The wall confronting large language models

There is a formal extensional equivalence between Markov chains & LLMs but the only person who seems to be saying anything about this is Gary Marcus. He is constantly making the point that symbolic understanding can not be reduced to a probabilistic computation regardless of how large the graph gets it will still be missing basic stuff like backtracking (which is available in programming languages like Prolog). I think that Gary is right on basically all counts. Probabilistic generative models are fun but no amount of probabilistic sequence generation can be a substitute for logical reasoning.

What does this predict about LLMs ability to win gold at the International Mathematical Olympiad?

Depends which question you're asking.

Ability to win a gold medal as if they were scored similarly to how humans are scored?

or

Ability to win a gold medal as determined by getting the "correct answer" to all the questions?

These are subtly two very different questions. In these kinds of math exams how you get to the answer matters more than the answer itself. i.e. You could not get high marks through divination. To add some clarity, the latter would be like testing someone's ability to code by only looking at their results to some test functions (oh wait... that's how we evaluate LLMs...). It's a good signal but it is far from a complete answer. It very much matters how the code generates the answer. Certainly you wouldn't accept code if it does a bunch of random computations before divining an answer.

The paper's answer to your question (assuming scored similarly to humans) is "Don’t count on it". Not a definitive "no" but they strongly suspect not.

The type of reasoning by the OP and the linked paper obviously does not work. The observable reality is that LLMs can do mathematical reasoning. A cursory interaction with state of the art LLMs makes this evident, as does their IMO gold medal scored like humans are. You cannot counter observable reality with generic theoretical considerations about Markov chains or pretraining scaling laws or floating point precision. The irony is that LLMs can explain why that type of reasoning is faulty:

> Any discrete-time computation (including backtracking search) becomes Markov if you define the state as the full machine configuration. Thus “Markov ⇒ no reasoning/backtracking” is a non sequitur. Moreover, LLMs can simulate backtracking in their reasoning chains. -- GPT-5

  > The observable reality is that LLMs can do mathematical reasoning

I still can't get these machines to reliably perform basic subtraction[0]. The result is stochastic, so I can get the right answer, but have yet to reproduce one where the actual logic is correct[1,2]. Both [1,2] perform the same mistake and in [2] you see it just say "fuck it, skip to the answer"

  > You cannot counter observable reality

I'd call [0,1,2] "observable". These types of errors are quite common, so maybe I'm not the one with lying eyes.

[0] https://chatgpt.com/share/68b95bf5-562c-8013-8535-b61a80bada...

[1] https://chatgpt.com/share/68b95c95-808c-8013-b4ae-87a3a5a42b...

[2] https://chatgpt.com/share/68b95cae-0414-8013-aaf0-11acd0edeb...

Why don't you use a state of the art model? Are you scared it will get it right? Or are you just not aware of reasoning models in which case you should get to know the field