User blog comment:DrCeasium/new hyperfactorial array notation/@comment-5529393-20130416153852/@comment-7484840-20130416173829

I believe that you may have misunderstood the definition of my notation. First of all, as you go further into an array (that is, with more rows, planes etc, this does not apply to longer linear arrays), the reduction rule becomes a lot stronger very quickly. This is due to the ath row being evaluated using the (a-1)th stage of the nesting heirarchy (see page for details), and more features of the nesting hierarchy in deeper planes, realms etc.

The other problem is that, as I said on the website, although BEAF and my notation look very similar, at least for linear arrays, the actual way an array evaluates is very different. This is because, for BEAF, when an array is nested, you just work out a value for this and replace the array with this value. With my arrays however, I specifically designed them to work in a similar way to ordinals, where applying the function twice is much more powerful. for example, f_{w^(f_{w^3}(n))}(n) >>  f_{w^3 + 1}(n). But how is my array similar to ordinals?

Unlike BEAF, when nesting arrays in my notation, you do not just calculate a value for them, but use their main entry as if it was the active entry. (see my response to the comment below for an example). This means that in cases like  n![[1,1,2],1,2], the one you mentioned, this would work just like n![1,1,2] (evaluating the main entry of the outer array), until it was completely evaluated, and then, whenever this got down to (n![1])@, this would be replaced by  (n![ 1,1,2] )@, and this entire process would happen agai, not to mention all of the other ![@]'s generated by the 1st evaluation. This comparison should show this (choosing n = 3 for simplicity's sake):

3![[1,1,2],1,2] =  ​3![[1, [1,1,1] ,1],1,2] =  3![[1, 3 ],1,2] =  3![[[1,1], 2 ],1,2] =  3![[3, 2 ],1,2] =  (( 3![[2, 2 ],1,2]) ![[2, 2 ],1,2]) ![[2, 2 ],1,2] = (replacing outer arrays and recursions with @)  ( 3![[2, 2 ],1,2])@ = (( ( 3![[1, 2 ],1,2]) ![[1, 2 ],1,2]) ![[1, 2 ],1,2]) @ = (and again)  ( 3![[1, 2 ],1,2]) @ =  ( 3![[3 ],1,2]) @ = (more recursions from the 3, skipped this step, so @ will have changed)  ( 3![[1 ],1,2]) @ =  ( 3![3 ,1,2]) @ = (more recursions)  ( 3![1 ,1,2]) @ etc.

f_{(w^2).2}(3) =  f_{(w^2)+ (w^2) }(3) =  f_{(w^2)+ (w.3) }(3)  =  f_{(w^2)+ (w.2)+w }(3)  =  f_{(w^2)+ (w.2)+3 }(3) = (recursions (actually 1 more than my array, but this is negliable) @ representing more functions)  @( f_{(w^2)+ (w.2) }(3)) =  @( f_{(w^2)+ (w)+3 }(3)) =  (more recursions from the 3, skipped this step, so @ will have changed)  @( f_{(w^2)+ w }(3)) =  @( f_{(w^2)+3 }(3))  = (more recursions)  @( f_{(w^2) }(3))  etc.

See any similarities? The arrays actually work in a very ordinal-esque way. Sorry if this was a bit long.