Re: EML: All elementary functions from a single operator
Brent Meeker <[email protected]>
| Newsgroups | gmane.comp.mathematics.maxima.general |
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| Message-ID | <[email protected]> |
I think it's interesting that such a compression of functions is possible. It's rather like Turing completeness; of no practical use but good to know. Brent On 4/15/2026 6:55 PM, Stavros Macrakis wrote: > On Wed, Apr 15, 2026 at 6:33 PM Richard Fateman <[email protected]> wrote: > > Seems to me to be uninteresting for any practical application > > > Not just practical applications ... I don't see how it's of > theoretical interest either. > > Of more economical use would be to allow > integers, i, > exp(x) .. which is, I suppose eml(x,1) > log(y) ... eml(-infinity, y) > and rational operations +,*,-, / . maybe abs(). maybe some > single-valued version of sqrt(). > > > Well, yes, but then you're back to standard operators. His claim is > that somehow squeezing everything into one function is a good idea. > > I can't see how. > > -s > > RJF > > On Wed, Apr 15, 2026 at 3:17 PM Stavros Macrakis > <[email protected]> wrote: > > The representation is as far as I can tell completely useless. > > I played around a bit with generating eml expressions. One > thing that's immediately clear is that the vast majority of > expression trees produce uninteresting functions. > > I don't find it surprising that you can represent all the > continuous scientific-calculator functions with > compositions of /eml/. After all, x^y / trig / arctrig / > hyperbolic all have well-known definitions in terms of exp and > log. > > What /is/ a bit shocking is how hard it is to write, say, the > number 2. The paper reports that it requires 19 instances of > /eml/, probably as "+"(1,1), since it takes 19 instances to > write "+". And /i/=sqrt(-1) requires >55 instances, so those > trig functions are going to be enormous. > > The /eml /form of standard functions quickly gets huge. > Conversely, most of the functions you can write with it are > utterly uninteresting. You ask whether Maxima can simplify > various arcane expressions back to their elementary forms. > Well, let's look at some examples. There are 2,090,918 trees > of depth 4 (or less). Most of them involve nested > exponentials, > like %e-log(%e^%e^x/x-log(%e^%e-log(%e-log(x)))), and I don > > As for "symbolic regression", that makes no sense. Most of the > search space consists of useless functions like > -(log(%e^%e^x-x*log(%e^%e-log(%e-log(x))))-log(x)-%e).If you > allow more constants, e.g., 0 and /i/, the trees get less > deep, but more broad. Might as well have a search space where > the density of "interesting" functions is large. > > I don't find this elegant at all. > > In fact, I wonder if it is elaborate April Fool's joke > published 5 days late. > > -s > > > On Wed, Apr 15, 2026 at 3:19 PM Barton Willis > <[email protected]> wrote: > > I kept scanning the paper for a table showing how to > express each elementary function in terms of /eml/. I > never found one. After that, my interest evaporated. It’s > cute, I suppose; useful, I'm not convinced. > The “phylogenetic” tree of the elementary functions is a > typesetting tour de force, but isn’t the English language > already sufficiently weirded? > It might be amusing to see whether Maxima can simplify > various arcane expressions in /eml/ back to their > elementary forms. And what about expressing the principal > branches? That looks like a nightmare. > > > --Barton > ------------------------------------------------------------------------ > *From:* Stavros Macrakis <[email protected]> > *Sent:* Wednesday, April 15, 2026 1:17 PM > *To:* <[email protected]> > <[email protected]> > *Subject:* [Maxima-discuss] EML: All elementary functions > from a single operator > > > Caution: Non-NU Email > > > The physicist Andrzej Odrzywołek (/adiunkt/ at > Jagiellonian University), announced > <https://urldefense.com/v3/__https://arxiv.org/html/2603.21852v2__;!!PvXuogZ4sRB2p-tU!CA_SQ_a-ptqXUR52sYmTFsZPoqM1rv8G8RRMpX7n0iMb36Rdj7JVWac5EfgBxFPePSiC6rswnszt5fE$>: > > a single binary operator, eml(x,y)=exp(x)-ln(y), > together with the constant 1, generates the standard > repertoire of a scientific calculator. > > and compared /eml/ to the Sheffer stroke (/nand/) as a > universal operator, in fact he calls it /EML Sheffer/. The > intermediate results are in general complex. > > This has gotten some attention in the press, e.g., in /The > Register > <https://urldefense.com/v3/__https://www.theregister.com/2026/04/14/two_button_calculator/__;!!PvXuogZ4sRB2p-tU!CA_SQ_a-ptqXUR52sYmTFsZPoqM1rv8G8RRMpX7n0iMb36Rdj7JVWac5EfgBxFPePSiC6rsw8-BevUA$>/. > > > Let's assume that his central claim is true -- it doesn't > seem implausible, although I wonder whether his use of the > principal branch runs into problems. > > Still, as far as I can tell, this has little interest for > mathematics or symbolic calculation. He claims two > applications for it: > > 1. analog computing"pure-EML form could possibly be > implemented efficiently in FPGA or analog circuits." > 2. discovering closed-form expressions from data > > For analog computing, the combination of *exp *and *log > *in the formulas means that intermediate results may have > a large range of magnitudes, and accuracy will be hard. > EML requires complex arithmetic; he has no concrete > proposal on how to implement that in analog circuits -- > perhaps as magnitude/phase in an AC circuit? And even > something as simple as *x*y* is a 12-node network. > > For discovering closed-form expressions from data points > ("symbolic regression"), he proposes using gradient-based > optimizers (like Adam) to train trees to recover > closed-form expressions from numerical data, and gives > some examples. I am skeptical, because (a) it's not clear > to me that having one operator and deep trees is better > than having many operators and shallower trees; (b) > combining *exp* and *log *is going to create expressions > that have large dynamic range and unstable behaviorwhich > I'd think would be unsuitable for gradient-based optimization. > > He also claims > > all the above difficulties (edge cases) are not much > different from those usually encountered in every kind > of floating-point or symbolic computation. > > > Comments? > > Does anyone see any value in this? > > -s > > All elementary functions from a single operator > Andrzej Odrzywołek > Institute of Theoretical Physics, Jagiellonian University, > 30-348 Krakow, Poland > E-mail: [email protected] > > https://arxiv.org/html/2603.21852v2#S0.SSx1.p1.7 > <https://urldefense.com/v3/__https://arxiv.org/html/2603.21852v2*S0.SSx1.p1.7__;Iw!!PvXuogZ4sRB2p-tU!CA_SQ_a-ptqXUR52sYmTFsZPoqM1rv8G8RRMpX7n0iMb36Rdj7JVWac5EfgBxFPePSiC6rsw0XFuLag$> > > > > _______________________________________________ > Maxima-discuss mailing list > [email protected] > https://lists.sourceforge.net/lists/listinfo/maxima-discuss > > > > _______________________________________________ > Maxima-discuss mailing list > [email protected] > https://lists.sourceforge.net/lists/listinfo/maxima-discuss _______________________________________________ Maxima-discuss mailing list [email protected] https://lists.sourceforge.net/lists/listinfo/maxima-discuss