[smeeth]

The main limitation of tokenization is actually logical operations, including arithmetic. IIRC most of the poor performance of LLMs for math problems can be attributed to some very strange things that happen when you do math with tokens.

I'd like to see a math/logic bench appear for tokenization schemes that captures this. BPB/perplexity is fine, but its not everything.

[cschmidt]

This paper has a good solution:

https://arxiv.org/abs/2402.14903

You right to left tokenize in groups of 3, so 1234567 becomes 1 234 567 rather than the default 123 456 7. And if you ensure all 1-3 digits groups are in the vocab, it does much better.

Both https://arxiv.org/abs/2503.13423 and https://arxiv.org/abs/2504.00178 (co-author) both independently noted that you can do this with just by modifying the pre-tokenization regex, without having to explicitly add commas.

[nielsole]

Isn't that the opposite of the bitter lesson - adding more cleverness to the architecture?

[jvanderbot]

Ok great! This is precisely how I chunk numbers for comparison. And not to diminish a solid result or the usefulness of it or the baseline tech: its clear that it we keep having to create situation - specific inputs or processes, we're not at AGI with this baseline tech

[Y_Y]

What do the vector space embeddings for digit strings even look like? Can you do arithmetic on them? If that's even desirable that it seems like you could just skip "embedding" altogether and intern all the numbers along one dimension.

[williamdclt]

Even if LLMs get better at arithmetic, they don't seem like the right tool for the job.

LLMs might never be able to crunch numbers reliably, however I expect they should be very good at identifying the right formula and the inputs for a problem ("i need the answer to x*y, where x=12938762.3 and y=902832.2332"). Then they can call a math engine (calculator or wolfram alpha or whatever) to do the actual computation. That's what humans do anyway!

[calibas]

It's a non-deterministic language model, shouldn't we expect mediocre performance in math? It seems like the wrong tool for the job...

[rictic]

Models are deterministic, they're a mathematical function from sequences of tokens to probability distributions over the next token.

Then a system samples from that distribution, typically with randomness, and there are some optimizations in running them that introduce randomness, but it's important to understand that the models themselves are not random.

[CamperBob2]

We passed 'mediocre' a long time ago, but yes, it would be surprising if the same vocabulary representation is optimal for both verbal language and mathematical reasoning and computing.

To the extent we've already found that to be the case, it's perhaps the weirdest part of this whole "paradigm shift."

[currymj]

thanks to training data + this being a popular benchmark, they're pretty good at grinding through symbolic mathematical derivations, which is often useful if you want an explanation of a mathematical concept. there's not really a better tool for this job, except for "a textbook which answers the exact question you have".

but from time to time, doing this does require doing arithmetic correctly (to correctly add two exponents or whatever). so it would be nice to be able to trust that.

i imagine there are other uses for basic arithmetic too, QA applications over data that quotes statistics and such.

[drdeca]

Deterministic is a special case of not-necessarily-deterministic.

[search_facility]

regarding “math with tokens”: There was paper with tokenization that has specific tokens for int numbers, where token value = number. model learned to work with numbers as numbers and with tokens for everything else... it was good at math. can’t find a link, was on hugginface papers

[samus]

Shouldn't production models already do this? They already tend to use tokenizers with complex rules to deal with a lot of input that would otherwise be tokenized in a suboptimal way. I recall a bug in an inference engine (maybe llama.cpp?) because of an implementation difference in their regex engine compared to the model trainer. Which means that the tokenizer used regex-based rules to chop up the input.

[vendiddy]

Do LLMs need to be good at math with the same approach?

To draw an a analogy, we've got our human brain specialized.

Why not implement a part of the AI brain that's not neural nets, but instead circuitry specialized to math?

Maybe a dumb question since I'm a layperson!

[js8]

It's not strange at all. I am playing with lambda calculus and combinatory logic now, as a base for mathematics (my interest is to understand rigorous thinking). You can express any computation using just S and K combinators, however, there is a price to that - the computations will be rather slow. So to make the computation faster, we can use additional combinators and rules to speed things up (good example is clapp() function in https://github.com/tromp/AIT/blob/master/uni.c).

Of course, the extra rules have to be logically consistent with the base S and K combinators, otherwise you will get wrong result. But if the inconsistent rule is complicated enough to be used only infrequently, you will still get correct result most of the time.

Which brings me to LLMs and transformers. I posit that transformers are essentially learned systems of rules that are applied to somewhat fuzzily known set of combinators (programs), each represented by a token (the term being represented by the embedding vector). However, the rules learned are not necessarily consistent (as it happens in the source data), so you get an occasional logical error (I don't want to call it hallucination because it's a different phenomenon from nondeterminism and extrapolation of LLMs).

This explains the collapse from the famous paper: https://ml-site.cdn-apple.com/papers/the-illusion-of-thinkin... One infrequent but inconsistent rule is enough to poison the well due to logical principle of explosion. It also clearly cannot be completely fixed with more training data.

(There is also analogy to Terry Tao's stages of mathematical thinking: https://terrytao.wordpress.com/career-advice/theres-more-to-... Pre-rigorous corresponds to soomewhat random set of likely inconsistent logical rules, rigorous to small set of obviously consistent rules, like only S and K, and post-rigorous to a large set of rules that have been vetted for consistency.)

What is the "solution" to this? Well, I think during training you somehow need to make sure that the transformer rules learned by the LLM are logically consistent for the strictly logical fragment of the human language that is relevant to logical and programming problems. Which is admittedly not an easy task (I doubt it's even possible within NN framework).