Re: [Maxima-discuss] simplify & radical form in Maxima

[Gunter K. <gu...@pe...>]

Common types of equation I normally need to optimize by hand (asking
maxima lateron if ratsimp(expand(new_equation - old_equation) = 0) are:

y^2-2*y+x^2+2*x+2

which is

(y-1)^2+(x+1)^2

factor() wouldn't find that because it can either factor out x, or y -
and even if it could do both at the same time it would need to get the
idea that it can do better if it splits the "+2" in "+1" "+1"... ...but
better is relative: Perhaps I just wanted to add two parables (one in x
and one in y direction) and wanted to move the result upwards by 2 units.

It might sound logical to teach factor() to do this. And this would
simplify

x^2/(x-1)+(2*x)/(x-1)

to

((x+1)^2-1)/(x-1)


But that would mean simplifying x^2+2x to (x+1)^2-1. Which is arguably
mind-boggingly wrong.

Normally the advantage of maths is that you can express a problem in a
way that doesn't require the computer to understand the purpose of the
calculation. For simplification it normally needs to, though.

---

Artificial intelligence might or might not having to do with natural
intelligence as it normally means:
 - We give a computer loads of pairs of input and correct output
 - then we tell a computer to try more or less random algorithms until
one with some probability predicts the right output
 - and then we tell the computer to play around with the algorithm's
coefficients and to look if that makes the results better.

Unfortunately
 - the resulting algorithm will most probably not be very efficient (and
maxima already now often faces the problem that it has to decide if to
use an algorithm that makes the user run out of time / the computer run
out of memory or if it is more reasonable to just use a noun form or an
algorithm that solves a smaller class of problems but does do so with
limited resources.
 - the algorithm is bound to take obviously wrong turns. For example an
algorithm that is trained to discern Stop signs from all other street
signs might in the end only look at the angle of the lower left edge and
will interpret any sign that is dented at the lower left edge as "Stop"
 - and there is a probability that there are cases in which the result
from the AI will be completely erroneous.

For an CAS one could try to limit the AI to only do things that are
guaranteed to not change the result and to provide the AI with hundreds
of variants of factor() that factor x^2+2*x to (x^2-1)^2-1, that do so
only for x but not for y, that factor x^2 to (x-1)^2+2x-1 (sometimes
that is useful, too) or with a more generalized form of playing with the
equation. But that would mean we need to train the AI on millions on
equations that each have a known "best" form. And if we generated the
input for training the AI automatically the AI will be optimized on
exactly these types of equations and maybe neglecting equations that
don't occur too often in the training data.

Normally the advantage of mathematics is that it doesn't require us to
explain the computer what it is for - just to explain what needs to be
calculated. For the purpose of simplification the computer would need to
know what we need the result for.

=> I wouldn't be astonished if we had 20 full-time programmers available
the AI still would need > 10 Years in order to get better than an
average student that leaves the grunt work to a traditional CAS. And it
would need considerably more time to match a traditional CAS in the
hands of a well-trained and creative student. Which on the other hand is
good news: If you study mathematics now the computer won't make you
obsolete not will it do so with your grandchildren. And even basic
mathematics is still fun: The computer helps you but working with the
computer still is more teamwork than being a computer that just needs
you to explain the problems.

Kind regards,

   Gunter.