The problem chosen seems to be
integrate((16*x^3-42*x^2+2*x)/sqrt(-16*x^8+112*x^7-204*x^6+28*x^5-x^4+1),x)
The integrand is of the form dy/sqrt(1-y^2)
and so the integral is arcsin(y).
An algorithm to look for this kind of pattern (in general) would not be
difficult to write, although it does assume you can compute the integral
of any polynomial (here,let y= the integral of the numerator of the
integrand above),
and also that you can compute 1-y^2 and compare it to the radicand in the
denominator.
I find it somewhat unlikely that the machine learning mechanism in that
paper
would come up with this generalization.
Even given 80 million specific examples of this.
e.g. for i from 1 to 80000000 do {h=arcsin( random_expression[i]),
g=diff(h,x), remember
the integral of g is h}
would it integrate other technically similar examples?
One could propose a mechanical artificial
intelligence that, given sufficient knowledge of axioms, logical methods,
etc, would derive all the theorems of mathematics past, present, and
future. This would presumably include any theorems of Robert Risch
relevant to symbolic indefinite integration in finite terms.
Hilbert thought about this.
It may not be possible -- Gödel's incompleteness results
are probably relevant. If we replaced the phrase "all the theorems"
in the paragraph above with the phrase "all the useful theorems",
or "all the necessary theorems to do symbolic indefinite integration" there
could be a debate of some sort. I don't know that Sage has incorporated
theorem provers, but there are research programs e.g. Coq.
It may be useful to see John McCarthy's paper from 1996
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.79.8613&rep=rep1&type=pdf
But this is wandering far afield from the presumed topic of this thread.
RJF
On Wednesday, December 25, 2019 at 7:54:47 PM UTC-8, Brent W. Baccala wrote:
>
>
> I know the Risch algorithm fairly well.
>
> I made two screencast videos describing how to use Axiom or Sage to
> simplify one of the integrals used in the Facebook paper.
>
> Quick summary - Axiom works quite well. Sage can't do it in one step, but
> the new function field features in Sage 9 allow the integral to computed,
> but you have to know something about the Risch algorithm to step through it.
>
> The second video is a whirlwind overview of the Risch theorem and shows
> how to do a Risch calculation (on the Facebook integral) using Sage.
>
> You can view the videos here:
>
> https://www.freesoft.org/blogs/soapbox/the-facebook-integral/
>
>
>
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