On Feb 11, 11:35 pm, boot...@u.washington.edu wrote:
> > Are you aware of the results of Daniel Richardson on the recursive
> > undecidability of
> > (rather simple) identities? He proved that in general there is no
> > algorithm possible.
>
> Yeah, and the halting problem is undecidable too, but you would still call
> the following program "stupid":
>
> while 1: continue
Actually, a program like that is very useful. Stupidity is usually
considered a characteristic of humans, so
as a first cut, I would say that no program is "stupid".
>
> There are a huge number of intractable problems. Determining whether sin^2 x
> + cos^2 x = 1 is not one of them.
No, it is not intractable. But the tools you use to apply such
identities or simplifications are not usually efficient,
since to be effective there is a combinatorial search to see which, if
any, of the identities can be effectively applied.
To say that one should search for, in particular, sin^2+cos^2 is to
assume that your audience consists largely
of high school students taking trigonometry, or calculus students
doing simple problems for homework.
While this may be (statistically speaking) probable for Sage, it is a
bad design assumption for someone writing
a program that is supposed to do serious applied mathematics, as an
assistant to someone who presumably
has some knowledge of the tool, and for example knows of the existence
of programs like "trigsimp" or "ratexpand"
or "factor" ...
>
> > Especially given that as background,
>
> > How much work do you think Maxima should do to try to determine for
> > arbitrary f, if f(x)>0 or not?
>
> Enough that it doesn't ask the user obviously stupid questions.
OK, here's your homework. Write a program to determine if a question
is "obviously stupid".
When you are finished, ask for a PhD.
>
> > Now Maxima does know, in many different contexts, that sin^2+cos^2 can
> > be simplified to 1
> > But looking for all such relationships that it is aware of (and there
> > are many such relationships),
> > at every decision point, is time consuming.
>
> Yes, but I'd bet that Maxima could answer 99% of such questions faster than I
> could find a piece of paper.
I don't know you, but I keep a pad of paper handy. Also a pencil.
> It's great that Maxima is "aware" of many relationships. It'd be awesome if
>it would effectively wield them.
Again, this could be made the core of a homework assignment, but could
lead to a PhD.
>
> > Mathematica has a function which tries searching for smaller
> > equivalent expressions.
> > Simplify, or maybe FullSimplify.
>
> > The difficulty is that the program tends to take too much time for any
> > but rather small expressions
> > to start with.
>
> Ah! So the problem is that Maxima is slow.
No, there is no such program in Maxima. Perhaps you meant to say
"the problem is that Mathematica is slow"?
Even so, your conclusion would be wrong. Read up on "combinatorial
explosion".
I complain about that a lot, actually.
Well, maybe are using the slow version of Maxima that is attached to
Sage, instead
of one of the fast versions?
>
> > Such a program could presumably be written in Sage,
> > where each subexpression
> > is repeatedly submitted to 15 or 20 or 30 different "simplifier-like"
> > programs to see which
> > equivalent expression is smaller. This is not a great idea.
>
> No, sending 20-30 requests to Maxima is not a great idea. Fewer is better,
> usually.
Perhaps what you are saying is "the problem is that Sage is too slow"?
>
> > But if you really really want to make sure that Maxima always knows
> > that sin^2+cos^2=1, you can consider
> > wrapping trigsimp() around every expression that you send to it.
>
> > By the way, Macsyma is not 30 years old. The first paper describing
> > it dates back to 1967.
> > So it is 42 years old or more.
>
> Neat! In a few years, it'll be an antique! I used to own a 1969 Ford 150.
> Nothing like old iron -- but the damned thing burned a gallon of gasoline in
> under 6 miles on a straight, flat road in a tailwind! Out with the old, in
> with the new -- my '05 pickup gets 25 miles to the gallon in hilly city
> conditions.
>
> > Richardson's results date to about 1968.
>
> Pythagoras's result dates back to about 500BC. Have you heard of it?
>
>
>
> > RJF
>
> > On Feb 11, 11:11 am, mabshoff <mabsh...@googlemail.com> wrote:
> >> On Feb 11, 9:45 am, kcrisman <kcris...@gmail.com> wrote:
>
> >> Hi,
>
> >>> There are of course several trac tickets related to this, so this is
> >>> not a bug report (for Sage or for Maxima), but I had to laugh when
> >>> this came up today in preparing for class - enjoy!
>
> >> Well, it would be truly funny if we didn't use Maxima for symbolics,
> >> but this is a sad, sad bug for a 30 year old system.
>
> >>> TypeError: Computation failed since Maxima requested additional
> >>> constraints (try the command 'assume(sin(t)^2+cos(t)^2>0)' before
> >>> integral or limit evaluation, for example):
> >>> Is sin(t)^2+cos(t)^2 positive or zero?
>
> >>> - kcrisman
>
> >> Cheers,
>
> >> Michael
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