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This isn't unique, or even the least compute way to do this. For example, let f(x,y) = 1/(x-y). This too is universal. I think there's a theorem stating for any finite set of binary operators there is a single one replacing it.

write x#y for 1/(x-y).

x#0 = 1/(x-0) = 1/x, so you get reciprocals. Then (x#y)#0 = 1/((1/(x-y)) - 0) = x-y, so subtraction.

it's common problem to show in any (insert various algebraic structure here ) inverse and subtraction gives all 4 elementary ops.

I haven't checked this carefully, but this note seems to give a short proof (modulo knowing some other items...) https://dmg.tuwien.ac.at/goldstern/www/papers/notes/singlebi...

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Good find.

It cites a paper from 1935: https://www.pnas.org/doi/10.1073/pnas.21.5.252

Here is a bit more: https://mathoverflow.net/questions/57465/can-we-unify-additi...

Ooh, that 2nd link has a nice construction by Terry Tao giving a clear way to show infinitely many such functions exist for pretty much any set of operations.

I’m fairly certain that the difference between the approaches is that the f(x,y) function you mentioned requires limits to represent certain concepts while the eml approach is essentially a tree or a chain of computations meant to represent a model of a system.

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

I understand your point. The paper is more about the depth of the tree to represent and audit a model versus the raw CPU clock cycles. It takes the exponent and logarithm as given since for all practical purposes, in a scientific context, they are.

To represent something like sin(x) with f(x,y) requires infinite steps. Conversely, with eml you get an exact result in around 4 using identities and such.

One could argue that we do Taylor Series approximations on the hardware to represent trigonometric functions, but that highlights the key aspect of the eml approach. You can write a paper with those four steps that describes an exact model, here sin(x). And people can take that paper and optimize the result. This paper is about an auditable grammar that you can compute with.

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 functions? Hardy proved over a century ago that real (+,-,/,*,exp,ln) can't do this (and answering the paper's unresolved question about similar real-valued functions in the negative). EML being able to build periodic functions is a lot less surprising for obvious reasons, but still pretty neat.

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.

You're completely missing the point here.

You can reduce all Boolean logic to NAND, but that doesn't actually mean that semiconductor fabs translate their HDL to NAND gates, because it is possible to build complex gates that directly implement higher level operations in a single gate.

Your "cost of computation" objection can be easily resolved by adding more operators, which makes it boring from a research perspective.

Meanwhile the loss of expressivity can only be compensated by encoding algorithms directly into the expression tree. Your objection that an infinite series is a bad thing rings hollow, since you now introduce the concept of an infinitely sized expression tree. That sounds much more impractical than implementing an algorithm for the exponential and logarithm functions.

Show me a way to physically compute exp or ln that is less gates than add. More gates means costlier in $, more energy in compute, and for these functions, higher latency.

You don’t get to make up free ops, claim there is no cost in reality, and hand wave away reality.

There are infinitely many ways to do what the paper did. There’s no gain other than it’s pretty. It loses on every practical front to simply using current ops and architectures.

The feasibility of memristor analog circuits is evident, and I believe this paper represents a valuable early exploration. We've been constrained by Boolean logic for quite some time now.

The world has had many types of logic before and after Boolean logic was created, many used in computing. Boolean logic isn’t a constraint; it’s used where it’s useful, and others are used where they’re useful.

> no gain other than it’s pretty

Conceptual elegance is worth something, isn't it? I don't mean just aesthetic pleasure, as in recreational mathematics, but there's often (not always) value in being able to succinctly describe a wide range of phenomena with a small number of primitives. It could open up new ways of understanding or working that wasn't possible before. Not saying this specific discovery fits the description, but it seems too early to dismiss the idea entirely based on its im/practicality compared to existing solutions.

Aren't there examples in the history of mathematics, where a new idea was criticized for being impractical, then later found to have applications or implications, possibly unexpected even to the person who discovered it?

I feel sad when pragmatics comes in scientific discussions... that's not what science should be (I think). But I value the discussion this paper is bringing to distinct contexts (even outside academia). That by itself adds value to this work.

i am outisde of academia, i am told there are significant findings here at monogate.dev . just started vibe coding from the paper.

appreciate any feedback

> Show me a way to physically compute exp or ln that is less gates than add.

IIRC a resistor in series to a capacitor does the trick, for exp.

No, it approximates exp poorly over an infinitesimally small interval compared to exp. Resistors and capacitors are no where ideal components, which is why they have spec sheets to show how quickly they diverge.

If we’re making sloppy approximations to a tiny range of exp, then I too can do it with a few terms.

I think this is the novel bit:

> This includes constants such as e, pi, and i; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions.

And those come from the infinite series needed to compute exp and ln. They’re just as much work either way. The exp and ln way are vastly costlier for every op, including simply adding 1 and 2.

It's not about being costly or not, this is completely irrelevant to the point being made. eml is just some abstract function, that maps ℝ² to ℝ. Same as every other mathematical function it is only really defined by the infinite set of correspondences from one value to some other value. It is NOT exp(x) - ln(y), same as exp is not a series (as you wrongfully stated in another comment). exp can be expressed (and/or defined) as a series to a mathematician familiar with a notion of series, and eml can be expressed as exp(y) - ln(y) to a mathematician familiar with exp and ln. They can also be expressed/defined multiple other ways.

I am not claiming this is better than 1/(x-y) in any way (I have no idea, maybe it isn't if you look closely enough), but you are simply arguing against the wrong thing. Author didn't claim eml to be computationally efficient (it even feels weird to say that, since computational efficiency is not a trait of a mathematical function, but of a computer architecture implementing some program) or anything else, only that (eml, 1) are enough to produce every number and function that (admittedly, somewhat vaguely defined) a scientific calculator can produce.

However, I want to point out that it's weird 1/(x-y) didn't appear on that graph in Figure 1, since if it's as powerful as eml, it should have all the same connections as eml, and it's a pity Odrzywołek's paper misses it.

His paper misses infinitely many such functions.

Oh, glad you are still here. Because I kept wondering about 1/(x-y), and came to the conclusion it actually cannot do nearly as much as eml. So maybe you could confirm if I understood your assumptions correctly and help me to sort it out overall.

In your original post you were kinda hand-wavy about what we have except for x # y := 1/(x-y), but your examples make it clear you also assume 0 exists. Then it's pretty obvious how to get: identity function, reciprocity, negation, substraction & addition. But I effectively couldn't get anywhere past that. In fact, I got myself convinced that it's provably impossible to define (e.g.) multiplication as a tree of compositions of # and 0.

So here's my interpretation of what you actually meant:

1. I suppose, you assumed we already have ℕ and can sample anything from it. Meaning, you don't need to define 5, you just assume it's there. Well, to level things out, (#, 0, 1) are enough to recover ℤ, so I assume you assumed at least these three. Is that right?

2. Then, I suppose you assumed that since we have addition, multiplication simply follows from here. I mean at this point we clearly have f(x) = 3x, or 4x, or 5x, … so you decided that the multiplication is solved. Because I couldn't find how to express f(x, y) = x⋅y, and as far, as I can tell, it's actually impossible. If I'm wrong, please show me x⋅y defined as a sequence of compositions of (#, 0, 1).

3. This way (assuming №2) we get (ℚ, +, -, ⋅, /). Then, I suppose, you assume we can just define exp(x) as its Taylor series, so we also have all roots, trig functions, etc., and then we obviously have all numbers from ℝ, that are values of such functions acting on ℚ. Exactly as we do in any calculus / real analysis book, with limits and all that jazz.

If that's what you actually meant, I'm afraid you completely missed the point, and 1/(x-y) in fact isn't nearly as good as eml for the purposes of Odrzywołek's paper. Now, I didn't actually verify his results, so I just take them for granted (they are totally believable though, since it's easy to se how), but what he claims is that we can use eml essentially as a gate, like Sheffer stroke in logic, and express "everything else" just as a sequence of such gates and constant 1 (and "everything else" here is what I listed in №3). No words, limits, sets and other familiar mathematical formalism, just one operation and one constant, and "induction" is only used to get all of ℕ, everything else is defined as a finite tree of (eml, 1).

All of these are standard fare in abstract algebra classes, and I didn’t care to write it all out. Once you have the “inverse” operations - and reciprocal, the entire structure follows, for a large set of objects, whether N or Q or R or C or finite fields or division rings, and a host of other structures. So I only wrote - and 1/x

Then, subtraction is (x#y)#0 = x-y. Reciprocal is x#0 = 1/x. Addition follows from x+y=x-((x-x)-y). This used the additive identity 0.

Multiplication follows from

x^2= x-1/(1/x + 1/(1-x)), so we can square things. Then -2xy = (x-y)^2 -x^2 - y^2 is constructible. Then we can divide by -2 via x/-2 = 1/((0-1/x)-1/x), and there’s multiplication. In terms of #, this expression only needed the constant 1, which is the multiplicative identity.

Now mult and reciprocal give x * 1/y = x/y, division.

Any nontrivial ring needs additive and multiplicative identities 1!=0, which are the only constants needed above. If you assume this is Q or R or C, it may be possible to derive one from the other, not sure. But if you’re in these fields, you know 0 and 1 exist.

Then any element of Q is a finite set of ops. R can be constructed in whatever way you want: Dedekind cuts, Cauchy sequences, whatever usual constructions. Or assume R exists, and compute in it via the f(x,y).

This also works over finite fields (eml does not), division rings, even infinite fields of positive characteristic, function fields (think elements are ratio of polynomials), basically any algebraic object with the 4 ops.

You are taking `exp` and `ln` to require infinite series, because you aren't taking his operator as primitive.

You forgot to create numbers. You need 0 (explicitly used in your construction), and you need another number so you generate numbers that are not 0 and infinity.

The OP only needs 1 number: 1.

I don't think this can do any of the "standard" constants or what we generally consider to be closed-form expressions, though ! (E.g., no e, pi, exp, log, etc.)

Yes it can, by using the same infinite series that exp and ln use to compute. This one just costs less in money, hardware, energy, and is faster for basically every basic op.

I think the point is that it is _finite_. if you allow infinite expressions then the basic monomial basis or quotients thereof are “even simpler”

It’s only finite by putting the infinite series into an operation.

And the basic monomial basis is not a single binary operation capable of reproducing the set of basic arithmetic ops. If you want trivial and basic, pick Peano postulates. But that’s not what this thread was about.

well, the statement is: is there a single operation, built from elementary operations, such that all _other_ elementary operations have finite representations.

this preprint answers that in the affirmative

otoh, (x, y) -> 1/(x-y) does not answer this question at all. you can argue that the preprint does so "via the infinite series in an operation" (which I have no idea what that means; surely if exp(x) qualifies then so must 1/(x-y) if we pick a monomial basis?) but ¯\_(ツ)_/¯

now, do I think that this is groundbreaking magical research (as I'm currently seeing on twitter) no... But it's neat!

> This too is universal

Could that be used to derive trigonometric functions with single distinct expressions?

The exp and ln are infinite series. Exp is roughly the infinite series for cos AND the infinite series for sin. Hiding that every op is an infinite series behind a name doesn’t make things free. It just makes even trivial ops like 1+2 vastly more work.

They are not infinite series per se. They can be represented by infinite series in several ways but there are standard ways to define them that do not involve infinite series. The logarithm in particular is not even represented by an infinite series (in form of Taylor expansion) defined in the whole complex plane. And knowledge/use of trigonometric functions greatly precedes such infinite series representations.

Moreover, the point is not always numerical computation. I don’t think anybody argues that eml sounds like an efficient way to compute elementary functions numerically. It may or may not still be useful for symbolic computations.

The article is about producing all elementary functions, which 1/(x-y) clearly doesn’t, as it doesn’t produce any transcendental function. Like many of such universality-style results it may not have practical applications, but may still be interesting on its own right.

Any transcendental function can be produced by arithmetic, since its complete for R.

Go ahead and show how to compute exp or ln without an infinite series without circular reasoning. You can’t, since they’re transcendental.

There are infinitely many ways to make these binary operators. Picking extremely high compute cost ones really doesn’t make a good basis for computation.

> Any transcendental function can be produced by arithmetic, since its complete for R.

Not without some form of limit process or construction. You can approximate e with the basic arithmetic operations but not actually get an exact form in finite steps. And you definitely cannot transverse an infinite binary tree, so the main point of the result in the article is missed by your arguments.

Again, you are mixing separate things. Nobody said that eml is some way to approximate elementary functions more efficiently. It is a way to express elementary functions in a finite amount of operations. Meaning, computing symbolically, not numerically. Eg I may care that exp(3)*exp(2)=exp(5) without caring to approximate exp(5) numerically. The paper is literally under "Computer Science > Symbolic Computation", not "numerical analysis" or "engineering" after all.

And to be precise:

> Go ahead and show how to compute exp or ln without an infinite series without circular reasoning. You can’t, since they’re transcendental.

You don't necessarily need "infinite series", you need some limit process. A basic example is that exp(x) can be approximated by (1 + x/n)^n for large n. For the logarithm you can use a formula involving the arithmetic–geometric mean which you can approximate using an iterative process/recursion without infinite series. You can also approximate the exponential by using Newton's method together with that, see [0].

[0] Fast Computations of the Exponential Function https://link.springer.com/chapter/10.1007/3-540-49116-3_28

Yep, I’ve written numerical methods papers, and am very well aware of the field.

A limit process is a definition. Try computing with it. You’ll end up with an infinite sequence, or an approximation.

An iterative process is an infinite series. They’re equivalent.

Newtons method is the same. Completely equivalent to an infinite series as you increase precision.

And both require constants, infinitely precise. So you’re still not doing anything the 1/(x-y) operation cannot do, and to do those series you’ll compute using things amenable to being done via ops easy to do by hand or machine via the 1/(x-y) op.

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.

yes, but are you currently experiencing both hypergraphia and chatbot AI induced psychosis while also thinking about this problem?

It's math. You can check it yourself instead of this (and many other) thoughtless posts.

i'm mocking the LLM-generated scientific article you're replying to, not you. i'm agreeing with you haha

Do you claim all things you don’t understand are LLMs? This is what I mean by these and many of your other comments being extremely poor quality to the point of deliberate ignorance.

The paper above was published in 2012 [1], so that’s quite a feat for an LLM. This takes about zero effort to check.

Put some thought or effort into your claims; they’ll look less silly.

[1] https://orcid.org/0000-0002-0438-633X

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