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All elementary functions from a single binary operator

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Re: All elementary functions from a single binary operator

#151

Post by PaulHoule »

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…

I can't say I'm surprised at this result at all, in fact I'm surprised something like this wasn't already known.

Re: All elementary functions from a single binary operator

#152

Post by yorwba »

DoctorOetker wrote: post #90
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'…

In my experience this exponential expression blow-up is less the result of the approach of decomposing into a minimum of primitives, but rather a result from repetition in expression trees: If we make the analogy from Bertrand Russel's Principia Mathematica, he derived fully expanded expressions, i.e. trees where the leaves only may refer to the same literals, everyone claimed this madness underscored how formal veri…

Yes, metamath uses a large collection of specialized but reusable building blocks, so it doesn't blow up exponentially. However, if you want to "just do gradient descent" on general trees built from a single universal primitive, you now have to rediscover all those building blocks on the fly. And while the final result may have a compact representation as a DAG by merging common subexpressions, you also need to be able to represent potential alternative solutions, and that's where the exponential blowup comes in.

Or you could accept that there's already a large collection of known useful special functions, and work with shallower trees of those instead, e.g. https://arxiv.org/abs/1905.11481

Re: All elementary functions from a single binary operator

#153

Post by mah4k4l »

Sharlin wrote: post #150
mah4k4l wrote: post #143

According to Gemini 3.1 Pro this would shoot the current weather forecasting power through the roof (and math processing in general): The plan is to use this new "structurally flawless mathematical primitive" EML (this is all beyond me, was just having some fun trying to make it cook things together) in TPUs made out of logarithmic number system circuits. EML would have DAGs to help with the exponential bloat problem…

Congrats, you made a hallucination machine successfully hallucinate?

I understand enough for it's arguments' symmetry to have an impact. Used Deep Research, it had the paper from the link as input plus some previous discussions about Tensor Logic and the new hardcoded neuroweb - like processors. Didn't make those up either.

Re: All elementary functions from a single binary operator

#155

Post by irchans »

ikrima wrote: post #46
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…

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'…

ikrima, Do you have any links to your research or paper titles?

Re: All elementary functions from a single binary operator

#157

Post by Twey »

layer8 wrote: post #149
evnix wrote: post #78

Can someone explain how is this different from lambda calculus, it seems like you can derive the same in both. I don't understand both well enough and hence the question.

Lambda calculus is about discrete computations, this is about continuous functions. You can’t reason about continuous functions in lambda calculus.

Depending on your lambda calculus! From a categorical perspective a lambda calculus is just a nice syntax for Cartesian closed categories (or similar, e.g. *-autonomous categories for linear lambda calculus) so you can use it to reason about anything you can fit into that mould. For example, Paul Taylor likes to do exactly this: https://www.paultaylor.eu/ASD/analysis#lamcra

Re: All elementary functions from a single binary operator

#158

Post by IAmBroom »

brucehoult wrote: post #68
gilgoomesh wrote: post #38

Earlier quoted context omitted.

> 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? Same reason all boolean logic isn't performed with combinations of NAND – it's computationally inefficient. Polynomials are (for their expressivity) very quick to compute.

The Cray 1 was built 100% from NOR gates and SRAM.

50 years ago.

Re: All elementary functions from a single binary operator

#159

Post by rnhmjoj »

lmf4lol wrote: post #154

Stupid question maybe (I am no mathematician), but aren't exp and ln really primitives? Aren't they implemented in terms of +,-,/,* etc? Or do we assume that we have an infinite lookup table for all possible inputs?

> aren't exp and ln really primitives? Aren't they implemented in terms of +,-,/,* etc?

They're primitive in the sense that you can't compute exp(x) or log(x) using a finite combination of other elementary functions for any x. If you allow infinite many operations, then you can easily find infinite sums or products of powers, or more complicated expressions to represent exp and log and other elementary functions.

> Or do we assume that we have an infinite lookup table for all possible inputs?

Essentially yes, you don't necessarily need an "implementation" to talk about a function, or more generally you don't need to explicitly construct an object from simpler pieces: you can just prove it satisfies some properties and that it is has to exist.

For exp(x), you could define the function as the solution to the diffedential equal df/dx = f(x) with initial condition f(0) = 1. Then you would enstablish that the solution exists and it's unique (it follows from the properties of the differential equation), call exp=f and there you have it. You don't necessarily know how to compute for any x, but you can assume exp(x) exists and it's a real number.

Re: All elementary functions from a single binary operator

#160

Post by xpe »

AlotOfReading wrote: post #24
wildzzz wrote: post #11

Earlier quoted context omitted.

Dreadfully slow for integer math but probably some similar performance to something like a CORDIC for specific operations. If you can build an FPU that does exp() and ln() really fast, it's simple binary tree traversal to find the solution.

You already have an FPU that approximates exp() and ln() really fast, because float integer conversions approximate the power 2 functions respectively. Doing it accurately runs face-first into the tablemaker's dilemma, but you could do this with just 2 conversions, 2 FMAs (for power adjustments), and a subtraction per. A lot of cases would be even faster. Whether that's worth it will be situational.

What are some of your favorite sources to dig into this field as a whole?
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