Hello everyone, this is my first blog entry here. The pressing topic I would like to discuss is whether or not Bowers' Exploding Array Function is a well-defined notation.
I have no problem with Bowers' Extended Array Notation, which has seven clear cut rules for evaluating arrays up to n dimesions. And I think it is fairly clear how to extend the notation to "exponential" arrays; see Bird's "Nested Array Notation", for example, for a set of completely explicit rules for arrays in which the dimensions are themselves arrays, whose dimensions are themselves arrays, etc. nested arbitrarily deep. But when we start bringing in tetration to the arrays (what Bowers for some reason calls "pentational arrays"; it seems to me that "tetrational arrays" would have made more sense.), things become much less clear. What exactly are the "structures" in pentational spaces? Bowers gives examples but doesn't give an explicit definition. For "tetrational spaces", a workable definition is as follows: the set of "entries" is the set containing 0 and closed under + and a -> X^a. (Note that X^0 = 1.) The set of "structures sizes" are of the form X^a where a is an entry. A "structure" is an ordered pair (a, b) where a is a structure size and b is an entry. So I have a couple of possible definitions for entries and structures in pentational spaces:
1) An entry is the set 0 closed under +, a -> X^a and a -> X^^a. A structure size is of the form X^a or X^^a, where a is an entry. A structure is an ordered pair (a, b) where a is a structure size and b is an entry.
2) An entry is the set 0 closed under +, *, a-> X^a and a -> X^^a. A structure size is the product of elements of the form X^a or X^^a, where a is an entry. A structure is an ordered pair (a, b) where a is a structure size and b is an entry.
The difference is the second version is closed under multiplication. So the second version has more structures, so would lead to larger numbers, but also has more potential complications. One of the potential pitfalls: do all the structures "fit" nicely within one another? It seems that a requirement for the array notation to work is that the structures form an increasing sequence such that each structure wholly contains the previous structures. This may be true of pentational space, but it's not clear to me.
Another problem is how to define "previous structures", a requirement for the "catastrophic rule" in BEAF. It's possible, although complicated, to define this for exponential arrays (what Bowers calls tetrational arrays); this is basically what Bird does in his "Nested Array Notation". But defining previous structures for pentational arrays seems considerably more complicated, and appears to get more complicated has you go to higher and higher Knuth operators.
Despite this difficulty, Bowers goes on to introduce the "array of" operator &. This appears to take an arbitrary array notation, including those defined using &, and makes a "space" out of that notation to define a much larger notation. How this is achieved is not even touched upon. What are the entries, structures, and previous structures of "3 & 3 & 3" space? Is this definable from knowing "3 & 3" space? If so, how? I won't talk about the higher order notations like L yet, although I have questions about those as well.
I'm not saying that a sensible definition can't be made for all the notations in BEAF. However, as it stands there are probably many such definitions that could be made. In such a case, numbers such as "triakulus" won't correspond to a single number, but a multitude of numbers corresponding to the multitude of ways to complete the definition of BEAF. So to get well-defined numbers, we must come up with an explicit definition for BEAF.
I'm very curious to hear what others have to say on this matter.
One final quibble: it seems that using numbers in place of X leads to ambiguity. For example, what is 3^10 & 3? Is it a X^10 structure, or a X^(3X+1) structure, or a X^(X^2 + 1) structure?