Talk:Bowers Exploding Array Function

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. FB100Z &bull; talk &bull; contribs 22:47, January 30, 2013 (UTC)

Further extension
How do we extend this? FB100Z &bull; talk &bull; contribs 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)

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}. FB100Z &bull; talk &bull; contribs 19:02, April 12, 2013 (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 ? &#123;hyp&#60;hyp··cos&#62;cos&#125; (talk) 15:55, July 13, 2013 (UTC)

BEAN and BEAF
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)

L-structures
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")? &#123;hyp&#60;hyp··cos&#62;cos&#125; (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(ΩΩ Ω )(or LVO)in FGH, and L2 ~ ψ(ΩΩ Ω 2), Lm ~ ψ(ΩΩ Ω m), LX ~ ψ(ΩΩ Ω ω), LL ~ ψ(ΩΩ Ω ψ(ΩΩ Ω )), and LL2 ~ ψ(ΩΩ Ω ψ(ΩΩ Ω 2)), LLL ~ ψ(ΩΩ Ω ψ(ΩΩ Ω ψ(ΩΩ Ω ))), (1)L ~ ψ(ΩΩ Ω Ω) = θ(ΩΩ+1), (2)L ~ θ(ΩΩ+ω), (0,1)L ~ θ(ΩΩ+ωω), and L array of L's (no comma) ~ θ(ΩΩ*2). How can we get these? &#123;hyp&#60;hyp··cos&#62;cos&#125; (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. &#123;hyp&#60;hyp··cos&#62;cos&#125; (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)

Weak BEAF
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} = {3,3,3,3,3,3,3,3,3 (1) 5} = {3,3+3+3+3+3+3+3+3 (1) 5} = {3,24 (1) 5}

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

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