> LLMs are not particularly good at arithmetic, counting syllables, or recognizing haikus

I suspect most of this is due to tokenization making it difficult to generalize these concepts.

There are some weird edge cases though, for example GPT-4 will almost always be able to add two 40 digits number but it is also almost always wrong when adding a 40 digit and 35 digit number.

It doesn't have anything to do with tokenization. You can define binary addition using symbols, e.g. a and b, and provide properly tokenized strings to GPT-4. GPT-4 appears to solve the arithmetic puzzles for a few bits, but quickly falls apart on larger examples.

What I was saying is that because you need to go out of your way to make sure it's tokenized properly, I wouldn't be surprised if there are enough non properly tokenized examples in the dataset.

If that was the case, it would make it difficult to generalize these concepts.

Could it also be that syllables are intrinsically mechanical? They are strongly related to how our mouths work. While it may be possible to extract syllables from written text - following the consonants and vowels - I'm not sure that many humans could easily count syllables without using their mouths.

Many humans are also often really bad at doing speech related things when writing.

I've known many native English speakers who write things like "an healthy" (because they learned to write "an" before words starting with "h") and write poems that don't rhyme because the words end with the same letters (e.g. "most" and "cost").

Yeah, I find it weird how LLMs make a lot of the kind of mistakes that people do, but somehow this is held up as being a reason why LLMs don’t work similarly to brains.

Since discovering LLMs I’ve become convinced that my brain works like them. I really don’t know the next word I’m going to say until it’s nearly out. And since learning about how LLMs work, I really can’t argue it away.

It’s a reasonably disturbing feeling.

> LLMs are not particularly good at arithmetic, counting syllables, or recognizing haikus, though, because (contrary to the thesis of the article) they don’t magically acquire whatever ability would “simplify” predicting the next token.

LLMs understand it to a certain extent. It's more then "predicting" the next token. When people ascribe "predicting the next token" it's a niave and unintelligent description to cover up what they don't understand.

I mean you can describe a human brain as simply wetware, a jumble of signals and chemical reactions that twitch muscles and react to pressure waves in the air and light. But obviously there is a higher level description of the human brain that is missing from that description.

The same thing could be said about LLMs. I can tell you this, researchers completely understand token prediction that much can be said. What we don't currently understand is the high level description. Perhaps it's not something we can understand as we've never been able to understand human consciousness at a high level either.

That's the thing with people. Nobody actually understands the high level description of a fully trained LLM. People are lambasting others because they "think" they understand when they only actually understand the low level primitives. We understand assembly, but you don't understand the Operating system written in assembly.

Take this for example:

     Me: 4320598340958340958340953095809348509348503480958340958304985038530495830 + 1
     chatGPT: 4320598340958340958340953095809348509348503480958340958304985038530495830 + 1 equals 4320598340958340958340953095809348509348503480958340958304985038530495831.

The chances of chatGPT memorizing or even predicting the next tokens here are in a probability too low to even consider. There are so many possible numbers here even numbers that aren't true but have a "higher probability" of being close to the truth from a token standpoint. It's safe to say, from a scientific standpoint, chatGPT in this scenario understands what it means to add 1.

But there are, of course, things it gets wrong. But again we don't truly understand what's going on here. Is it lying to us? Perhaps it can't differentiate between just a generated statistical token or a actual math equation. It's hard to say. But from the example above, by probability, we know that an aspect of true understanding and ability exists.

I do think that LLMs have emergent properties that do some interesting things, however I would like to point out simple next token prediction would work on your example quite well.

<numbers>0 + 1 -> <numbers>1

Even simple attention mechanisms would handle that quite well with enough examples of <numbers>

I agree with you, but it also works for

    4320598340958340958340953095809348509348503480958340958304985038530999999 + 1 ?

    The sum of 4320598340958340958340953095809348509348503480958340958304985038530999999 and 1 is 4320598340958340958340953095809348509348503480958340958304985038531000000.

which is more complex.

I'm too lazy to get it to add two large numbers together.

Also, I've never been convinced that "ability to do arithmetic" has any relationship to intelligence. We don't expect regular humans to be able to add two large numbers together reliably, either.

This number is tokenized as a list: 43 20 59 83 409 58 340 9 58 340 95 30 95 809 34 850 9 34 850 34 809 58 340 9 58 30 49 850 385 30 49 58 30. If GPT recognizes the context "+ 1 equals" through the attention mechanism, it can predict that the next number in the sequence should be 31: ... 58 30 -> ... 58 31

Author here. First off, thank you for reading and for your thoughts. I provided examples that I thought would be intuitive for humans to help folks understand that an understanding of the underlying phenomena is useful for next token prediction (I've added this as a note). Could you share what part of the article came across as suggesting that LLMs "magically" acquire whatever ability helps them to predict? I'd like to make that section clearer, so that doesn't come across.

Re: "LLMs are not particularly good at arithmetic". There are published results that show that LLMs using certain techniques reach close to 100% accuracy on 8-digit addition: https://arxiv.org/pdf/2206.07682.pdf. There are also recent results from OpenAI where their model obtained solid results on high school math competition problems, which are harder than arithmetic: https://openai.com/research/improving-mathematical-reasoning... I haven't looked into counting syllables or recognizing haikus but I bet that this is a result of tokenization and not an inability of the model to create a representation of the underlying phenomena.

Thanks for responding to my comment.

I'm not an expert in the field, but, there are lots of previous algorithms for predicting the next token in a series (Markov chains, autocomplete). None of them felt so much pressure to make an accurate prediction that they had no alternative but to teach themselves arithmetic! It seems what is different about LLMs (as far as the post goes) is that we can anthropomorphize them.

More seriously, I guess I just feel like a meaningful sketch of an explanation for why algorithm X (where X is LLMs in this case) for continuing a piece of text is good at problem A should involve something about X and A. Because it is clearly highly dependent on the exact values of X and A, not just whether A can be posed as a text completion problem and humans would prefer the computer learn to solve the underlying problem to produce better text. For example, it could help to imagine a mechanism by which algorithm X could solve problem A. The closest thing to a mechanism (something algorithm X, i.e. LLMs, might be doing that's special) in the post is the talk of necessity being the mother of invention and "a deeper understanding of reality simplifies next-token prediction tasks," and the suggestion that if you were an LLM you might want to use "the rules of addition."

It's true that modeling arithmetic in some way could help a LLM account for known arithmetic problems in the training data, which could help it on unseen arithmetic problems, but what problems an LLM can solve is a function of what it can model. Anything an LLM can't model or can't do, it just doesn't. LLMs are really bad at chess, for example. The patterns of digits in addition may be similar enough to the hierarchical patterns in language the LLM is modeling. But it's not clear if the LLM is using the "rules of addition" or not. As far as I know, we don't actually understand why LLMs are able to store so much factual information, produce such coherent stories, and do the specific things they can do.

> what problems an LLM can solve is a function of what it can model.

Well said. The model that LLM has is very simple: If text X precedes current conversation then the most likely continuation of discussion is, according to the model held by LLM, Y. Right?

So the point is LLM does not create models. It has only a single model based on probabilities of text-sequences, created by its programmers. So it can (mostly?) only solve the problem of what would be a good textual response to an earlier text. It can do it well but most difficult problems don't fall into that category of "having a great chat".

What would be an example of a difficult problem?

A lot of things that LLMs can already do reliably don't fall into the category of "having a great chat" either. Examples include retrieving data from external sources using commands (known as "plugins" in ChatGPT / Langchain) or writing working code to calculate information needed for answers or to create artifacts, such as charts.

Yes, all of this stems from the task of continuing text. However, more and more, this is veering into the category of behavior. I don't mean "conscious behavior," but "behavior" nevertheless. It's surprising, but it is also the reality in which we currently live.

Hmm, yea, I agree with you on several points. For one, we don't fully understand the internal mechanisms of LLMs. I'm also with you on Markov chains and autocomplete tools not having an understanding of the underlying concepts. They merely use statistical patterns in the data.

Based on what you've said, it sounds like your take is that unless we can specify the exact mechanism by which LLMs understand, we have no business saying that they understand. In a lot of cases, this is a reasonable approach. In many areas, if someone tells you X, and you ask for a mechanism of action, and they can't produce one, you have solid grounds for thinking they're bullshitting.

But this case isn't quite the same. We know that LLMs learn to represent their inputs in a high-dimensional vector space (embeddings) and learn the relationships between those vectors. We also see them effectively solve problems in a variety of domains using this representation. I think these two ingredients: having a semantic representation and being able to effectively solve problems amount to something like "understanding." The lack of both properties is why I'd say Markov chains and autocomplete tools don't "understand" -- they haven't learned an effective representation of the underlying phenomena. (I'd also argue this is similar to us as humans. We don't have a good understanding of the human brain or precise mechanisms of action underlying thought. All we know is we as humans have semantic representations and can effectively solve problems.)

small note on your chess point: it now looks like chat gpt 3.5 can achieve draws against stockfish 8: https://marginalrevolution.com/marginalrevolution/2023/06/th...

bigger note on your chess point: this example illustrates that LLMs are "semi-decidable." We thought they were bad at chess, but we just hadn't discovered the right way to prompt. More generally, we can confirm when an LLM is good at X because we feed it a prompt that produces performance in X, but given the size of the input space we're dealing with here, we can't confirm that LLMs are bad at X just because we haven't seen them do well at it. Maybe we just haven't discovered the right prompt. (These input spaces are massive, by the way. ChatGPT-3.5, for example, has a context window of 4,096 tokens, so if we were considering only the English alphabet, we're looking at more than 26^{4,096} possible inputs.)

Two points in response to this:

I think it's a category error to call word embedding in a vector space "semantic" representation when discussing concepts like understanding. Semantics deals with the referents of words, but in this case there are no referents, merely a list of representational tokens which are defined as being "close in meaning" to the original due to proximity in text or some other structural characteristic. We call the embedding "semantic" because it is useful for human semantic purposes as we can mechanize some translations from one vector to another and receive a useful response that we then assign meaning to, but that usefulness doesn't indicate that the machine itself has any access to the referents of the tokens it's processing or semantic understanding. Put more simply, "semantics" does not merely mean the relationship between several ungrounded tokens, but that is all a vector embedding can accomplish.

Secondly, I think in the chess thread, the prompt being "engineered" in the example is extremely complex and constrains the output space sufficiently to produce high-quality results, but you start to wonder at what point the LLM is not doing most of the work. Meanwhile deeper in the thread we learn that even this prompting is not reliable and occasionally requires giving feedback that the move was bad(!) and repetition to achieve good results "the majority of the time in less than 3 tries". You can see where the practical problem arises, if we want to rely on LLMs for answers we don't already know. Claiming that we have a "general" function that "just" requires arbitrarily varying the input over an uncountably large space until you achieve the desired result is akin to saying f(x) = rand() * x is a universal computer as long as you find the right x. The ad absurdum version of the chess example is running Stockfish, sending a prompt that contains the Stockfish move and a request to repeat it, and then claiming that the LLM draws against Stockfish. However as we have seen with tokens like "SolidGoldMagikarp", LLMs are not even yet capable of reliably implementing the identity function, so I am not sure we can even say this.

Transformers don’t predict next tokens, right? They predict sequences based on their self-attention to some preceding token sequence?

No, what they do is predict a single token that follows the preceding token sequence (which was indeed analyzed using self-attention). Longer output sequences are created by repeating this simple task multiple times, where the previously output tokens become part of the preceding token sequence.

I'm surprised they're bad at predicting haikus.

I assume because there's little documentation about how many syllables every word has on the internet?

> LLMs are not particularly good at arithmetic

I'm reminded of "Benny's Rules", where someone sat down with a "self-directed" 6th grader of high IQ who had been doing okay in math classes... but their success so far was actually based on painstakingly constructing somewhat-lexical rules about "math", mumbo-jumbo that had been just good enough to carry them through a lot of graded tests.

> Benny believed that the fraction 5/10 = 1.5 and 400/400 = 8.00, because he believed the rule was to add the numerator and denominator and then divide by the number represented by the highest place value. Benny was consistent and confident with this rule and it led him to believe things like 4/11 = 11/4 = 1.5.

> Benny converted decimals to fractions with the inverse of his fraction-to-decimal rule. If he needed to write 0.5 as a fraction, "it will be like this ... 3/2 or 2/3 or anything as long as it comes out with the answer 5, because you're adding them" (Erlwanger, 1973, p. 50).

[0] https://blog.mathed.net/2011/07/rysk-erlwangers-bennys-conce...