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...
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.
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.
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.
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.
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?
IIRC a resistor in series to a capacitor does the trick, for exp.
If we’re making sloppy approximations to a tiny range of exp, then I too can do it with a few terms.
> 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.
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.
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).
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.
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!
Could that be used to derive trigonometric functions with single distinct expressions?
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.
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.
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
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.
The OP only needs 1 number: 1.
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.