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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

#111

Post by theanonymousone »

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…

I changed the prompt to this:

""" Consider a mathematical function EML defined as `eml(x,y)=exp(x)−ln(y)`

Please produce `sin(x)/x` as a composition on EMLs and constant number 1 (one). """

Re: All elementary functions from a single binary operator

#112

Post by fxwin »

geocar wrote: post #76
prvc wrote: post #57

This is neat, but could someone explain the significance or practical (or even theoretical) utility of it?

Read the paper. On the third page is a "Significance statement".

eh, i didnt find that paragraph very helpful. it just restates what it means do decompose an expression into another one only relying on eml, and vaguely gestures at what this could mean, i was hoping for something more specific.

Re: All elementary functions from a single binary operator

#113

Post by fxwin »

psychoslave wrote: post #64

Very nice, though I'm not found of the name. What comes to my mind as an alternative which I would subjectivity finer is "axe". Think axiom or axiology. Anyone with other suggestions? Or even remarks on this one?

i think eml is fine, names should be connected to the thing they represent so 'exponential minus log' makes sense to me

Re: All elementary functions from a single binary operator

#115

Post by hyperhello »

freehorse wrote: post #97
hyperhello wrote: post #37

Earlier quoted context omitted.

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

Okay, I’m tired. Not quite inverse but per the title , must be a way.

Re: All elementary functions from a single binary operator

#116

Post by WilcoKruijer »

prvc wrote: post #57

This is neat, but could someone explain the significance or practical (or even theoretical) utility of it?

From the paper:

> Everyone learns many mathematical operations in school: fractions, roots, logarithms, and trigonometric functions (+, −, ×, /, sqrt, sin, cos, log, …), each with its own rules and a dedicated button on a scientific calculator. Higher mathematics reveals that many of these are redundant: for example, trigonometric ones reduce to the complex exponential. How far can this reduction go? We show that it goes all the way: a single operation, eml(x, y), replaces every one of them. A calculator with just two buttons, EML and the digit 1, can compute everything a full scientific calculator does. This is not a mere mathematical trick. Because one repeatable element suffices, mathematical expressions become uniform circuits, much like electronics built from identical transistors, opening new ways to encoding, evaluating, and discovering formulas across scientific computing.

Re: All elementary functions from a single binary operator

#117

Post by thaumasiotes »

adrian_b wrote: post #108
thaumasiotes wrote: post #6

Earlier quoted context omitted.

Have you gone through The Little Schemer ? More on topic: > No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations. I was taught that these were all hypergeometric functions. What distinction is being drawn here?

Hypergeometric functions are functions with 4 parameters. When you have a function with many parameters it becomes rather trivial to express simpler functions with it. You could find a lot of functions with 4 parameters that can express all elementary functions. Finding a binary operation that can do this, like in TFA, is far more difficult, which is why it has not been done before. A function with 4 parameters can a…

> Hypergeometric functions are functions with 4 parameters.

Granted, but the claim in the abstract says:

>> computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations

And I don't see how this is true as to hypergeometric functions in a way that isn't shared by the approach in the paper.

> Finding a binary operation that can do this, like in TFA, is far more difficult, which is why it has not been done before.

> A function with 4 parameters can actually express not only any elementary function, but an infinity of functions with 3 parameters, e.g. by using the 4th parameter to encode an identifier for the function that must be computed.

These statements seem to be in direct conflict with each other; you can use the second parameter of a binary function to identify a unary function just as you can use the fourth parameter of a quaternary function to identify a trinary one.

Re: All elementary functions from a single binary operator

#119

Post by vintermann »

I'm way too unschooled to say if it's important or not, but what really excites me is the Catalan structure ("Every EML expression is a binary tree [...] isomorphic to well-studied combinatorial objects like full binary trees and Catalan objects").

So, what happens if you take say the EML expression for addition, and invert the binary tree?

Re: All elementary functions from a single binary operator

#120

Post by adrian_b »

thaumasiotes wrote: post #117
adrian_b wrote: post #108

Earlier quoted context omitted.

Hypergeometric functions are functions with 4 parameters. When you have a function with many parameters it becomes rather trivial to express simpler functions with it. You could find a lot of functions with 4 parameters that can express all elementary functions. Finding a binary operation that can do this, like in TFA, is far more difficult, which is why it has not been done before. A function with 4 parameters can a…

> Hypergeometric functions are functions with 4 parameters. Granted, but the claim in the abstract says: >> computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations And I don't see how this is true as to hypergeometric functions in a way that isn't shared by the approach in the paper. > Finding a binary operation that can do this, like in TFA, is far more difficu…

There is an essential difference between binary functions and unary functions.

With binary functions you can compose them using a very complex composition graph.

With unary functions you can compose them only linearly, so in general it is impossible to make a binary function with unary functions.

You can make binary functions from unary functions only by using at least one other binary function. For instance, you can make multiplication from squaring, but only with the help of binary addition/subtraction.

So the one function that can be used to generate the others by composition must be at least binary, in order to be able to generate functions with an arbitrary number of parameters.

This is why in mathematics there are many domains where the only required primitives are a small number of binary functions, but there is none where strictly unary functions are sufficient. (However, it may be possible to restrict the binary functions to very simple functions, e.g. making a tuple from components, for instance the CONS function of LISP I.)

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