What should "solving" really mean, some literature.

(This discussion started in the "maxima" mailing list,
but seems to be more than about Maxima..)

Daniel Lemire wrote (to maxima..):

....
> 
> >>The truth, it seems, is that Maxima cannot solve
> >>
> >> exp(a * x) = C * exp(b * x)
> >
> >
> > Ah, here's an example.  Indeed, Maxima doesn't solve it.  Does Maple?
> > Does Mathematica?  What results do they give?

RJF: 
I'm not sure I mentioned it previously, but the commercial Macsyma
solves this without any fuss, providing [x=-log(c)/(b-a)].

Of course if C= -1, then Macsyma provides i*pi/(b-a), but
if I understand Daniel's preferences, he would like it to
return "no solutions exist" in this case.

How hard would it be to change Maxima?  Probably not so
hard. My guess is that what you need is a "heuristic program"
that attempts to collect all occurrences of the unknown on
one side. A program to do this, and other
similar things, called PRESS, was written in Prolog by
some distinguished researchers in Artificial Intelligence
The following paper is a review of Press (by Fateman), perhaps
more of an attack than a review:) 
followed by a defense of PRESS (by Bundy, O'Keefe, Sterling).
followed by a rejoinder by Fateman, in one case.


See 
@Article{Fateman:1988:CSS,
  author =       "Richard Fateman and Alan Bundy and Richard O'Keefe and
                 Leon Sterling",
  title =        "Commentary on: solving symbolic equations with
                 {PRESS}",
  journal =      j-SIGSAM,
  volume =       "22",
  number =       "2",
  pages =        "27--40",
  month =        apr,
  year =         "1988",}

Some comments...

PRESS solves this equation:
cos(x) +cos(3*x) +cos(5*x)=0.
It gets arccos(0)/3 and arccos(-1/2)/2, which are probably simplified to pi/6
and pi/3
Macsyma gets 10 not necessarily distinct solutions to this.
Like 4*pi*(n-1)/2, 4*pi(n+1)/2, pi*(12*n+5)/6  etc..
What would you like here?


PRESS also solves exp(3*x)-4*exp(x)+3*exp(-x)=0, giving
log(sqrt(3), 0, log(-1), log(-sqrt(3)).
Two of these are wrong.
Macsyma gives 4 parameterized solution sets.

I say "Can PRESS combine the merits of searching/matching with
mathematical methods?  Probably by discarding most of PRESS
and adding some additional heuristics to any of the well-known systems: but
generally at the risk of introducing extraneous roots (difficult to
check), or possibly missing roots."


I just noticed that I wrote a paper (10 years ago) after this
paper of 1988, called Notes on Computer Systems for Solving
Symbolic Equations.   I re-ran it through tex and it lives as
a pdf file... http://www.cs.berkeley.edu/~fateman/papers/solving.pdf

It was never published (until now.)
RJF

RJF