[Top][All Lists]

[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]

[Axiom-developer] Re: Embedding Axiom (Hickey and fold/unfold) Folding a

From: Tim Daly
Subject: [Axiom-developer] Re: Embedding Axiom (Hickey and fold/unfold) Folding and generalization
Date: Sat, 21 Nov 2009 21:08:40 -0500
User-agent: Thunderbird (Windows/20090302)

Martin Baker wrote:
I think it helps to get a wider perspective.

Part of what prompted my question is when I was thinking about how to implement the exterior product I thought about possible options, for instance rules:

<e1, e2…en | ei/\ei=0, ei/\ej= -ej/\ei>

or an algorithm:

And(b1::SINT,b2::SINT) ~= 0 => z
c := c1 * c2
bz := Or(b1::SINT,b2::SINT)
for i in 0..n-1 | bit?(b1,i) repeat
  k := 0
  for j in i+1..n-1 | bit?(b1, j) repeat k := k+1
  for j in 0..i-1   | bit?(bz, j) repeat k := k+1
  if odd? k then c := -c := + c

or a multiplication table:

0      e1^e2
-e1^e2 0

Out of these options, the rules just seems to be operating at a higher level? In that it would seem relatively easy to translate from the rules to the algorithm but a lot more difficult to go from the algorithm to the rules (could Concordia do it?). Also, unlike the algorithm, there is no need to introduce the concept of time. I wonder which type would be the best for making a test for equivalence?

I take your point that the rules are not necessarily better for parallel processing and they don't scale up very well, I wonder if there would be some way to get the best of all these approaches? lots of small rule bases or something like that?

Another point is that there are no 'side effects' to this function but we are trying to add one in by caching the multiplication table!

I don't have any answers but its interesting to speculate,

I like the declarative form a lot, which would fit rules rather well. I think you understand
my feelings on rules being non-algebraic, though, and I'd advise against it.

The table form could get very large (unless you only cache computed entries). The table form would also require some matching since "e1" could be something
more complex than a symbol.

I think that the procedural form gives you the most flexible version since
(a) it fits into the spad coding scheme and (b) allows you to manipulate the
internal representations in efficient ways that can "violate causality" within the function without being visible to the user. It also gives you the opportunity to coerce to more specific forms (like linear algebra forms) in special cases.

In addition, the coding form will force you to be explicit about the categories that need to be defined. I think one of the key strengths of Axiom is the organization of the system into categories. If you started with Operads (per William Sit's comments) as categories and derived your Grassmann/Clifford/Octonion forms in a disciplined way then everyone wins. I will admit that this is VERY hard. I tried to do this for the Infinite Group Theory area and failed. (Hardly a surprise since I'm not an
Infinite Group Theorist by any stretch of the imagination.)

William may have some useful insights here as he is a mathematician and an
Axiom algebra author.


reply via email to

[Prev in Thread] Current Thread [Next in Thread]