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Re: Replacement Rule with Sqrt in denominator

On 12/4/2010 3:12 AM, Daniel Lichtblau wrote:

>> (RJF wrote) Thus the burden falls on
>> you to look at the data representation underlying the abstraction
>> to make your program work."
> (DanL wrote) I'd change "abstraction" to "representation". At which point I would
> agree, regarding the burden.

No, I think the abstraction is something like  "a/b"  or possibly 
"a^(-n)".  Or "An expression x such that Numerator[x]==a, Denominator[x]==b.

Of course the representation, deep down, is using electrons and stuff.

But one layer down the representation can be Rational[a,b] if a,b, are 
numbers, and gcd(a,b)=1 and a>0.  Or it can be Times[a,Power[b,-1]] or
yet other options.

The fact that Mathematica requires you to deal with this representation 
layer is the problem.
>> It's not so hard to make the system smarter.
> I'll indicate below why I am not convinced what you want equates to
> smarter software.
>> But you pay for what
>> you get.
> Here I agree.
> Actually one of the things I really appreciate about Mathematica, and
> for which, alas, I can take no credit, is that rule replacement works so
> reliably well. When I get frustrated with it failing to do what I intend
> (about once every month or two), there is the inevitable session with
> MatchQ to sort things out.

You see, the abstraction (in your mind) and the representation are 
different.  I too find the pattern issues occasionally frustrating, and
I have the advantage of (reverse engineering) the Mathematica pattern
matcher (version 3.0) or so, in Lisp.  So there were things I had to
figure out so I could program them. Things I would not have ever tried,
and probably no one had ever used.

>> RJF
> I think any nonsyntactic rule replacement will need very clear
> semantics.
Yes, I agree.
The notion "replace it if it shows up visually" is not
> terribly clear. It relies, among other things, on specifics of the
> format settings, vagaries of e.g. StandardForm, and other details.
I agree.  That's why notions like
"coefficient of I"  and "numerator of X" are needed.
> To give an idea of what is meant by the formatting vagaries, consider
> that we could represent 1/Sqrt[x+1] explicitly as (x+1)^(-1/2) (mentally
> format that so it looks pretty, with the exponent raised, no caret,
> etc.) So the substitution semantics being proposed would depend on
> formatting?!

Not really. The notion of sqrt(x) is fairly clear, and it seems to 
include the denominator of 1/sqrt(x).  This is not, I think, rocket science.

  Or should it apply to any power of (x+1) when that power
> visually has a 2 in the denominator? (Does it then apply to
> (x+1)^(a/2+b/4)? Or only to the equivalent (x+1)^(1/2*(a+b/2))?)

Possibly.  It seems to me that this is kind of like "Simplify" vs
> These are the questions that would need to be answered before even
> embarking on this journey.

Note that I have shown that the improved pattern matching can be built
using syntactic pattern matching plus some other programs.  I am not
requiring that the syntactic pattern matching be removed.

> I will add that about the last thing anyone wants is that StandardForm
> formatting be influenced by the desire to avoid any possibility of
> showing "misleading" expression structure. I don't think you are
> proposing otherwise, but it would be a natural outcome to any effort to
> try to curtail this misunderstanding.

I am not proposing altering the display.

> I have not raised the I->3*I issue. I do wonder what should be the
> outcome of Sin[x] /. I->3*I .

Well, you could also claim that   -1/.  I->x   should be  x^2.  This
kind of issue can be thought through by fairly simple means.
If you can divide an expression  H by P  to compute Q,R in H=Q*P+R,  then
replacing P by M results in the expression Q*M+R. Done.
So we are talking about division.   (This does not quite cover all
the cases, but it is close).

  Maybe the trig equivalent of
> (Exp[3*I*x] - Exp[-3*I*x]) / (6*I), which is Sin[3*x]/3? (Maybe we
> should make it case insensitive, and return S3in[x]?)

String substitution and regular expressions and the like are of course
a different matter. :)

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