,a(0,0,2)2} ~ {a,a,a,a}
{a,a(1,0,2)2} ~ {a,a,1,1,2}
{a,a((1)1)2} ~ {a,a(1)2}
So, yes, it looks like that. Wythagoras (talk) 07:44, May 18, 2014 (UTC)
Well, I thought it would catch at {X,X / 2} & n because if normal BEAF works like in FGH, then weak BEAF works like in SGH. Ikosarakt1 (talk ^ contribs) 08:10, May 18, 2014 (UTC)
That was actually my idea. But it's ok -- A Large Number Googologist -- 01:16, October 21, 2014 (UTC)
Oh and also, i still think Weak BEAF catches up at X^^X & n -- A Large Number Googologist -- 01:18, October 21, 2014 (UTC)
Why {a,b}=a+b for more naturalness? Because then {a}=>{a,1}=a+1. Why not \(\{a,b\}=a^b\)? 14:57, December 29, 2017 (UTC)

Legion array tier 2?

If I know {a,b/c}= {a&a...a&a/c-1} with b a's, then what is {a/b}

Bubby3 (talk) 20:10, June 25, 2014 (UTC)

{a / b} = {a,1 / b} = {a,1 #} = a. Ikosarakt1 (talk ^ contribs) 20:19, June 25, 2014 (UTC)

Isn't {L,1}a,b = a◆b = b&b&b&...&b (a times)? I'm talkig to the most recent talker.Antares.I.G.Harrison (talk) 12:00, February 7, 2015 (UTC)

:Actually, {L,1}a,b = {a,b/2} = a&a&a...a&a (b times). By the way, where did you get the ◆ symbol? Wythagoras (talk) 12:41, February 7, 2015 (UTC)

He probably got it here: http://googology.wikia.com/wiki/Introduction_to_BEAF Fluoroantimonic Acid (talk) 09:34, August 2, 2015 (UTC)


Could I/we remove all relations BEAF (ill-defined part) / FGH / other notations? e.g. on the Big Hoss page the approximation f_ψ(ψ_Ι(0))(100) makes me struggle xP same for the "Limit of BEAF is f_LVO(n)" on the list of googological functions (multiple interpretations => we also could put, say, f_ψ(ψ_α->Ι_α(0))(n) or some other very large ordinal, it would surely match with some interpretation of BEAF (again, I'm only talking of the ill-defined part) Fluoroantimonic Acid (talk) 08:34, July 31, 2015 (UTC)

Hmm... someone? Fluoroantimonic Acid (talk) 09:30, August 2, 2015 (UTC)

Growth rate

@cf: why keeping the growth rate thing? in some interpretations BEAF does not even reach Γ(1), while in some other it gets beyond ordinals like the TFB! Fluoroantimonic Acid (talk) 15:08, August 9, 2015 (UTC)


Does anyone know how to solve {3,3(((0,1),1),1)2}? I mean, is there a clear ruleset? As far as I know there isn't, he only properly defined {a,b(c)d}. I have no idea from where does this idea of "tetrational arrays are well-defined" but I am sure its not, Bowers never defined it Fluoroantimonic Acid (talk) 15:58, October 14, 2015 (UTC)

Under the most technical definition tetrational arrays wouldn't be "well-defined". However, there's an agreed-upon definition of those arrays based on the Bird's Array Notation equivalent of tetrational arrays; I believe that they are practically identical except that in BAN the separators have each argument increased by 1. Cookiefonster (talk) 16:55, October 14, 2015 (UTC)
"there's an agreed-upon definition of those arrays..." What is this definition? Where can we find it? LittlePeng9 (talk) 17:10, October 14, 2015 (UTC)
No, just no, you cannot take """agreed-upons""" as definitions, it is NOT Bower's work! it is not BEAF, no matter what you can say! BEAF is what Bowers _defined_, i.e. just dimensional {a,b(c)d,...} arrays, everything else is just ideas. But ideas are not mathematics, just sequels to something that would be defined. Tetrational arrays and so forth are not "mathematics", from a mathematical point of view (not a strict point of view, just mathematical, i.e. normal) it does not properly exist Fluoroantimonic Acid (talk) 15:32, October 15, 2015 (UTC)
Tetrational arrays ARE well-defined as \(^\omega\omega=\varepsilon_0\), and so are pentational, hexational,..., expandal arrays. Probably even beyond. 18:45, December 30, 2017 (UTC)

Mathematics 1001

Has anyone read this book? Will mention this is because Wikipedia shortly before the emergence of a Bowers' operators article, which is the only source of reference. Khankao1 (talk) 16:49, October 15, 2015 (UTC)

There's a used copy selling for $5 on amazon -- vel! 01:10, October 16, 2015 (UTC)
Thanks. Khankao1 (talk) 16:43, October 16, 2015 (UTC)


I just made a Google search and I found out where (apparently) term "BEAN" was coined: link. However, it seems that the author uses name BEAN to denote Extended Array Notation (not whole BEAF), and throughout the page makes clear distinction between BEAN and BEAF.

Just thought you might find it interesting. LittlePeng9 (talk) 19:06, November 2, 2015 (UTC)

Huh. I always wondered why some people say BEAN instead of BEAF. Cookiefonster (talk) 22:48, November 2, 2015 (UTC)

Rename to "Bowers' Exploding Array Function"

BEAF as an acronym is more common than the full name, but it seems to be wiki tradition to prefer full titles to abbreviated ones: Extensible-E System, Busy beaver function, Subcubic graph number, Peano arithmetic, Zermelo-Fraenkel set theory. Discuss please -- vel! 23:24, November 2, 2015 (UTC)

Yeah I think we should. It also makes it a bit more clear what exactly this BEAF thing mentioned on every article is, it confused me when I always saw "BEAF BEAF BEAF" in my early lurking days of the wiki. Cookiefonster (talk) 23:40, November 2, 2015 (UTC)

👍 Fluoroantimonic Acid likes this

Comperison to FGH

I know that tetrational arrays (the most powerful ones that Bowers has actually properly well defined) have a growth rate of about fε₀(n),but what about legions,lugions,ligions,lagions...whatever. Can someone tell me what the growth rate of beyond tetrational arrays is comparable to? —Preceding unsigned comment added by Boboris02 (talkcontribs) 16:24, October 26, 2016 (UTC)

Pentational have f2φ0 growth rate and are well-defined. 15:00, December 29, 2017 (UTC)


Why are first "," separators, then after go dimensional ones? Just why? 19:19, November 29, 2017 (UTC)


What exactly causes BEAF to become ill defined at a higher size?

Testitemqlstudop (talk) 06:56, October 21, 2019 (UTC)

The reason is explicitly written in the article. It is simply because the creator Bowers could not define it. Bowers just explained how it was supposed to work. Several others gave alternative formulations of pentation levels separatedly, and several others "analysed" BEAF of higher levels without definitions under the assumptions that BEAF worked as Bowers desired.
p-adic 09:10, October 21, 2019 (UTC)
It lacks a rigorous definition?
Testitemqlstudop (talk) 00:16, October 22, 2019 (UTC)
Right. For example, you personally have never found it, have not you?
p-adic 00:25, October 22, 2019 (UTC)

New Function which based on BEAF

I have created A new Function, Vista Function


Formalization of BEAF up to X^^^X using formal strings

It seems to me that I finished formalizing BEAF up to X^^^X in similar way with Pound-Star notation (something like "delete this part of expression and append n copies to the end". I almost feel need to edit main page myself, but I need verifying check.


You should not add your original work without external source to the main space. Also, you should try to check it hard by yourself, as you always ask others to check the definition before you seriously check it by yourself. Please remember how many times I checked for you, and how many times you made mistakes. This wiki is not a place for personal volunteers for you, and I recommend you to seriously check it by yourself. (Not in a day. Since your check in days was not so effective, please try to check the well-definedness and the compatibility more than a month.)
p-adic 23:48, 29 December 2020 (UTC)
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