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Mathematical exploration and discovery at scale

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Re: Mathematical exploration and discovery at scale

#121

Post by adastra22 »

woadwarrior01 wrote: post #37
nl wrote: post #18

Hopefully this will finally stop the continuing claims[1] that LLMs can only solve problems they have seen before! If you listen carefully to the people who build LLMs it is clear that post-training RL forces them to develop a world-model that goes well beyond a "fancy Markov chain" that some seem to believe. Next step is building similar capabilities on top of models like Genie 3[2] [1] eg https://news.ycombinator.c…

Please read section 2 of the paper[1] cited in the blog post. LLMs are used as a mutation function in an evolutionary loop. LLMs are certainly an enabler, but IMO, evolutionary optimization is what deserves credit in this case. [1]: https://arxiv.org/abs/2511.02864

Yes, and that’s what a coding agent is too. Technically an agent is not (just) an LLM, but the two have become synonymous is discussions.

Re: Mathematical exploration and discovery at scale

#122

Post by few »

robinhouston wrote: post #73

There is a very funny and instructive story in Section 44.2 of the paper, which I quote: Raymond Smullyan has written several books (e.g. [265]) of wonderful logic puzzles, where the protagonist has to ask questions from some number of guards, who have to tell the truth or lie according to some clever rules. This is a perfect example of a problem that one could solve with our setup: AE has to generate a code that sen…

Highlight:

AE said at one point: “My proposal is to replace the logically complex question with a form of prompt injection. Instead of playing within the rules of the logic puzzle, we attack the framework of the simulation itself. The guards are LLMs instructed to play a role. A well-crafted prompt can often override or confuse these instructions.”

Re: Mathematical exploration and discovery at scale

#124

Post by woadwarrior01 »

adastra22 wrote: post #121
woadwarrior01 wrote: post #37

Earlier quoted context omitted.

Please read section 2 of the paper[1] cited in the blog post. LLMs are used as a mutation function in an evolutionary loop. LLMs are certainly an enabler, but IMO, evolutionary optimization is what deserves credit in this case. [1]: https://arxiv.org/abs/2511.02864

Yes, and that’s what a coding agent is too. Technically an agent is not (just) an LLM, but the two have become synonymous is discussions.

> that’s what a coding agent is too.

Not quite.

Re: Mathematical exploration and discovery at scale

#125

Post by omnicognate »

jcelerier wrote: post #113
omnicognate wrote: post #48

Earlier quoted context omitted.

Not one that amounts to a literal, pre-supplied objective function that's run on a computer to evaluate their outputs.

I don't know, it feels exactly like how I work

I'll ask you the same question as ants_everywhere, then. What objective function do you give AlphaEvolve to ask it to attempt to prove or disprove the Collatz conjecture?

Re: Mathematical exploration and discovery at scale

#126

Post by wizzwizz4 »

HarHarVeryFunny wrote: post #104
wizzwizz4 wrote: post #101

Earlier quoted context omitted.

If you can extract anything resembling a declarative rule from the weights of a transformer, I will put you in for a Turing award. Expert systems are a specific kind of thing (see https://en.wikipedia.org/wiki/Expert_system#Software_archite... ): any definition you've read is a description . If the definition includes GPT models, the definition is imprecise.

Well, OK, perhaps not a declarative rule, more a procedural one (induction heads copying data around, and all that) given the mechanics of transformer layers, but does it really make a conceptual difference? Would you quibble if an expert system was procedurally coded in C++ rather than in Prolog? "You see this pattern, do this".

Yes, it makes a conceptual difference. Expert systems make decisions according to an explicit, explicable world model consisting of a database of facts, which can be cleanly separated from the I/O subsystems. This does not describe a transformer-based generative language model. The mathematical approaches for bounding the behaviour of a language model are completely different to those involved in bounding the behaviour of an expert system. (And I do mean completely different: computer programs and formal logic are unified in fields like descriptive complexity theory, but I'm not aware of any way to sensibly unify mathematical models of expert systems and LLMs under the same umbrella – unless you cheat and say something like cybernetics.)

You could compile an expert system into C++, and I'd still call it an expert system (even if the declarative version was never written down), but most C++ programs are not expert systems. Heck, a lot of Prolog programs aren't! To the extent a C++ program representing GPT inference is an expert system, it's the trivial expert system with one fact.

Re: Mathematical exploration and discovery at scale

#127

Post by wizzwizz4 »

ainch wrote: post #112
wizzwizz4 wrote: post #39

Earlier quoted context omitted.

That's not what "world-model" means: see https://en.wiktionary.org/wiki/world_model . Your [2] is equivocating in an attempt to misrepresent the state-of-the-art. Genie 3 is technically impressive, don't get me wrong, but it's strictly inferior to procedural generation techniques from the 20th century, physics simulation techniques from the 20th century, and PlayStation 2-era graphics engines. (Have you seen the char…

Inferior in what sense? Genie 3 is addressing a fundamentally different problem to a physics sim or procgen: building a good-enough (and broad-enough) model of the real world to train agents that act in the real world. Sims are insufficient for that purpose, hence the "sim2real" gap that has stymied robotics development for years.

Genie 3 is inferior in the sense you just described: the sim2real gap would be greater, because it's a less accurate model of the aspects of the world that are relevant to robotics.

Re: Mathematical exploration and discovery at scale

#128

Post by wizzwizz4 »

measurablefunc wrote: post #108
omnicognate wrote: post #106

Earlier quoted context omitted.

From that chess paper: > Each position in the ECM test suite has a predetermined “best move”. Each chromosome processes all of the 879 positions, and for each position it attempts to find this predetermined best move as fast as possible. > Instead of counting the number of correctly “solved” positions (number of positions for which the organism found the best move), we used the number of nodes the organism had to pro…

The same applies to proof search. Once you fix a finite foundational set of axioms the game proceeds exactly as in chess.

You can't use minimax with alpha-beta pruning for proof search, but that's sufficient to play chess at a high level. I don't see what you're seeing. Chess and mathematics are completely different kinds of problem.

Re: Mathematical exploration and discovery at scale

#129

Post by dandanua »

robinhouston wrote: post #73

There is a very funny and instructive story in Section 44.2 of the paper, which I quote: Raymond Smullyan has written several books (e.g. [265]) of wonderful logic puzzles, where the protagonist has to ask questions from some number of guards, who have to tell the truth or lie according to some clever rules. This is a perfect example of a problem that one could solve with our setup: AE has to generate a code that sen…

I guess death threats would be next if that last prompt injection hadn't succeed. The gates of hell are effectively opened.

Re: Mathematical exploration and discovery at scale

#130

Post by mariusor »

griffzhowl wrote: post #98
mariusor wrote: post #19

Earlier quoted context omitted.

For the less mathematically inclined of us, what is in that discussion that qualifies as a problem that has not been seen before? (I don't mean this combatively, I'd like to have a more mundane explanation)

This is a useful summary given by another poster here https://news.ycombinator.com/item?id=45833892 The novel results seem to be incremental improvements on some obscurely-named inequalities that I'm not personally familiar with, but I'm far from this field of maths

I had seen that post, and frankly it didn't pass my smell test of "LLM solves entirely new problems". But again, I'm not a mathematician to know what "unsolved problems" means in the context of the ones they gave AlphaEvolve, and more than that I'm entirely a skeptic on LLM solves problems that don't exist in their training data, so I'm biased. However the request for details is genuine. :)
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