decidability does not distribute over pointwise question asking on sets, or if you believe it does, show us the proof.

Telling if an EML(x,y),1 constructed expression is identically 0 is in the gray zone, as far as I can tell, it has neither been proven decidable nor been proven undecidable.

Nevertheless regardless of decidability the authors clearly show the multipoint sampling/testing is a decent filter, and the shorter resulting expressions have been proven correct in the results for the construction at least.

1) is there some value x such that some function F(x)=A(x)-B(x)=0?

2) is there some value x such that F(x)>0?

while you asked:

> I'm pretty sure it's not decidable if two EML trees describe the same function.

that would be

3) is for every x F(x)=A(x)-B(x)==0?

which Miklós Laczkovich's extension does not provide.

And you ignore the fact that Miklós Laczkovich's extension applies to real numbers and functions...

> 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.

Perhaps, perhaps not, same function so basically is this question solvable:

A(x[,y,...]) = f(x[,y,...])-g(x[,y,...]) == 0 everywhere?

if a user brings EML functions f and g; given their binary EML trees; can we decide if they represent the same function, so the question form is

A(x)=0 EVERYWHERE?

(like given 2 fractions a/b == c/d ? do the fractions represent the same fraction?)

From Wikipedia link reikonomusha gave:

> Miklós Laczkovich removed also the need for π and reduced the use of composition.[5] In particular, given an expression A(x) in the ring generated by the integers, x, sin xn, and sin(x sin xn) (for n ranging over positive integers), both the question of whether A(x) > 0 for some x and whether A(x) = 0 for some x are unsolvable.

Here the question forms are

1) exist x such that A(x) > 0 (does there exist an x where A(x) becomes positive?)

2) exist x such that A(x) = 0 (does there exist a value such that A(x) becomes 0? or basically find real roots

so at least the forms on WikiPedia don't generate the results both of you claim it does.

it does present undecidability results, but not straightforwardly in the context of this EML work.

second the Richardson's theorem is about the function on the reals, not complex functions (I mean the roots must lay somewhere)

I'm surprised traditional NLP being better than ML models for this task, can you point me to a benchmark analysis pointing out that non-neural Espeak-ng is better than ML models?

Also, I asked for a neural model for another reason as well, I still want semantic knowledge present, I want more than pronunciation, but before I use myself as a test subject, I want to make sure I get the proper pronunciation in case the highly speculative "uploading game" works... I don't want to early systematically mis-train myself on pronunciation...

I want to test a hypothesis for "uploading" neural network knowledge to a user's brain, by a reaction-speed game.

If legal arms dealers want the state to step in because of some decentralizing technology, then for the government it would be yet another cost center to combat this phenomenon. So lobbyists need to come up with a kind of reward, and design more "palatable" proposals, so that income can be derived by somehow initiating government control into the whole decentralized technology instead of just the illegitimately decentralized subsection...

but punishing "the rest" for actions of a few would mean financing it with taxes, instead of scapegoating the legitimate majority of 3D printer users.

I remind the trivial results that both E ⊆ EML and EML ⊆ E and hence EML = E

apart from construction: which is minimal for EML but highly redundant for E.

the EML paper shows that this minimal construction for EML is not unique so other binary operations may be found with perhaps more interesting properties, or admitting shorter binary trees for commonly used functions and values (which may reflect subjective "simplification" of expressions in mathematics.

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...)

> 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.

2)

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:

3)

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.

I still consider the article important, as it demonstrates techniques to conduct searches, and emphasizes the very early stage of the research (establishes non-uniqueness for example), openly wonders which other binary operators exist and which would have more desirable properties, etc.

Sometimes articles are important not for their immediate result, but for the tools and techniques developed to solve (often artificial or constrained) problems. The history of mathematics is filled with mathematicians studying at-the-time-rather-useless-constructions which centuries or millennia later become profound to human interaction. Think of the "value" of Euclid's greatest common divisor algorithm. What starts out as a curiosity with 0 immediate relevance for society, is now routinely used by everyone who enjoys the world wide web without their government or others MitM'ing a webpage.

If the result was the main claimed importance for the article, there would be more emphasis on it than on the methodology used to find and verify candidates, but the emphasis throughout the article is on the methodology.

It is far from obvious that the tricks used would have converged at all. Before this result, a lot of people would have been skeptical that it is even possible to do search candidates this way. While the gradual early-out tightening in verification could speed up the results, many might have argued that the approach to be used doesn't contain an assurance that the false positive rate wouldn't be excessively high (i.e. many would have said "verifying candidates does not ensure finding a solution, reality may turn out that 99.99999999999999999% of candidates turn out not to pass deeper inspection").

It is certainly noteworthy to publish these results as they establish the machinery for automated search of such operations.

Thats not really necessary, imagine somewhere near the top of the binary tree a leaf ("1" or "x" or ...), with the current brute force method thats a whole binary subtree with parameters going unused.

One could just as well use that whole culled binary subtree as a DAG node.

It does require more complex routing, but selecting input nodes is a sparse task, so those routing parameters can use sparse virtual parameters, say inner products of dense vectors in some vector space, so it doesn't need to take up much memory...

For extreme regularization, one can even go down to 10 arbitrary precision numbers: if we have a single vector of 10 dimensions, we can re-order the components 10! different ways.

10! = 3 628 800

so we can retrieve ~3M vectors from it, and we can form about 10 T inner products.

LLM's are capable of producing code that passes formal verification.

The writing is on the wall: in the future more and more software on the abstract or platonic side of our computing base will be hermetically sealed against bugs and exploits. This quenching of bugs in the assured side will shift the mean location of bugs closer to the hardware side: at some point bugs and exploits will rely more and more on hardware quirks, and simply unspecified hardware.

Afterwards we can expect a long exponential decay of preventable safety violations: people mistakenly or surreptitiously disengaging the formal verification steps and shipping malicious or unverified code. Each such event will be its own big or small scandal, at some point there will be no deniability left: something must be on purpouse, either a malicious vulnerability or intentional disengagement of safety measures.

As the attack surface recedes towards the lower level hardware stack, it will open the debate that the community needs proper formal hardware descriptions (at least at the interface initially, not necessarily how the hardware has implemented it). As interface bugs get formalized 3 things can happen:

either vulnerabilities go extinct, and full formal hardware descriptions are not released

or vulnerabilities remain in each new generation of hardware, and malicious intent or negligence on behalf of the manufacturer can only be presumed, this will set up the community against manufacturers, as they demand full hardware descriptions (verilog, VHDL,...).

or vulnerabilities are tactically applied (vulnerabilities appear extinct to the bulk of the population, but only because manufactured vulnerabilities are sparingly exploited by the manufacturing block)

It is hard to predict what is more valuable: embedding HW vulnerabilities for the status quo and being able to exploit it for while before the public demands full hardware IP descriptions (verilog, VHDL) etc. or facing the end of vulnerabilities a little sooner but keeping hardware IP private (if the bugs stop with full interface descriptions).

If one doesn't have free time, consider half the budget for the RPi+camera, then just find the closest makerspace or kid with an interest in electronics and ask them if they would like to hook it up for you, you saved half your expense, the kid has some more experience and some money to show for it.

Same with the amplifier knob, just needs a cheap wireless microcontroller, and sufficient low pass filtering on a PWM signal, perhaps a quad op-amp IC for the filtering and the voltage buffer...

Unless you had hit upon a very magical binary function where certain special values of the second parameter happens to coincide with useful unary functions, without those values trampling on a useful binary mode or region of your binary function, but the search space for such a special binary function is so large that you shouldn't demand us to disprove the existence, but rather employ your non-surprisal at the EML result and challenge you to present such a binary function, so we can challenge you to demonstrate how it captures binary functions like addition,products, exponentiation with arbitrary base etc.

So, can we see your construction, or if you refuse to present one, we may conclude you have implicitly reconsidered your position and understand the theoretical elegance this EML (and presumably many other) basis brings?

I did however keep thinking there was a lot of attention to trying to include special constants even though we don't know that much about these constants yet, while comparatively little attention went to say elliptic integrals etc.

When aiming for a wide catch, you'd include some of those esoteric functions, or erf() etc...

I also wished they had attempted to find a representation for derivative and integrals.