Traditionally they were done in FORTRAN or C or maybe bespoke DSP code, usually using ordinary boring complex numbers. Sometimes you’ll see FFTs over rings, called “number theoretic transforms”, used to multiply very large integers.

I suppose these days FFTs are used for highly power optimized, very much real-time OFDM transceivers. I’m guessing they’re done in carefully written assembly, possibly automatically generated (by conventional code, not LLMs, at least for existing solutions), possibly with a little bit of hardware help.

I think it's important to point out that DeepSeek was basically soft-launching their v4 model, and they weren't emphasizing it as some sort of SOTA-killer but more as proof of a potentially non-NVIDIA serving world, and as a venue for their current research approaches.

I think/hope we'll see a 4.2 that looks a lot better, same as 3.2 was quite competitive at the time it launched.

That's not what their latest post shows.

I can also implement compliant IEEE 754 floating point arithmetic, with all rounding modes and exceptions, in Conway’s Game of Life, and I can implement that on a Turing machine and emulate that Turing machine in Brainfuck. This does not mean it’s sensible — it would be a serious engineering project that should be decomposed, built in pieces, and tested or verified in pieces. Which even a Chinese LLM ought to be able to do in an appropriate environment with appropriate resources and an appropriate prompt.

But that is not what this is testing. And I’m not even sure which number system the mentioned roots of unity are in. Maybe it’s supposed to be generic over number systems in a delightfully untyped lambda calculus sort of way?

edit: after pasting the entire problem including the solution, ChatGPT is able to explain “GN numbers”. I suspect its explanation is sort of correct. But I get no web search results for GN numbers and I can’t really tell whether ChatGPT is giving a semi-hallucinated description or figuring it out from the problem and its solution or what.

Lambda calculus people have the phrase "adequate numeral system" because they've discovered many different numeral systems.

So interesting, thank you, I'm glad I asked.

I've been looking into tree calculus, which can "dissolve" lambda calculus and SKI combinators into natural (unlabelled) binary trees. I was wondering if you had any thoughts on it. It's a curious model of computation with unique properties, that I think can also convert to interaction nets or vice versa. I saw that Taelin tweeted about it once, dismissing it as misguided somehow ("not expressive nor efficient") but I feel like there's more to it.

EDIT: Oh I see you've commented on it before. Right, it's equivalent to SKI but worded differently. And this "intensionality", I guess no one has yet found practical use for that property.

> It's not based on a single combinator. It's based on the two standard basis combinators S and K, and a non-combinator that can inspect normalized terms to distinguish between degrees 0,1,and 2.

I quoted his tweet below, I'll keep it in case it's of any interest.

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it is not even related to kind or interaction nets, this calculus is simply not expressive nor efficient. it comes from a reasonable but incorrect intuition: "the less axioms you have, the more natural it must be!". that's not true. take nand logic, for example; it is complete, but expressing circuits with nand gates only will take more space than separating and/or/not, because you're just forcing ands/ors/nots to be inserted and computed where they're not needed. SKI combinators suffer from a similar trap; they're inherently slower and less expressive than the λ-calculus, because there are λ-terms that can't be expressed without "a lot of bloat" in them

this calculus does a similar thing: it packs many operations into a single one, and, by doing so, it bloats the program space and forces useless computations. yes, this makes the language feel more natural, because there are less axioms; but the axiom is just an amalgamation of many different functions in one. you want it the other way around: you want each axiom to add a key "expressivity" to your language, as tersely as possible, and in a way that makes the programs written on top of them as clean as possible, because everything you need to express is expressive from these axioms. yes, it is good to have as few axioms as possible (to avoid redundancy), but the expressivity of your axioms is what matters the most. the add function in tree calculus is a great example of a program that has been forced to be far from optimal due to the underlying languages constraint - that's terrible.

interaction combinators do the opposite: yes, the core language is extremely small and elegant - in fact, smaller than tree calculus - but, what is most important, it that its axioms allow terms written in it to be expressed in their shortest, cleanest, and most efficient formulation. interaction combinators provide exactly 3 interaction rules: erasure, which erases data; commutation, which copies data incrementally; and annihilation, which observes and moves data. from these 3, all else follows. loops and recursion, for example, can be optimally implemented (both in time AND space) by commutations. beta-reduction (i.e., λ-calculus) and pattern-matching (i.e., branching) can also be done optimally through annihilations. tuples and datatypes can also be encoded space-optimally with the same axioms. there is no waste, there is no bloat, which is make it so good to compute and to search.

in fact, you can't invent an axiomatic interaction system that is more efficient than interaction combinators, because they're optimal. and if, for some reason, interaction combinators scare you, the same holds for the recursive linear λ-calculus with addition of incremental copying, for example. in many of these senses, even this language is better to express algorithms on, than this proposed tree calculus

There is no doubt that for many tasks the SotA models of OpenAI and Anthropic are better than the available open weights models.

Nevertheless, I do not believe that either OpenAI or Anthropic or Google know any secret sauce for better training LLMs. I believe that their current superiority is just due to brute force. This means that their LLMs are bigger and they have been trained on much more data than the other LLM producers have been able to access.

Moreover, for myself, I can extract much more value from an LLM that is not constrained by being metered by token cost and where I have full control on the harness used to run the model. Even if the OpenAI or Anthropic models had been much better in comparison with the competing models, I would have still been able to accomplish more useful work with an open-weights model.

I have already passed once through the transition from fast mainframes and minicomputers that I was accessing remotely by sharing them with other users, to slow personal computers over which I had absolute control. Despite the differences in theoretical performance, I could do much more with a PC and the same is true when I have absolute control over an LLM.

I am desperate for the tooling that puts me back in charge. And just has the models as advisor. In which case the "smart level" is just a dial.

I'm probably going to have to make it myself.

How many samples you need depends on the difference you want to be able to measure (0% to 1% is different from 50% to 51% is different from 0% to 10% is different from 50% to 60%), the significance level at which you will declare a difference (conventionally, p < 0.05) and how likely you want this to happen when there is indeed such a difference (statistical power, conventionally 80%). Of course you can also just sample an arbitrary number of times and compute confidence intervals after the fact, but doing a statistical power computation helps clarify what it is you want to know, how certain you want to be, and whether you can realistically achieve such knowledge with the budget you have.