Re: Question on rewriting and program specification

[Andrej Bauer <Andrej.Bauer@andrej.com>]

Jacques Carette wrote:
> Of course, if you happen to know exactly
> which semi-decision procedures are being used, you can fool them and get
> the system to either go into an infinite loop or give a wrong answer.
> But that requires a huge amount of knowledge to do so.  The average user
> would be unable to manufacture such examples.

I have a rather unfortunate experience with "average users" who get
wrong answers from CAS's, namely our undergraduate math majors. In their
first-year analysis course they learn how to compute limits. Invariably,
they are given some limits which Mathematica gets wrong, e.g.:

Limit[((1 + 4 x^2)^(1/4) - (1 + 5 x^2)^(1/5))/(a^(-x^2/2) - Cos[x]),
      x -> 0]

The answer is 0, _except_ when the parameter a equals e, in which case
the answer is 6. Yes, this is a nasty limit pulled out of a hat, but it
is precisely the sort of thing we test our students on. It is rather
disappointing that Mathematica falls into exactly the same sort of trap
as the average student.

Another example is the use of l'Hospital rule, which is used by every
CAS. There is a side condition which is not checked by them, which makes
them give wrong answers. (The side condition is very nasty to check,
namely, whether the zero of a derivative is isolated.)

The situation is even worse when engineers and physicsts use CAS. They
trust them blindly (I suspect). One day they're going to build a nuclear
power plant based on a faulty limit computed by Mathematica or Maple.

Andrej Bauer