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All elementary functions from a single binary operator

arxiv.org

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Re: All elementary functions from a single binary operator

#91

Post by DoctorOetker »

canjobear wrote: post #44
DoctorOetker wrote: post #30

EDIT: please change the article link to the most recent version (as of now still v2), it is currently pointing to the v1 version which misses the figures. I'm still reading this, but if this checks out, this is one of the most significant discoveries in years. Why use splines or polynomials or haphazardly chosen basis functions if you can just fit (gradient descent) your data or wave functions to the proper computati…

> Why use splines or polynomials or haphazardly chosen basis functions if you can just fit (gradient descent) your data or wave functions to the proper computational EML tree? Because the EML basis makes simple functions (like +) hard to express. Not to diminish this very cool discovery!

consider how a wavefunction might be stored for numerically searching the ground state of a molecule, or a crystal lattice, humans appreciate 2D imagery that is about 1 kilopixel x 1 kilopixel; so expanding this to 3 dimensions that means the wavefunction would have to store 10 ^ 9 complex numbers (easily 4 GB at 16 bit precision for real and imaginary compnents so 4 bytes per complex number), do we really believe that a DAG variant of the EML construction would consume a bigger value to represent the analytically correct solution? do we really believe that the 4GB DAG variant of EML would produce a less accurate representation (i.e. less fidelity with the schroedinger equation?) If the ground state hydrogen atom is any indication, my money is on EML-style constructions, not naive 3D arrays modelling the wavefunction by brute force.

This also re-opens a lot of "party pooper" results in mathematics: impossibility of representing solutions to general quintic (fine print: if we restrict ourselves to arithmetic and roots/radicals). In mathematics and physics there have been a lot of "party pooper" results which later found more profound and interesting positive results by properly rephrasing the question. A negative result for a myopic question isn't very informative on its own.

Re: All elementary functions from a single binary operator

#92

Post by fc417fc802 »

dang wrote: post #65
DoctorOetker wrote: post #30

EDIT: please change the article link to the most recent version (as of now still v2), it is currently pointing to the v1 version which misses the figures. I'm still reading this, but if this checks out, this is one of the most significant discoveries in years. Why use splines or polynomials or haphazardly chosen basis functions if you can just fit (gradient descent) your data or wave functions to the proper computati…

What URL should we change it to?

IMO arxiv links should pretty much always be to the abstract (ie .../abs/...) as opposed to particular versions of the html or pdf.

Re: All elementary functions from a single binary operator

#93

Post by fc417fc802 »

eugene3306 wrote: post #61

This makes a good benchmark LLMs: ``` look at this paper: https://arxiv.org/pdf/2603.21852 now please produce 2x+y as a composition on EMLs ``` Opus(paid) - claimed that "2" is circular. Once I told it that ChatGPT have already done this, finished successfully. ChatGPT(free) - did it from the first try. Grok - produced estimation of the depth of the formula. Gemini - success Deepseek - Assumed some pre-existing knowl…

> Once I told it that ChatGPT have already done this, finished successfully.

TIL you can taunt LLMs. I guess they exhibit more competitive spirit than I thought.

Re: All elementary functions from a single binary operator

#95

Post by meindnoch »

gopalv wrote: post #58
ikrima wrote: post #46

Earlier quoted context omitted.

From my experience of working in this problem domain for the last year, I'd say it is pretty powerful but the "too good to be true part" comes from that EML buys elegance through exponential expression blow-up. Multiplication alone requires depth-8 trees with 41+ leaves i.e. minimal operator vocabulary trades off against expression length. There's likely an information-theoretic sweet spot between these extremes. It'…

> Multiplication alone requires depth-8 trees with 41+ leaves i.e. minimal operator vocabulary trades off against expression length. That is sort of comparable to how NAND simplify scaling. Division is hell on gates. The single component was the reason scaling went like it did. There was only one gate structure which had to improve to make chips smaller - if a chip used 3 different kinds, then the scaling would've re…

Are you under the impression that CPUs are made exclusively from NAND gates? You can't be serious.

Re: All elementary functions from a single binary operator

#96

Post by peyton »

meindnoch wrote: post #95
gopalv wrote: post #58

Earlier quoted context omitted.

> Multiplication alone requires depth-8 trees with 41+ leaves i.e. minimal operator vocabulary trades off against expression length. That is sort of comparable to how NAND simplify scaling. Division is hell on gates. The single component was the reason scaling went like it did. There was only one gate structure which had to improve to make chips smaller - if a chip used 3 different kinds, then the scaling would've re…

Are you under the impression that CPUs are made exclusively from NAND gates? You can't be serious.

Might’ve gotten mixed up with CMOS dominance, or I’m ignorant.

Re: All elementary functions from a single binary operator

#97

Post by freehorse »

hyperhello wrote: post #37
thaumasiotes wrote: post #25

Earlier quoted context omitted.

> isn't the operation its own inverse depending on the parameter? This is a function from ℝ² to ℝ. It can't be its own inverse; what would that mean?

eml(1,eml(x,1)) = eml(eml(1,x),1) = exp(ln(x)) = ln(exp(x)) = x

eml(1,eml(x,1)) = e + x

and

eml(eml(1,x),1) = e^e * x

Re: All elementary functions from a single binary operator

#98

Post by adrian_b »

samus wrote: post #87
Nevermark wrote: post #50

Earlier quoted context omitted.

They are done with transistors though. Transistors form an efficient, single element, universal digital basis. And are a much less arbitrary choice than NAND, vs. NOR, XOR, etc. Using transistors as conceptual digital logic primitives, where power dissipation isn't a thing, Pass Logic is "The Way".

Single transistors aren't yet logic gates by themselves; they are amplifiers with a very specific gain function that makes it possible to use them as switches. Logic gates usually consist of at least two transistors. See https://en.wikipedia.org/wiki/CMOS for an example of how it is done in CMOS technology.

That is correct.

Moreover, the amplifying function must exist at least in some gates, to restore the logic levels, but there is no need for it to exist in all gates.

For instance, any logic circuit can be implemented using AND gates and/or OR gates made with diodes, which have no gain, i.e. no amplifying function, together with 1-transistor inverters, which provide both the NOT function and the gain needed to restore the logic levels.

The logic functions such as NAND can be expressed in several way using simpler components, which correspond directly with the possible hardware implementations.

Nowadays, the most frequent method of logic implementation is by using parallel and series connections of switches, for which the MOS transistors are well suited.

Another way to express the logic functions is by using the minimum and maximum functions, which correspond directly with diode-based circuits.

All the logic functions can also be expressed using the 2 operations of the binary finite field, addition (a.k.a. XOR) and multiplication (a.k.a. AND).

This does not lead to simpler hardware implementations, but it can simplify some theoretical work, by using algebraic results. Actually this kind of expressing logic is the one that should have been properly named as Boolean logic, as the contribution of George Boole has been precisely replacing "false" and "true" with "0" and "1" and reinterpreting the classic logic operations as arithmetic operations. It is very weird to see in some programming languages a data type named Boolean, whose possible values are "false" and "true", instead of the historically correct Boolean values, "0" and "1", which can also be much more useful in programming than "false" and "true", by simplifying expressions, especially in array operations (which is why array-oriented languages like APL use "0" and "1", not "false" and "true").

Re: All elementary functions from a single binary operator

#99

Post by DoctorOetker »

dang wrote: post #77
lioeters wrote: post #71

Earlier quoted context omitted.

The URL is already pointing at v2, which apparently is the newest one requested by the comment above. > Submitted on 23 Mar 2026 (v1), last revised 4 Apr 2026 (this version, v2)

Thanks! that's what it looked like to me too.

I agree: I don't understand what happened, but the first "View PDF" resulted in a PDF where the hyperlinks to the figures didn't work. Upon closer inspection it wasn't v1 at all, thats a PNAS article. I am unable to remove the EDIT:... line in my original comment at this time...

Re: All elementary functions from a single binary operator

#100

Post by spuz »

eugene3306 wrote: post #61

This makes a good benchmark LLMs: ``` look at this paper: https://arxiv.org/pdf/2603.21852 now please produce 2x+y as a composition on EMLs ``` Opus(paid) - claimed that "2" is circular. Once I told it that ChatGPT have already done this, finished successfully. ChatGPT(free) - did it from the first try. Grok - produced estimation of the depth of the formula. Gemini - success Deepseek - Assumed some pre-existing knowl…

So what is the correct answer?
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