Not all elementary functions can be expressed with exp-minus-log

https://www.stylewarning.com/posts/not-all-elementary/

by mmastrac • 3 months ago

146 118 3 months ago

lotaezenwa2• 3 months ago

The author essentially says that the quintic has no closed form solution which is true regardless of the exp-minus-log function. The purpose of this blog post is lost on me.

Can anyone please explain this further? It seems like he’s moving the goalposts.

DevelopingElk• 3 months ago

His claim is that we exp-minus-log cannot compute the root of an arbitrary quintic. If you consider the root of an arbitrary quintic "elementary" the exp-minus-log can't represent all elementary functions.

I think it really comes down to what set of functions you are calling "elementary".

throwanem• 3 months ago

The author discusses this in his third paragraph, and states explicitly in his fourth that he considers the result faulty for its unrealistically narrow definition of elementarity.

(I'm not a mathematician, so don't expect me to have an opinion as far as that goes. But the author also writes well in English, and that language we do share.)

petters• 3 months ago

Yes, that blog post could have been much shorter….

AlotOfReading• 3 months ago

The argument is that a universal basis would be capable of solving arbitrary polynomial roots. The rest is an argument that the group constructed by eml is solveable, and hence not all the standard elementary functions.

It wouldn't be a math discussion without people using at least two wildly different definitions.

markgall2• 3 months ago

Can anyone provide a link that "Some are going as far as to suggest that the entire foundations of computer engineering and machine learning should be re-built as a result of this", or anything similarly grandiose?

I am a professional mathematician, though nowhere near this kind of thing. The result seems amusing enough, but it doesn't really strike me as something that would be surprising. I confess that this thread is the first I've heard of it...

saithound4• 3 months ago

It's a fun, but unsurprising undergrad-level result. It got picked up and overhyped on HN [1] and /r/math [2] earlier this week.

Some of my favorites:

DoctorOetker: "I'm still reading this, but if this checks out, this is one of the most significant discoveries in years."

cryptonektor: "Given this amazing work, an efficient EML operator HW implementation could revolutionize a bunch of things."

zephen: "This is about continuous math, not ones and zeroes. Assuming peer review proves it out, this is outstanding."

[1] https://news.ycombinator.com/item?id=47746610

[2] https://www.reddit.com/r/math/comments/1sk63n5/all_elementar...

reikonomusha15• 3 months ago

"The quintic has no closed form solution" is a theorem that is more precisely stated (in the usual capstone Galois proof) as follows: The quintic has no closed form solution in terms of arbitrary compositions of rational numbers, arithmetic, and Nth roots. We can absolutely express closed form solutions to the quintic if we broaden our repertoire of functions, such as with the Bring radical.

The post's argument is different than the usual Galois theory result about the unsolvability of the quintic, in that it shows a property that must be true about all EML(x,y)-derived functions, and a hypothetical quintic-solver-function does not have that property, so no function we add to our repertoire via EML will solve it (or any other function, elementary or not, that lacks this property).

lotaezenwa2• 3 months ago

Cool explanation, thanks!

cyberax4• 3 months ago

Bring radicals are just cheating.

You can't solve an equation? Why not just introduce a function that is equal to the solution of the equation! Problem solved.

saithound4• 3 months ago

The original article explicitly acknowledged this limitation, that while in "the classical differential-algebraic setting, one often works with a broader notion of elementary function, defined relative to a chosen field of constants and allowing algebraic adjunctions, i.e., adjoining roots of polynomial equations," the author works with the less general definition.

Neither the present article, nor the original one has much mathematical originality, though: Odrzywolek's result is immediately obvious, while this blog post is a rehash of Arnold's proof of the unsolvability of the quintic.

vintermann4• 3 months ago

Yes, this article is kicking in open doors, the original article was quite clear about the scope.

The present article could rather have spent time arguing why this isn't like NAND gate functional completeness.

I would have thought the differences lie in the other direction: not that trees of EML and 1 can describe too little, but that they can describe too much already. It's decidable whether two NAND circuits implement the same function, I'm pretty sure it's not decidable if two EML trees describe the same function.

reikonomusha15• 3 months ago

You are correct, it is undecidable by Richardson's theorem [1].

[1] https://en.wikipedia.org/wiki/Richardson%27s_theorem

zephen2• 3 months ago

It's decidable whether two NAND circuits implement the same function

Well, sure. At least, until you have a loop that starts clocking for you, and now you've got the halting problem.

empath753• 3 months ago

It has more problems. See my other comment in this thread.

reikonomusha15• 3 months ago

Arnold (as reported by Goldmakher [1]) does prove the unsolvability of the quintic in finite terms of arithmetic and single-valued continuous functions (which does not include the complex logarithm). TFA's result is stronger, which is something about the solvability of the monodromy groups of all EML-derived functions. So it doesn't seem to be a "rehash", even if their specific counterexample could have been achieved either in fewer steps or with less machinery.

[1] https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.p...

saithound4• 3 months ago

Arnold's proof can be used to show that certain classes of functions are insufficient to express a quintic formula.

These classes can always safely include all single-valued continuous functions (you cannot even write the _quadratic_ formula in terms of arithmetic and single-valued continuous functions!), but also plenty of non-single-valued functions (e.g. the +-sqrt function which appears in the well-known quadratic formula).

Applying Arnold's proof to the class given by arithmetic and all complex nth root functions (also multivalued) gives the usual Abel-Ruffini theorem. But Arnold's proof applies to the class "all elm-expressible functions" without modification.

cubefox3• 3 months ago

Odrzywolek's result is immediately obvious

Many things that in retrospect seem immediately obvious weren't obvious before, let alone immediately obvious.

DoctorOetker9• 3 months ago

and its depressing when the rare actual progress is made, a collection of jealous practitioners comes to party-poop all over the place, for bringing the insights that make the result from then on immediately obvious.

eru5• 3 months ago

Neither the present article, nor the original one has much mathematical originality, though: Odrzywolek's result is immediately obvious, [...]

Maybe. But I found it a nice piece of recreational mathematics nevertheless.

DoctorOetker9• 3 months ago

Odrzywolek's result is immediately obvious

This may or may not be true; but the burden of proof should not lay with the reader.

Please provide (in absence of which every reader can draw their own conclusions) a reference which simultaneously:

1) predates Odrzywolek's result

2) and demonstrates the other unary and binary operations typically tacitly assumed can be expressed in terms of a single binary operation and a constant.

(in other news: I can spontaneously levitate, I just don't feel like demonstrating it to you right now...)

zephen2• 3 months ago

Upvoted back to not-greyed-out. You must have struck a nerve.

saithound4• 3 months ago

Questions which have never been asked or answered before, but to which practitioners have immediately obvious answers, are dime a dozen in mathematics.

You can find thousands of such questions on Math StackExchange. Take e.g. [1]: never been asked anywhere else, interesting enough, yet answered pretty much immediately by two separate mathematicians.

"Is there a single constant and function with connected domain that can express all of $\log, \exp, \sin, \dots$?" would have made a fine question there too, the type that gets a thorough answer very quickly if anyone bothers to ask it.

the burden of proof should not lay with the reader

You were the one who made the claim that "this is one of the most significant discoveries in years". Feel free to substantiate that claim first, according to the same standards. Are there any authors who ask this question, and/or suggest that they don't know an answer?

[1] https://math.stackexchange.com/questions/2308587/is-the-set-...

empath753• 3 months ago

His derivations are genuinely pretty, but it isn’t a result when everything he claims is actually just wrong.

avmich2• 3 months ago

I'd really like more details on the terminology used.

Also I'd be glad to see a specific example of a function, considered elementary, which is not representable by EML.

It could be hard, and in any case, thanks for the article. I wish it would be more accessible to me.

markgall2• 3 months ago

I only skimmed the article, but I think the idea is to use some variation on:

f(a,b,c,d,e) = the largest real solution x of the quintic equation x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0

There's not a simple formula for this function (which is the basic point), but certainly it is a function: you feed it five real numbers as input, and it spits out one number as output. The proof that you can't generate this function using the single one given looks like some fairly routine Galois theory.

Whether this function is "considered elementary" depends on who you ask. Most people would not say this is elementary, but the author would like to redefine the term to include it, which would make the theorem not true anymore.

Why any of this would shake the foundations of computer engineering I do not know.

avmich2• 3 months ago

I've thought something like that, but I'm interested more in details of the argument.

As for why this could be important... we sometimes find new ways of solving old problems, when we formulate them in a different language. I remember how i was surprised to learn how representation of numbers as a tuple (ordered list of numbers), where each element is the remainder for mutually prime dividers - as many dividers as there are elements in the tuple - reduces the size of tables of division operation, and so the hardware which does the operation using thise tables may use significantly less memory. Here we might have some other interesting advantages.

Aardwolf• 3 months ago

But can you even express this function with the elementary operator symbols, exp, log, power and trig functions? It seems to me like no, you can't express "largest real solution" with those (and what's the intended result for complex inputs?)

At least eml can express the quintic itself, just like the above mentioned operators can

teo_zero• 3 months ago

I feel that saying that EML can't generate all the elementary functions because it can't express the solution of the quintic is like saying that NAND gates can't be the basis of modern computing because they can't be used to solve Turing's halting problem.

bawolff4• 3 months ago

Elementary functions typically include arbitrary polynomial roots

Admittedly this may be above my math level, but this just seems like a bad definition of elementary functions, given the context.

js82• 3 months ago

I would agree, it makes them anything but elementary. I am honestly not even sure if there is a finite constructible basis of the functions that can express any solution of single-variable integer polynomials.

And for multivariate polynomials, the roots are uncomputable due to MRDP theorem.

reikonomusha15• 3 months ago

It is not known, and the model problem for this is Hilbert's 13th [1].

Nonetheless, "elementary function" is a technical term dating back to the 19th century; it's very much not a general adjective whose synonym is "basic".

[1] https://en.wikipedia.org/wiki/Hilbert%27s_thirteenth_problem

SabrinaJewson• 3 months ago

Related is the paper [What is a closed-form number?], which explores the field E, defined as the smallest subfield of ℂ closed under exp and log. I believe the set of numbers that can be generated using exp-minus-log is a strict subset of this.

In a similar vein to this post, the paper points out that general polynomials do not have solutions in E, so of course exp-minus-log is similarly incomplete.

What is intriguing is that we don’t even know whether many simple equations like exp(-x) = x (i.e. the [omega constant]) have solutions in E. We of course suspect they don’t, but this conjecture is not proven: https://en.wikipedia.org/wiki/Schanuel%27s_conjecture

What is a closed-form number?: http://timothychow.net/closedform.pdf omega constant: https://en.wikipedia.org/wiki/Omega_constant

DoctorOetker9• 3 months ago

Related is the paper [What is a closed-form number?], which explores the field E, defined as the smallest subfield of ℂ closed under exp and log. I believe the set of numbers that can be generated using exp-minus-log is a strict subset of this.

is that a typo / accidental mis-phrasing?

exp-minus-log construction is closed for the operations it supports, and spans both exp and log, so E must be either identical to or a subset of exp-minus-log; not the other way around.

EML is spanned by a single binary operator, while the article you reference describing ("what is a closed-form number") just tacitly assumes +, -, x, / are available for free, so even in just this sense the EML construction is superior. Since EML can construct the larger presumed basic operations of E, E must be contained in it, but since the E implicitly has +, - besides exp(x) and ln(x) the reverse can also be said, so the sets and functions spanned by E and EML should be equivalent. So what is novel? precisely what the recent article describes: all the tacitly (+,-,x,/) and explicitly assumed (exp and ln) operations can be spanned with just 1 (non-unique) binary operation; and on top of that:

the recent article describes freely available code to conduct such searches and find alternative binary operations, search for functions or constants.

The EML paper provides code and machinery to conduct a search for the value x in exp(-x)=x : use a multiprecision library to get an arbitrarily precise representation, and search for some EML expression to find candidates.

xigoi2• 3 months ago

exp-minus-log construction is closed for the operations it supports, and spans both exp and log, so E must be either identical to or a subset of exp-minus-log; not the other way around.

Since E is by definition closed under exp, log and subtraction, it is clearly also closed under EML.

zarzavat• 3 months ago

This is a bit like invalidating a result based on 0^0 := 1 because you work in a field of mathematics where 0^0 is an indeterminate form. Not very interesting.

AFAIU the original paper is a result in the field of symbolic regression. What definition of elementary function do they use?

renewiltord3• 3 months ago

If this is true, then this blog post debunking EML is going to up-end all of mathematics for the next century.

rnhmjoj3• 3 months ago

My concern is that the word “elementary” in the title carries a much broader meaning in standard mathematical usage, and in this meaning, the paper’s title does not hold.
Elementary functions typically include arbitrary polynomial roots, and EML terms cannot express them.

If you take a real analysis class, the elementary functions will be defined exactly as the author of the EML paper does.

I've actually just learnt that some consider roots of arbitrary polynomials being part of the elementary functions before, but I'm a physicist and only ever took some undergraduate mathematics classes. Nonetheless, calling these elementary feels a bit of stretch considering that the word literally means basic stuff, something that a beginner will learn first.

reikonomusha15• 3 months ago

The definition of "elementary function" typically includes functions which solve polynomials, like the Bring radical. The definition was developed and is most fitting in algebraic contexts where algebraic structure is meaningful, like Liouvillian structure theorems, algorithmic integration, and computer algebra. See e.g.

- Page 2 and the following example of https://billcookmath.com/courses/math4010-spring2016/math401... (2016)

- Ritt's Integration in Finite Terms: Liouville's Theory of Elementary Methods (1948)

It's not frequent that analysis books will define the class of elementary functions rigorously, but instead refer to examples of them informally.

thaumasiotes2• 3 months ago

See e.g. Page 2 and the following example of https://billcookmath.com/courses/math4010-spring2016/math401... (2016)

There appears to be a typo in that example; I assume "Essentially elementary functions are the functions that can be built from ℂ and f(x) = x" should say something more like "the functions that can be built from ℂ and f(x) = y".

Joker_vD• 3 months ago

The definition of "elementary function" typically includes functions which solve polynomials, like the Bring radical.

What. Does that "typical definition" of elementary function includes elliptic functions as well, by any chance?

burnished• 3 months ago

All I know is that when a class starts with 'elementary' or 'fundamentals of' you had best buckle up.

TeMPOraL2• 3 months ago

Introduction to ...

quchen• 3 months ago

Algebraic too.

There's also the opposite in physics though, "modern" means from the 60s with square roots drawn in manually.

chii• 3 months ago

jargon are words being used that don't carry the typical laymen definition, but a specific one from the domain of said jargon.

If a written piece is intended for an audience who knows the jargon, then it's fine to use jargon - in fact it's appropriate and succinct. If it was intended for the laymen, then jargon is inappropriate.

But it seems you're lamenting that this jargon is wrong and that it shouldn't be jargon!?

mcmoor• 3 months ago

I don't know if I read this right, but I thought it's proven that "elementary functions" can't solve 5th degree or higher polynomial, so I'm confused how it's interpreted if elementary functions also include arbitrary polynomial roots. Or is it different elementary functions?

eru5• 3 months ago

The term 'elementary function' doesn't really have a single universally agreed on strict definition.

Definitions are either a bit fuzzy, or not universally agreed on.

Though interestingly https://en.wikipedia.org/wiki/Elementary_function says "More generally, in modern mathematics, elementary functions comprise the set of [...]". Though at least Wikipedia thinks that 'modern mathematics' has a consensus; of course, there's no guarantee that whoever you are talking to uses the 'modern mathematics' definition that Wikipedia brings up.

adrian_b2• 3 months ago

That theorem is not formulated about "elementary functions".

It says that polynomial equations of the 5th degrees or higher cannot, in general, be solved using "radicals".

While something like "polynomials" or "radicals" has a clear meaning, which are the "elementary functions" is a matter of convention.

The usual convention is to include all algebraic functions and a few selected transcendental functions.

In "all algebraic functions", are included the rational functions, the radicals and the functions that compute solutions of arbitrary polynomial equations.

Some conventions used for "elementary functions" describe the expressions that you can use to write such "elementary functions", in which case not all algebraic functions are included, but only those written by combining rational functions with radicals.

For an algebraic function that computes a solution of a general polynomial equation, which cannot be expressed with radicals, you cannot write an explicit formula, but you can write the function only implicitly, by writing the corresponding polynomial equation.

So the difference between the 2 kinds of conventions about which are "the elementary functions" is usually based on whether only explicitly-written functions are considered, or also implicit functions.

SideQuark• 3 months ago

If you take a real analysis class, the elementary functions will be defined exactly as the author of the EML paper does.

I just looked through many of the best known real analysis texts, and not a single one defines them this way. This list included the texts by

Royden, Terence Tao, Rudin, Spivak, Bartle & Sherbert, Pugh, and a few others....

Can you cite a single text book that has this definition you claim is in every real analysis course? I find all evidence points to the opposite.

rnhmjoj3• 3 months ago

I guess you're right, I was probably mislead this whole time. I went through my old analysis class book [1] and there doesn't seem to be an explicit definition of elementary functions. The best I can find is this paragraph (I translate from italian):

The elementary functions of analysis, that is powers, roots, exponentials, logarithms and their inverses, functions obtained from the former by arithmetic operations or composition, admit the limit f(p) for x → p, for any p in their set of definition. The study of such functions, which is not limited to the sole real functions of real variable, is carried out naturally in the setting of metric spaces.

That said, I'm relatively sure that a definition was given in class and it didn't include arbitrary roots: despite being notoriously difficult, the exam didn't require students to draw the graph of any elementary function including implicitly-defined algebraic roots.

I picked up another one of the old recommended books [2] and it seems to be similarly vague; while the book currently taught in my university [3], gives this definition:

The following functions (from ℂ to ℂ) are called the elementary functions of the Analysis:
1) Rational functions (integral or fractional)
2) Algebraic functions (explicit or implicit)
3) The exponential function
4) The logarithm function
5) All those functions that can be obtained by combining a finite number of times the functions of kind 1)...4).

So, roots of arbitrary polynomials implicitly defined are indeed considered elementary. I never knew this.

[1] https://search.worldcat.org/title/1261811544

[2] https://search.worldcat.org/title/801297519

[3] https://search.worldcat.org/title/935666878

fnordpiglet• 3 months ago

In math elementary usually means fundamental or foundational not elementary school. The root word is element and the relationship to “simple subject” is tangential and more related to its teaching the elemental topics for a lifetime education than definitionally cross discipline.

derriz2• 3 months ago

When I first read the exp-minus-log paper, I found it extremely surprising - even shocking that such a function could exist.

But the fact that a single function can represent a large number of other functions isn't that surprising at all.

It's probably obvious to anyone (it wasn't initially to me), but given enough arguments I can represent any arbitrary set of n+1 functions (they don't even have to be functions on the reals - just as long as the domain has a multiplicative zero available) as a sort of "selector":

g(x_0, c_0, x_1, c_1, ... , x_n, c_n) = c_0 * f_0(x_0) + ... + c_n * f_n(x_n)

The trick is to minimize the number of arguments and complexity of the RHS - but that there's a trivial upper-bound (in terms of number of arguments).

kqr• 3 months ago

This is similar to the idea of generating functions, if you would like more to read!

eru5• 3 months ago

You might like https://www2.math.upenn.edu/~wilf/gfology2.pdf

cyberax4• 3 months ago

Why would it be surprising?

And if you want something truly surprising, Riemann's zeta function can approximate any holomorphic function arbitrarily well on the critical strip. So technically you need only _one_ argument.

derriz2• 3 months ago

Why would it be surprising?

Because I don’t know as much mathematics as you.

adrian_b2• 3 months ago

When you may use functions of 3 or more arguments, it becomes trivial to find a single function that can be used to express large classes of other functions.

These tricks break when you are restricted to use one binary function, like in the EML paper.

The second argument cannot be used as a selector, because you cannot make binary functions from unary functions (while from binary functions you can make functions with an arbitrary number of parameters, by composing them in a tree).

If you used an argument as a function selector in a binary function, which transforms the binary function into a family of unary functions, then you would need at least one other auxiliary binary function, to be able to make functions with more than one parameter.

The auxiliary binary function could be something like addition or subtraction, or at the minimum a function that makes a tuple from its arguments, like the function CONS of LISP I.

The EML paper can also be understood that the elementary functions as defined by it can be expressed using a small family of unary functions (exponential, logarithmic and negation), together with one binary function: addition.

Then this set of 4 simple functions is reduced to one complex function, which can regenerate any of those 4 functions by composition with itself.

This is the same trick used to reduce the set of 2 simple functions, AND & NOT, which are sufficient to write any logical function, to a single function, NAND, which can generate both simpler functions.

aixpert• 3 months ago

tangenially it would be interesting to analyze what infinite structures can be represented with infinite EML trees

throwaway815233• 3 months ago

It's news to me that "elementary functions" include roots of arbitrary polynomials, but the wiki article in fact says that they're included at least some of the time. I remember reading about the Risch algorithm (for finding closed form antiderivatives) a long time ago and elementary functions were just the ordinary ones found on calculators.

Interestingly, the abs (absolute value) function is non-elementary. I wonder if exp-minus-log can represent it.

traes• 3 months ago

abs(x) = sqrt(x*x), no?

throwaway815233• 3 months ago

I think the issue might be the branch cut in the sqrt function. Per the wiki article, elementary functions have to be differentiable in the complex plane at all but a finite number of points.

gus_massa• 3 months ago

The origianl article use complex numbers, in particular to get sin and cos from eml:

e^{iφ} = cosφ + i sin φ

So x may be a complex number and sqrt(x*x) is a complex number that sometimes is equal to x and sometimes to -x depending on how lucky you were selecting the branches of sqrt.

reikonomusha15• 3 months ago

EML can represent the real absolute value, so long as we agree with the original author's proviso that we define log(0) and exp(-∞), by way of sqrt(x^2) as f(x) = exp((1/2)log x). Traditionally, log(0) isn't defined, but the original author stipulated it to be -∞, and that all arithmetic works over the "extended reals", which makes

    abs(0)
    = f(0)            ; by defn
    = exp(1/2 log 0)  ; by defn
    = exp(-∞/2)       ; log 0 rule
    = exp(-∞)         ; extended real arith
    = 0               ; exp(-∞) rule

If we don't agree with this, then abs() could be defined with a hole punched out of the real line. The logarithm function isn't exactly elegant in this regard with its domain restrictions. :)

throwaway815233• 3 months ago

It's ok for elementary functions to have singularities, like 1/x at x=0. But I'm not sure what happens with your version of abs, since the log function has branches. log(1) is any of 0, 2*pi*i, 4*pi*i, etc.

ogogmad• 3 months ago

On a tangent: I've tried to connect Euclid's Elements with quantifier elimination theorems. It looks like most of the geometry follows from QE of real-closed fields. Some of the number theory relates to Presburger arithmetic. Some other number theory, including the irrationality of sqrt(2), is down to Skolem. The Pythagorean triples relate to extending Skolem to the Gaussian integers. I suspect some of the "embryonic" integral calculus could be related to holonomic functions, which seem like they admit a form of QE.

Don't have anything for the perfect numbers though.

cubefox3• 3 months ago

Where would EML expressions sit in this fascinating table?

https://en.wikipedia.org/wiki/Template:Mathematical_expressi...

arowthway2• 3 months ago

Closed-form expressions, I guess?

"For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions."

cubefox3• 3 months ago

Maybe I'm misunderstanding, but the table says that the "root of a polynomial that is not an algebraic solution" doesn't count as closed-form.

shiandow• 3 months ago

That's a kind of weak criticism. What functions are considered elementary was always going to be arbitrary, picking the set you can generate from exp, log, and some complex algebra is not the worst choice.

If nothing else you could solve simple differential equations with them. And it gives you the 'power' function.

The very fact that the set of functions is largely arbitrary is a much bigger issue. Or at least it limits the use of the fact that you can represent those functions.

Edit: I feel the need to add that just because it is a weak critique doesn't mean the argument itself is not interesting.

timschmidt• 3 months ago

Codex and I whipped this up based on the paper: https://github.com/timschmidt/emlmath

Tests for the trig functions aren't passing yet due to an issue with the derived eml form in some mirrored cases.

dukezzz• 3 months ago

How can an edit possibly be made on April 16 when today is the 15th?

mmarx• 3 months ago

It's already past midnight in New Zealand, e.g.

empath753• 3 months ago

I wish i had seen this a few days ago i spent days on this before arriving at the following:

It’s completely wrong, for multiple reasons.

First, and immediately, none of the derived total functions are the functions they appear to be, because they are all partial functions.

Worse than that the domain over which these trees are defined is undecidable by Richardson. You can encode Hilbert’s 10th with elementary functions. Which means you can’t algorithmically decide if a tree evaluates to zero so you never know if ln(y) is undefined

So the central claim is done right off the bat.

Secondly and probably more importantly, these trees have exponential blow up problems because even fairly shallow trees involve tetrations of e that don’t cancel. Even in float64 you can‘t add 800+800 without overflow. Clamping it just gives you wrong answers and clamped exp A) isn’t elementary and B) gives bounded growth so can’t generate all elementary functions.

The consequences for his claimed practical application is disastrous. the loss function is NaN everywhere once you get to around depth 6 in his trees. If you clamp it you just get a flat plateau. Even if you had infinite precision the gradient would be unusable and wouldn’t converge.

As a second note, he frequently in the paper talks about using positive reals as inputs. First, positive reals aren’t even part of his generating set (and if they are his whole central claim is wrong, it has uncountably many constants), but even if they are, fundamentally his operations are complex, and you have subtraction so it is not hard to construct trees that should be total but which never the less are undefined at undecideably many points.

Using extended reals doesn’t save it either. There are just different undefined values that lead to undecideability.

None of these arguments are really specific to EML either. Any binary operator that hopes to generate elementary functions must include exp and ln, must be partial and must be able to encode hilbert’s 10th. There is no fix for this.

You might ask why his tests didn’t find any of this but if you look at his code, they do. He just carefully restricts them to a narrow domain where they happen to work and even then he does things like drop imaginary components and filters out undefined results. He doesn‘t even really hide this in the paper, he just dismisses them as implementation details that can be fixed but the problems are structural

So what we are left with is a generator that can generate exp(x) cleanly and essentially nothing else.

Not all elementary functions can be expressed with exp-minus-log

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