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Re: Replacement Rule with Sqrt in denominator
*To*: mathgroup at smc.vnet.net
*Subject*: [mg114474] Re: Replacement Rule with Sqrt in denominator
*From*: Richard Fateman <fateman at cs.berkeley.edu>
*Date*: Sun, 5 Dec 2010 21:52:27 -0500 (EST)
*References*: <idd7nd$nb9$1@smc.vnet.net>
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
"FullSimplify".
>
> 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. :)
RJF
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