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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

#311

Post by empath75 »

empath75 wrote: post #309
Rubicund wrote: post #74

Earlier quoted context omitted.

On page 11, the paper explicitly states: > EML-compiled formulas work flawlessly in symbolic Mathematica and IEEE754 floating-point… This is because some formulas internally might rely on the following properties of extended reals: ln 0 = −∞, e^(−∞) = 0. And then follows with: > But EML expressions in general do not work ‘out of the box’ in pure Python/Julia or numerical Mathematica. Thus, the paper’s completeness cl…

I spent a few days playing around with this work in Lean and his central claim is provably wrong.. The main problem is that singularities infect everything, and you can't have an equational theory without rewrite/substitution rules that aren't grounded unless you can decide if an arbitrary EML tree is zero, which is undecidable in elementary functions (because of sine). Basically it's only valid on a undecideable sub…

Thinking about this some more. It having a NAND style combinator for elementary functions is probably impossible. ln(x) is inescapable as part of the composition in the generator somewhere, and you can't generate subtraction without passing through it which means the whole thing doomed as a project.

Re: All elementary functions from a single binary operator

#312

Post by scotty79 »

AlotOfReading wrote: post #262
SideQuark wrote: post #252

Earlier quoted context omitted.

Computing exp or ln is an infinite series, and vastly more compute. Hiding series behind a name doesn’t make them free to compute.

They're not series, that's just a convenient way to think about defining and calculating them. I've never found it particularly useful to deal with the series definitions either, and none of the (good) approximation methods I'm aware of actually take that approach. Moreover, EML is complete in a way that your suggested function isn't: If you take a finite combination of basis functions, can it build periodic function…

EML goes through complex numbers and infinities internally.

Previous commenter meant that there are infinite series inside log and exp.

To achieve a thing like sin(x) from his universal expression, yes you'd need infinite series of those, not a finite set of operations, but to get all 36 basic function you'd need a finite set of different infinite series.

So exp and log just happen to be the labels for two element set of such infinite series that you'd have to use to make 36 functions out of the operator that pervious commenter proposed.

All that said, I still think it's pretty neat too.

Re: All elementary functions from a single binary operator

#313

Post by monirmamoun »

this is really cool! i am going to try this in some of my own research. another cool thing is trying units different than 1 like adding imaginaries or other funky things, or different bases entirely. you can combine that concept with this for a lot of powerful math techniques.

Re: All elementary functions from a single binary operator

#314

Post by bastahimself »

this guy just took it from here: Chaotic Systems as Transcendental Structures: A Spiral Calculus Proof Framework https://www.researchgate.net/publication/391848560_Chaotic_S... He took my proposed equation Σ(t) = e −Dt C ix(t) Log to both sides and called eml

Re: All elementary functions from a single binary operator

#315

Post by peter_d_sherman »

>"Single, reusable primitives play a disproportionately large aesthetic and practical role in mathematics, engineering, and even biology. Widely known classical examples include the NAND gate (and its dual, Peirce Arrow, logical NOR) for Boolean 0/1 logic [2, 12], the operational amplifier [13] for positive and negative feedback processes, and, more recently, the rectified linear unit (ReLU) ”ramp” activation function [14] in deep learning [15]. We also mention Wolfram’s single axiom [16], K,S combinators from combinatory logic [17, 18], Interaction Combinators [19], and fuzzy versions of the Sheffer stroke [20]. Other wellknown examples are one-instruction set computers (OISC), e.g. SUBLEQ [21], Conway’s FRACTRAN [22] and the Rule 110 cellular automaton [16, 23]."

Re: All elementary functions from a single binary operator

#316

Post by applicative »

SideQuark wrote: post #305
applicative wrote: post #304

Earlier quoted context omitted.

1/(x-y) has nothing remotely like the power of his operator. The guy is not stupid.

It generates the same class of functions. Read the comments and links in this thread.

I did.
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