Re: Should facts() show duplicate facts?

Re CAD, double exponentials, etc.:


If I were to work on a PhD in *symbolic* algebra today, I'd work on utilizing the massive capabilities of modern computers -- particularly GPU's -- to evaluate various expressions *numerically* at huge numbers of points in order to gain insight into the symbolic characteristics of those expressions.


E.g., when given a high order polynomial over the complex numbers, we simply evaluate it at millions of points, which will enable us to rather quickly focus in on where the zeros are.


Consider the analogy with archeology.  Instead of digging by hand in tens of places, why not hire an airplane with ground-penetrating radar to map out millions of square miles of territory?


Even better, we can use DFT-like algorithms to quickly evaluate such expressions on entire *circles* of points in the complex plane.


Perhaps someone has already started down this path ?


-----Original Message-----
From: Richard J. FATEMAN via Maxima-discuss <[email protected]>
Sent: Aug 27, 2026 1:40 PM
To: David Scherfgen <[email protected]>
Cc:  <[email protected]>
Subject: Re: [Maxima-discuss] Should facts() show duplicate facts?


I think that addressing the creation and querying of a 'database of algebraic inequalities' may require an implementation of a system we have avoided for decades, but has been the topic of discussion in some other systems. Cylindrical algebraic decomposition would allow a program to determine, given a set of (algebraic) equations defining a region, whether a point is inside that region.  This doesn't address log, exp, etc functions.
The 'assume' database works for very simple cases, but I think 'debugging' it is not going to address serious shortcomings.  
see https://en.wikipedia.org/wiki/Cylindrical_algebraic_decomposition
You might read that CAD 
 is Fundamental to CAS,  which of course makes one wonder how we've done without it.

Read the wikipedia article and some references.
  One issue is that -- whatever it is you want to do with CAD, it comes with a cost, namely the algorithms are generally doubly-exponential in the number of variables.  The interest in it (by G.E.Collins) was sparked by finding an alternative for Tarski's decision procedure.
RJF


On Thu, Aug 27, 2026 at 6:56 AM David Scherfgen via Maxima-discuss <[email protected] (mailto:[email protected])> wrote:

I found this while working on bug #5120, and it surprised me, so I suspect it will surprise others too: The fact database can hold the same fact more than once, and facts() does not tell you.

assume(notequal(a, 0))$
assume(notequal(a*b, 0))$
facts();
  [notequal(a, 0), notequal(b, 0)]


forget(notequal(a, 0))$
facts();
  [notequal(a, 0), notequal(b, 0)]


The fact notequal(a, 0) existed twice, but facts() shows it only once. One must call forget() twice for the fact to actually disappear.


The reason is that some assumptions are filed as several facts. notequal(a*b, 0) is stored as notequal(a, 0) together with notequal(b, 0), because that is the form the database can actually answer questions from. These internal facts are stored even if the database already contained them. On the other hand, trying to establish the same fact twice using the user-level function assume() will answer [redundant] from the second time on.


That second copy of the fact is not a bug, and I want to be clear about that, because my first instinct was to call it a bug. It works as a "count of reasons", and it is what makes this case come out right:


assume(notequal(a*b, 0))$
assume(notequal(a*c, 0))$
forget(notequal(a*b, 0))$
facts();
  [notequal(a, 0), notequal(c, 0)]


notequal(a, 0) correctly survives, because notequal(a*c, 0) still says so. If the two assumptions shared a single entry, that forget would have taken it away.


The one thing that does seem wrong is facts() not showing the duplicate facts, and because it is the only window users have, the whole thing looks inconsistent: forget() says it removed something, but then facts() says it is still there. And forget(facts()) doesn't work as one might reasonably expect.


My proposal is simply that facts() should stop suppressing duplicates. forget(facts()) then becomes exact, and facts() agrees with what forget was already reporting.


Duplicates appear only where there genuinely are two independent reasons for the same statement, which is exactly the situation a user needs to know about.


This doesn't cause any test failures. But it is a user-visible change, so it would want a line in the manual saying a statement can appear more than once when more than one assumption supports it. It changes nothing about what assume means or what the database stores, only about what you are allowed to see.


Is anyone opposed, or relying on facts() returning each fact at most once?


There is a larger question behind this about whether the database should record what the user said or what Maxima knows, and the counting above is an approximation of the former. I would rather leave that for another thread and just stop hiding the count.


Best regards
David Scherfgen

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