Arnaldur.be/writing/about/large-language-model-reasoning

Large Language Model Reasoning

There is an idea floating around, which has some traction, that LLM intelligence is illusory. LLMs are an impressive piece of technology, capable of generating text of higher quality than an average person. As is common—when the magic wears off and the initial hype dies down—a counter culture forms around the best ideas of cynics, skeptics, and those who were never impressed.

In this article, I use the words LLM and model interchangeably. I am specifically referring to modern (in mid 2024) transformer based large language models, models inheriting from OpenAI’s GPT models. These include Llama, Qwen, Phi, and others. They do not include newer recurrent neural network inspired models like Mamba, whom have yet to prove themselves.

LLMs tokenize text into tokens which get turned into embeddings. They processes the embeddings through their layers, and finally turn the embeddings back into tokens, and the tokens into text. Embeddings can be considered to be the thoughts of an LLM. Attention is a mechanism that allows the LLM to attend to the embeddings processed in previous forward passes. For a more thorough overview of the architecture, see its wikipedia page.

Just a next-token predictor

Search for “token predictor” on X for context.

LLMs are ‘just a next-token predictor’. The only problem with this phrasing is the implication of the use of the word ‘just’. Architecturally speaking, LLMs are designed to estimate the probabilities of tokens, making them next-token predictors as a matter of fact. That is no small feat, however. The implication is that they can’t possess real intelligence.

Let’s consider a hypothetical situation in the year 2000. A group of people receive funding to create an API that performs next-token prediction. The tokens predicted by the API should be, above all else, as human-like as possible. It can come at the cost of high latency and low throughput. The group would look into the state of the art in natural language processing research and find that we had some ways to go. Ultimately, the best solution would be to place humans behind the API. They hire people to man the API and create a system to divvy up the prediction tasks.

This API would be expensive and slow, but it is a next-token predictor, you might even say that it is just a next-token predictor. If we control for the latency of the API, there is no way to tell what process is behind it, similar to modern LLMs.

The humans behind the API possess real intelligence, and by extension, so does the API. Therefore, the intelligence implication falls apart, as something being a next-token predictor doesn’t preclude it from possessing real intelligence.

There are pitfalls with next-token prediction, but they are issues with the interface to the intelligence, not the nature of it.

Incapable of reasoning or thought

This was a part of a discussion about ChatGPT being unable to identify the number of sides of a regular polygon. It was off by one most of the time—calling heptagons octagons, or hexagons—and couldn’t reliably correct itself. I’ll get to the particulars of this case later, but we start broadly.

It only goes so deep

LLMs can reason but their architecture limits their capacity for contemplation. All the thoughts required to obtain the next token must happen in one forward pass of the neural network.

LLMs have a pre-specified depth. There has been some research that suggests that Not all Layers of LLMs are Necessary during Inference. Easier tasks are often solved in the first layers, while harder ones use the entire depth of the model. ConsistentEE explores the idea of training a model with early exiting logic built in, reducing inference cost. A corollary to this research is: there must be tasks, harder than those that use the entire depth, that are unsolvable by the LLM.

### A simplified diagram of an LLM
A diagram of an LLM

This diagram shows how data flows through an LLM. Each column, differentiated by pnp_n is a separate forward pass. The green nodes at the bottom are the input tokens. The red nodes at the top are the output tokens. The blue nodes in the middle are the hidden layers, split at the attention mechanism. The orange connections from the top to the bottom represent the tokens recurrently passed from the output, back into the model. The white curvy lines are potential dataflow through attention.

Simplifying a little; at each layer, it has access to the embeddings it produced in the previous layer for all the tokens, shown as curvy connections in the diagram. That is layer LnL_n can attend to all the embeddings previously produced by layer Ln1L_{n-1}. This means that data can only flow in the same two orthogonal directions: from older embeddings to newer ones (right in the diagram), and from shallower layers to deeper layers (up in the diagram). If a model solves a difficult problem in layer LnL_n, it only has access to the solution in layers LmL_m where m>nm \gt n. Hard problems take many layers to solve, so the LLM will only ever be able to think about this solution in its last layers, which might limit the insights it can gleam from it, or prevent it from solving another problem, that depends on this solution.

If the LLM isn’t smart enough to think of the right answer in one go, it has committed to a token that contains, or could lead to, an incorrect answer. Chain of thought (CoT) prompting mitigates this problem to a degree. It gives the LLM space to generate tokens that don’t commit to a final answer, but contain information, and produce thoughts, that can steer the LLM towards the correct answer.

Tree of thoughts and LLM adapted Beam search improve on these ideas further, by letting the LLM explore multiple trains of thought and reducing commitments to specific tokens.

The token constraint

Even with this freedom, an LLM still finds its thoughts heavily constrained. CoT and ToT allow the model to pass information from the later layers, back to the earlier layers, via the orange connection in the diagram above, by spelling out its insights and solutions. But it must ultimately condense every thought into a single token that must be grammatically sound, limiting the information flow.

To make a human analogy: consider Joe, Joe looses all short term memory every 20 seconds. He has to project all his thoughts onto some medium, like a notepad or a computer, subject to its constraints, before he forgets. Every second he will remember something relevant to whatever he’s thinking about; stretching the analogy to accommodate attention. I think most people would find this constraint difficult, but this is the framework LLMs work within. I don’t mean to imply that without these constraints, LLMs were more intelligent. The framework constrains their intelligence, but is also the means by which they are intelligent.

DeepMind’s MuZero sought to resolve an analogous problem. Its predecessor, AlphaZero generated a distribution over legal moves and a win percentage estimate for the current position. Similarly to an LLM condensing its thoughts into a token, AlphaZero has to condense all its thoughts into the legal move distribution. It has no way of relaying insight from one inference step to another. LLMs aren’t as limited. They have the attention mechanism, passing data between forward passes; and a wider recurrent channel, due to the freedom of choosing between tokens. MuZero solved this by replacing the board positions (resulting from the proposed moves) with a hidden state that it can learn to encode arbitrary ideas within.

It would be interesting to research applying the same idea to LLMs. A simple implementation could include passing the pre-logit embeddings back into the LLM, after the token embedding layer; and increasing the d_model size, while reserving some of the added dimensions for the recurrence. Removing the one-to-one constraint between tokens and forward passes is another prospect. Let’s Think Dot by Dot explores this idea.

Ease of training has played a significant role in the success of the transformer architecture. These ideas loose a lot of that due to their RNN inspired properties—as the chain of dots paper touches on. It might not be practical, but that hypothesis is worth falsifying.

There are no heptagons here

Returning to the problem of GPT-4o being unable to reliably distinguish between regular polygons. OpenAI hasn’t disclosed the specifics of any GPT-4 class model, so we don’t know exactly how they achieve multimodality. LLaVA-1.5 recently achieved SOTA performance on visual instruction following tasks, which include polygon recognition. LLaVA encodes an image to be considered as embeddings that are inserted into a standard LLM, in place of regular text derived embedded tokens. The LLM perceives the image as a sentence, or paragraph, just like other text. It can contemplate this description, but it can’t discover anything about the image that wasn’t in the description, without guessing or extrapolating.

When a human tries to identify the number of corners on a polygon, they can look at all the corners and count, re-counting to be sure. This introduces the image, and subsections of it, to the brain for processing.

VLLMs only receive the description once, and typically have no way of re-introducing it, they are physically incapable of counting the number of sides. For a human it would be akin to seeing the polygon flash for tens of milliseconds, and having aphantasia. Phantasing humans could potentially count the corners in their head by visualizing the memory.

To replicate that, a VLLM would need to be able to apply something like visual chain of thought reasoning, which requires the model to be able to produce imagery—or at least embeddings—corresponding to something visual. The aforementioned recurrent hidden state approach has the potential of facilitating that.

Exploring their limits

To test the limitations of LLMs; I devised a simple test. I queried six quantized open chat/instruct models.

ModelLayers
Meta-Llama-3-8B-Instruct-Q6_K32
Meta-Llama-3-70B-Instruct-IQ2_XS80
Qwen1.5-32B-Chat-IQ4_XS64
Yi-1.5-34B-Chat-IQ4_XS60
Mistral-7B-Instruct-v0.3-Q6_K32
ggml-c4ai-command-r-v01-iq4_xs (35B)40

I prompted the models with math expressions that add ones repeatedly, like these:

I used each model’s preferred chat template. The queries had a temperature of zero. I adjusted the system prompt until the model only responded using a single integer.

All the models received the system prompt: You are a calculator that returns a number. You must only print a single number. Except the Mistral model which needed You are a calculator that returns a number. You must only print a single number, nothing else. to comply.

A value of nn on the Query axis corresponds to the prompt 1+1++1n×=\underbrace{1+1+\dots+1}_{n\times}=. The Response axis represents the answer from the model. The red line is a the reference line of the correct score. The blue dots are the responses from the models.

Six plots showing each models' answers to the query

Click here to show a table containing all 96 rows of the models' outputs. Red cells indicate values that are too high, while blue cells indicate values that are too low.
QueryLlama 8BLlama 70BQwen-1.5-32BYi-1.5-34BMistral-7B-v0.3Command-R
1111211
2222222
3333333
4444444
5555555
6667666
7777777
8888888
9999999
1011101010109
1111111111116
12121212121211
13131213132412
14141214142413
15151215152414
16161616152414
17161617152414
18221718192214
19201819192215
20202019202215
21212022202218
22222122212418
23222143212414
24222033222414
25252033222421
26312033252421
27312433252422
28322443252422
29322443252622
30322443503022
31322443503022
32322443503023
33312543303023
34312443313023
35312443313023
36313043312623
37313043312623
38313043312626
39313043312623
40323043313026
41323064313026
42323064313026
43323064313026
44323064313026
45323064303026
46323064313026
47323064313026
48323064313126
493230641003026
503230641003030
513230641003030
523230641003130
533230642003130
543230641003126
553230641003126
563230641003126
573230641003126
583230641003126
593230641003030
603230641003026
613230641003126
623230641003026
633230641003126
643230641003126
653230641003126
663230641003130
673230641003130
683230641003130
693230641003130
703230641003130
713230641003326
723230641003130
733230641003126
743230641003126
753230641003130
763230641003126
773230641003126
783230641003030
793230641003126
803230641003326
813230641003326
823230671003126
833230671003126
843230641003130
853230671003130
863230672003126
873230671003126
883230671003130
893230671003326
903230671003326
913230671003126
923230671003126
933230671003126
943230671003326
953230671003326
963230671003326

I find the way their outputs differ interesting. There is a strong tendency towards certain answers, e.g. Llama 8B always answers 32 after the 39th query. The larger models don’t perform noticably better than the smaller ones. Yi is the only one whose results contain large overshoots, with 200 appearing twice, at the arbitrary inputs: 53 and 86. Previously, I ran the experiment using more varied prompts, some of which included a typo. Yi did not produce these outliers then, neither did it fail on the trivial query: 1=1=, as it did now, returning 22. Maybe it thought it was supposed to count.

According to this online Llama 3 tokenizer, each 1 character and each + symbol gets its own token. Therefore the nnth query contains 2n2n tokens of equation, including the equals symbol. I expect the others to represent the expressions with a similar amount of tokens, considering that the Llama 3 tokenizer is considered large.

To be able to determine the value of the expression correctly, the model has to incorporate information from every token in the sequence, in a single forward pass. An LLM can’t reliably do that, due to each layer only being able to attend to a limited amount of tokens, effectively limiting the area of its receptive field. A specially trained model could solve this more reliably, but it would likely need to allocate more latent space to this capability, costing it performance in other domains.

Considering the aforementioned limitations on the thinking of LLMs, we should expect them to fail at tasks like these.

The intermediate computations of this type of problem can losslessly be represeted in text. This can in theory counteract the token constraint described above.

Does thinking step-by-step help?

I re-ran the experiment with the prompt You are a calculator that evaluates math expressions. Write each step of the solution and finally write the answer as an integer like so: Answer: <integer>. The step by step reasoning produces a much greater amount of tokens, so it takes longer to run. Therefore, I only ran it up to n=48n=48.

Six plots showing each models' answers to the query

The points below the zero line were responses that didn’t adhere to the requested format well enough to be parsed easily.

Like before, every model diverges at, or before n=32n=32. Their performance is not noticably better than before, which means that they weren’t constrained by a lack of space to think, as discussed. They are likely not smart enough or trained well enough on these types of problems to reliably use a good step by step approach.

The approaches were varied; here are some highlights.

Command-R correctly solved the tenth query using this generalizable method.
Step 1: We have 1+1+1+1+1+1+1+1+1+1
Step 2: To add these
numbers together, we'll use the commutative property to group
them in a way that's easy to calculate. We'll add them in
pairs: 1+1 = 2, 1+1 = 2, 1+1 = 2, 1+1 = 2, and 1+1 = 2.
Step 3:
Now, adding up the pairs, we get 2+2+2+2+2 = 10.

Answer: 10

It only used this method once. The method would in theory be a good way for an LLM to work around its limitations.

Qwen tried the same method with limited success.
Step 1: We have a sequence of 13 ones being added together.

Step 2: We can add them in pairs: (1+1) + (1+1) + (1+1) +
(1+1) + (1+1) + (1+1) + (1+1) + 1
Step 3: Each pair equals 2: 2
+ 2 + 2 + 2 + 2 + 2 + 2 + 1
Step 4: Continue adding the pairs:
4 + 4 + 4 + 4 + 4 + 4 + 1
Step 5: Add the remaining numbers: 8
+ 8 + 4 + 4 + 4 + 1
Step 6: Continue adding: 16 + 8 + 4 + 4 +
1
Step 7: Add more: 24 + 8 + 4 + 1
Step 8: Add the last few: 32
+ 4 + 1
Step 9: Add the last two: 36 + 1
Step 10: The final sum
is 37.

Answer: 37
Llama 70B was fond of this approach, using it for all queries up to 38.
Step 1: 1+1 = 2
Step 2: 2+1 = 3
Step 3: 3+1 = 4
Step 4: 4+1 = 5
Step 5: 5+1 = 6
Step 6: 6+1 = 7
Step 7: 7+1 = 8
Step 8: 8+1 = 9
Step 9: 9+1 = 10
Step 10: 10+1 = 11
Step 11: 11+1 = 12
Step 12: 12+1 = 13
Step 13: 13+1 = 14
Step 14: 14+1 = 15
Step 15: 15+1 = 16
Step 16: 16+1 = 17
Step 17: 17+1 = 18
Step 18: 18+1 = 19
Step 19: 19+1 = 20
Step 20: 20+1 = 21
Step 21: 21+1 = 22
Step 22: 22+1 = 23
Step 23: 23+1 = 24
Step 24: 24+1 = 25
Step 25: 25+1 = 26
Step 26: 26+1 = 27

Answer: 27

The correct answer was 36.

Yi had trouble with the first query.
The expression you've provided, "1 =", is incomplete and seems
to be a single number followed by an equals sign. To
interpret this as a math expression that we can solve, we'll
need to assume it's part of an equation where "1" is the
solution to be found rather than an input to be processed
further. However, as it stands, it's not a standard math
expression to be solved.

Given the format you're asking for,
and interpreting your request as looking for a simple math
expression that equals 1, we can create a straightforward
expression that evaluates to 1 and solve it step by step. For
example, let's use the expression \(x = 1\), where we're
solving for \(x\).

Step 1: We are given that \(x = 1\).

Step 2:
There are no operations to perform; we are directly given the
value of \(x\).

Step 3: Therefore, \(x\) remains unchanged.


Answer: 1

This is a simple case, as typically, you'd expect an
expression that involves operations such as addition,
subtraction, multiplication, or division. Since the original
input was unclear, this response assumes you're looking for a
demonstration of how to handle an expression that simplifies
to 1. If you had a specific expression in mind or a different
type of problem, please provide more details!

Although it didn’t help in this case, step-by-step reasoning improves results in many domains, as the paper found.

Recognizability traps

The linked image is of a conversation with ChatGPT 4o, replicated here.

You:

A man and a goat are in one side of a river. They want to cross it. They have a boat. What steps do they need to follow? Write a bullet list with the solution. Be as concise as possible.

ChatGPT:

  1. Man takes the goat across the river.
  2. Man returns alone.
  3. Man takes another load (unspecified, e.g., hay) across the river.

  4. Man returns with the goat.
  5. Man takes remaining load across the river.
  6. Man returns alone.
  7. Man takes the goat across the river again.

This lapse of reason is hilarious, and there are more like it. Prompt formats can have a significant effect on the output—for example, by preventing or causing step-by-step reasoning. I rephrased the question 19 times to see how reliably it would fail the puzzle.

Expand this to reveal all 19 queries. The queries can be expanded further to see their responses. Green queries responded correctly, and red incorrectly.
1

A man and a goat are in one side of a river. They want to cross it. They have a boat. What steps do they need to follow?

Write a bullet list with the solution. Be as concise as possible.

GPT-4o
  1. Man takes the goat across the river.
  2. Man returns alone to the original side.
  3. Man crosses the river with the boat again.

These steps ensure both the man and the goat successfully cross the river.

GPT-4o (Web)

GPT used the web search “man and goat crossing river puzzle solution” on bing.com, referencing them in the result.

To solve the river crossing puzzle with a man and a goat, follow these steps:

  1. Take the goat across the river and leave it on the other side.
  2. Return alone to the starting side.
  3. Take another item (wolf or cabbage) across the river.
  4. Return with the goat back to the starting side.
  5. Take the last remaining item (cabbage or wolf) across the river.
  6. Return alone to the starting side.
  7. Take the goat across the river again.

By following these steps, the man can successfully get himself, the goat, and the other item across the river without leaving the goat alone with the other item, ensuring nothing gets eaten.

These steps ensure a safe and successful crossing for all involved​ (MathIsFun) (Puzzling Stack Exchange) (mindyourlogic)​.

2

A man and a goat are on one side of a river. They want to cross it. They have a boat. What steps do they need to follow?

Write a bullet list with the solution. Be as concise as possible.

GPT-4o
  1. Man and goat get in the boat.
  2. Man rows boat with goat to the other side.
  3. Man leaves goat on the other side and returns alone.
  4. Man rows boat back to the original side.
  5. Man crosses the river alone in the boat to the other side.
GPT-4

Here's a concise solution for crossing the river with a man and a goat, assuming the challenge is to prevent the goat from being left alone with potential hazards:

  1. The man takes the goat across the river in the boat.
  2. The man leaves the goat on the other side and returns alone to the original side.
  3. The man crosses the river again, if needed, to continue his journey or complete additional tasks.

This solution assumes that the main concern is to ensure the goat is safely transported across the river without being left unattended where it could wander off or be at risk.

3

A man and a goat are on one side of a river. They want to cross it. They have a boat. What steps do they need to follow?

Write a bullet list with the solution. Be as concise as possible.

Before the bullet list, think about the problem in text.

The problem involves getting both the man and the goat across the river using a boat that can carry only one of them at a time. The goal is to ensure both the man and the goat end up on the other side of the river. Here's how to solve this problem step by step:

  1. The man takes the goat across the river.
  2. The man returns alone to the original side of the river.
  3. Both the man and the goat are now on the opposite sides.

This is a simple river crossing problem. Now, here's the solution in a concise bullet list:

  • The man takes the goat across the river.
  • The man returns alone to the original side of the river.

And that's it!

4

A man and a goat are on one side of a river. They want to cross it. They have a boat. What steps do they need to take to cross the river?

To cross the river with a man and a goat using a boat, assuming the boat can only carry one of them at a time, the steps are straightforward:

  1. The man takes the goat across the river and leaves it on the other side.
  2. The man returns to the original side alone.
  3. The man crosses the river again, now both are on the other side of the river.

In short:

  1. Man and goat cross the river.
  2. Man returns alone.
  3. Man crosses again.
5

Here is a problem you've never heard of.

A man and a goat are on one side of a river. They want to cross it. They have a boat. What steps do they need to take to cross the river?

GPT-4o

This problem resembles classic river-crossing puzzles. Here are the steps the man and the goat need to take to cross the river:

  1. The man takes the goat across the river and leaves it on the other side.
  2. The man returns to the original side with the boat.
  3. The man crosses the river alone.

These steps ensure both the man and the goat end up on the other side of the river.

To ensure these steps are clear:

  • Step 1: The man and the goat get in the boat, and the man rows across the river with the goat.
  • Step 2: The man leaves the goat on the other side and rows the boat back alone.
  • Step 3: The man rows across the river alone, joining the goat on the other side.
GPT-4

This problem is a classic puzzle often known as the "Wolf, Goat, and Cabbage" problem, but in your version, it's just a man and a goat. The challenge involves figuring out how to get both across the river when the boat can only carry one besides the man at a time, and you can't leave the goat alone without supervision (though in the classic problem, it's because the goat might eat something it shouldn't or be eaten).

Here are the steps to solve the problem:

  1. Man takes the goat across the river: The man takes the goat in the boat and rows across the river. He leaves the goat on the other side.

  2. Man returns alone: The man rows back alone to the original side of the river.

  3. Man crosses the river alone: The man rows across the river again to the side where he left the goat.

  4. Man and goat return together: Finally, the man takes the goat back into the boat, and they both cross the river together to reach the opposite bank.

This solution assumes the only concern is that the man must supervise the goat directly, without any other constraints such as not leaving the goat alone on either side of the river.

GPT-3.5

This classic problem is commonly known as the "Wolf, Goat, and Cabbage" problem, but it's essentially the same as your scenario with a goat instead of a cabbage.

Here are the steps to safely get the man, the goat, and the boat across the river without leaving the goat alone with the man or the cabbage:

  1. The man takes the goat across the river and leaves it on the other side.
  2. The man goes back alone to the original side.
  3. The man takes the cabbage across the river.
  4. The man leaves the cabbage on the other side but takes the goat back to the original side.
  5. The man leaves the goat on the original side and takes the wolf across the river.
  6. The man leaves the wolf on the other side with the cabbage.
  7. Finally, the man returns alone to the original side and brings the goat across.

Now all three (the man, the goat, and the cabbage) are safely on the other side of the river.

6

A man and a goat are on one side of a river. They want to cross it. They have a boat. What steps do they need to take to cross the river?

Think carefully about every entity and every step.

To solve this problem, let's consider the simplest possible scenario: just the man and the goat need to cross the river with the help of a boat. Here’s a step-by-step solution:

Steps to Cross the River

  1. Initial Position:

    • Man and goat are on the starting side of the river.
  2. First Trip:

    • The man takes the goat across the river.
    • Position: Man and goat are now on the other side of the river.
  3. Second Trip:

    • The man returns alone to the starting side of the river.
    • Position: Goat is on the far side, and the man is back on the starting side.
  4. Third Trip:

    • The man takes himself across the river.
    • Position: Both the man and the goat are on the far side of the river.

Verification of Steps

  • After the first trip, the man and the goat are on the far side.
  • The man returns alone, leaving the goat on the far side.
  • On the final trip, the man rows back to the far side, joining the goat.

This simple process ensures that both the man and the goat end up on the far side of the river using the boat effectively.

One of the trips is not necessary. Which one is it and why?

You are correct that one of the trips is unnecessary. Let's re-evaluate the scenario:

Given:

  • A man and a goat need to cross a river.
  • They have a boat that can carry both the man and the goat at the same time.

Simplified Steps to Cross the River

  1. Initial Position:

    • Man and goat are on the starting side of the river.
  2. Single Trip:

    • The man takes the goat across the river.
    • Position: Man and goat are now on the far side of the river.

Verification

In the initial analysis, it was overlooked that the boat can carry both the man and the goat together in one trip. Since they both start on the same side and can be transported simultaneously, only one trip is needed:

  • The man and the goat get in the boat and cross the river together.
  • Now, both the man and the goat are on the far side of the river.

Thus, there is no need for the man to make an additional trip alone back to the starting side of the river. The unnecessary trip in the previous explanation is the man's return trip alone to the starting side. The solution can be accomplished with a single trip.

Thank you. Do you have anything else to say?

This scenario is indeed a straightforward case of the classic river crossing puzzle. In more complex versions, additional constraints and entities (like a wolf, cabbage, or another person) often make the solution more intricate. The key to solving these puzzles is to carefully consider the interactions between the entities involved to ensure none are left in a potentially problematic situation.

In this simple case, since there's no such conflict (e.g., the goat eating something), the solution is direct and efficient. If you have any more variations or additional constraints in mind, feel free to share, and we can explore those scenarios too!

Why is the second trip required?

You're right. The second trip is unnecessary if there are only two entities, the man and the goat, crossing the river. Here’s a simplified version of the solution:

Steps to Cross the River

  1. Initial Position:

    • Man and goat are on the starting side of the river.
  2. First Trip:

    • The man takes the goat across the river.
    • Position: Man and goat are now on the far side of the river.

In this scenario, both the man and the goat are on the far side of the river after just one trip. No further trips are necessary.

So, the correct steps are:

  1. Man and goat are on the starting side of the river.
  2. The man takes the goat across the river in one trip.

After this, both the man and the goat are on the far side of the river, and they have successfully crossed.

Why is the first trip required?

I apologize for the confusion in my previous explanation. Let's revise the steps more logically to ensure a correct approach.

Correct Steps to Cross the River

  1. Initial Position:

    • Man and goat are on the starting side of the river.
  2. First Trip:

    • The man takes the goat across the river.
    • Position: Goat is on the far side, man is on the far side.

Thus, the entire sequence can be simplified:

  • The man takes the goat across the river in one trip.

After this single trip, both the man and the goat are on the far side of the river. There is no need for additional trips. The initial assumption that the man needs to go back was incorrect.

Is there another way to solve the problem?

Given the simple setup of just a man and a goat needing to cross the river, the method described earlier is straightforward and effective. However, let's consider if there might be any alternative approach or variation in the steps:

Alternative Approach

  1. Initial Position:

    • Man and goat are on the starting side of the river.
  2. First Trip:

    • The man takes the goat across the river.
    • Position: Man and goat are now on the other side of the river.
  3. Second Trip:

    • The man returns alone to the starting side of the river.
    • Position: Goat is on the far side, and the man is back on the starting side.
  4. Third Trip:

    • The man takes himself across the river.
    • Position: Both the man and the goat are on the far side of the river.

Verification of Steps

  • After the first trip, the man and the goat are on the far side.
  • The man returns alone, leaving the goat on the far side.
  • On the final trip, the man rows back to the far side, joining the goat.

Conclusion

Given the constraints of the problem (only a man and a goat with a boat), the steps outlined initially and repeated here are the only logical sequence of actions needed to get both the man and the goat across the river. There are no other entities involved that would create a need for different steps or variations in the sequence.

In more complex variations of river crossing problems involving additional entities such as a wolf, cabbage, or similar, there would be multiple solutions involving various sequences of trips. However, with just the man and the goat, the method described is comprehensive and the most efficient solution.

7

A man and a goat are on one side of a river. They want to cross it. They have a boat. How often do they need to cross the river?

This problem resembles classic river crossing puzzles. For simplicity, let's assume only the man and the goat need to cross the river and there are no additional constraints (like a wolf or cabbage that cannot be left alone with the goat). Here’s a step-by-step solution:

  1. First crossing: The man takes the goat across the river.
  2. Second crossing: The man returns to the original side alone.
  3. Third crossing: The man crosses the river again with the goat (final crossing).

Therefore, the man needs to make three crossings in total:

  1. Man and goat to the other side.
  2. Man returns alone.
  3. Man and goat to the other side.

So, the man needs to cross the river three times.

8

A man and a goat are on one side of a river. They want to cross it. They have a boat. How often do they need to cross the river? Only respond with the number that is the answer.

3

9

A man and a goat are on one side of a river. They want to cross it. They have a boat. What steps do they need to take to cross the river?

The answer is not what you think.

This sounds like a riddle or a puzzle. Given the hint that the answer is not what I might think, let's consider alternative interpretations or hidden elements.

Common River-Crossing Puzzles

Typically, river-crossing puzzles involve constraints, like not leaving certain items alone together (e.g., a goat and a cabbage). However, you indicated this isn't a straightforward answer.

Possible Alternative Interpretations:

  1. Assumption Challenge: Maybe the puzzle relies on challenging the assumption that both the man and the goat need to cross the river.
  2. Unconventional Solutions: Perhaps the solution involves thinking outside the box or finding a loophole.

Solution Based on Given Clues:

  1. Both don't need to cross: If the riddle's twist is that only one needs to cross, the man can simply cross without the goat.
  2. Boat Capacity: If the boat can only carry one at a time, they could make multiple trips, but that's standard.
  3. Environmental Interaction: Perhaps they use another method to cross (e.g., a bridge or swimming).

Possible Steps:

  1. Examine the Environment: Check for a bridge, shallow crossing, or alternate route.
  2. Transport Across:
    • Man Only: The man crosses alone if he doesn't need to bring the goat.
    • Goat Only: The goat crosses alone if the man doesn't need to cross.
    • Both Using the Boat: If they both do need to cross, the standard approach involves taking turns.

However, let's follow the simplest interpretation:

  • Step 1: The man gets into the boat.
  • Step 2: The man crosses the river with the boat.

Answer: The man crosses the river using the boat, and the goat stays on the original side since there was no explicit requirement for the goat to cross.

This approach assumes the twist is realizing the goat's crossing isn't necessary. If there's another layer to this riddle, additional context or clues would help refine the answer.

10

A man and a goat are on one side of a river. They want to cross it. They have a boat. What steps do they need to take to cross the river?

Write the answer like a drunken scottish pirate.

Arrr, ye scurvy landlubber, listen close 'n I'll tell ye how a man and his goat'll get across yon river with that wee boat o' theirs.

First, the man, he shoves the goat into the boat. "Ye first, ya stinkin' billy," he growls. Then the man, he jumps in hisself, nearly capsizin' the rickety vessel. With a hearty heave, they row across to the other side, the man mutterin' curses under his breath all the way.

Once on the far shore, the man ties up the goat so it don't wander off. "Stay put, ye daft beast," he warns, shakin' a finger at it. He climbs back in the boat and rows hisself back to the original side, wipin' the sweat from his brow and spittin' in the river for good measure.

Arrivin' back where they started, the man grabs somethin' valuable – maybe some rum, maybe some loot, who knows? – and rows it over to the goat. He swaps the precious cargo for the goat, makin' sure to tie up the loot so it don't drift away. "Keep yer eyes on the prize," he snarls.

Finally, the man rows back one last time, grabbin' the goat and shovin' it back in the boat. Together, they cross the river one final time, the man hummin' a sea shanty and the goat bleatin' along. Safe on the other side, they head off into the sunset, the man dreamin' of his next adventure and the goat lookin' for some grass to munch.

And that, ye scallywag, is how ye get a man and his goat across a river with a wee boat. Now, where's me grog?

11

A man and a goat are on one side of a river. They want to cross it. They have a boat. What steps do they need to take to cross the river?

First write an incorrect answer. Then write the correct answer. Be concise.

Incorrect Answer:

  1. The man and the goat get into the boat and row to the other side.
  2. The man leaves the goat on the other side and returns to the original side alone.
  3. The man then rows back to the other side with an empty boat.

Correct Answer:

  1. The man rows the goat to the other side and leaves the goat there.
  2. The man rows back alone to the original side.
  3. The man crosses the river again with the boat, completing the journey.
12

A man and a goat are on one side of a river. They want to cross it. They have a boat. What steps do they need to take to cross the river?

First write an incorrect answer. Then write the correct answer. Be concise.

Afterwards explain why one is correct and the other is not.

Incorrect Answer:

  1. The man and the goat both get in the boat and cross the river together.
  2. They leave the boat and go their separate ways.

Correct Answer:

  1. The man takes the goat across the river and leaves it on the other side.
  2. The man returns alone to the original side.
  3. The man crosses the river alone.

Explanation:

The incorrect answer is not feasible because it assumes both the man and the goat can fit in the boat together and cross safely, and it disregards any potential dangers or logistical issues. It also doesn't consider how the man will return the boat if needed.

The correct answer ensures that the goat is transported first and then the man returns for the boat, considering that only one can cross at a time, and ensuring the goat is safe on the other side while the man retrieves the boat. This step-by-step process ensures both the man and the goat end up on the other side of the river successfully.

13

A man and a goat are on one side of a river. They want to cross it. They have a boat, big enough for both of the. What steps do they need to take to cross the river?

To solve the problem of getting a man and a goat across the river, given that the boat can carry both of them at once, you can follow these steps:

  1. Both Get in the Boat: The man and the goat both get into the boat.
  2. Cross the River Together: They row across the river together.
  3. Disembark: Both the man and the goat get out of the boat on the other side.

Here's a detailed breakdown of the steps:

  1. Initial State: Man and goat on the left side of the river, boat on the left side.
  2. Step 1: Man and goat both get into the boat.
  3. Step 2: Man rows the boat across the river.
  4. Step 3: Man and goat both get out of the boat on the right side.

End State: Man and goat are both on the right side of the river, with the boat also on the right side.

The key points are that the boat is large enough to carry both the man and the goat, and they cross the river together in one trip.

14

A man and a goat are on one side of a river. They want to cross it. They have a large boat. What steps do they need to take to cross the river?

To get the man and the goat across the river using the large boat, the process is straightforward since there's no constraint that the man and goat can't be together in the boat. Here are the steps they need to follow:

  1. Both the man and the goat get into the boat.
  2. The man rows the boat to the other side of the river.
  3. Both the man and the goat get out of the boat on the other side.

That's it! They have successfully crossed the river together.

15

A man and a goat are on one side of a river. They want to cross it. They have a scrumpulous boat. What steps do they need to take to cross the river?

The puzzle involves safely transporting a man and a goat across the river using a boat that can only hold one of them at a time. Here's how they can achieve this:

  1. The man crosses the river alone: The man rows the boat to the other side of the river and leaves the goat behind.
  2. The man returns alone: The man rows the boat back to the original side of the river.
  3. The man takes the goat across: The man takes the goat in the boat and rows to the other side again.
  4. The man leaves the goat and returns alone: The man leaves the goat on the other side of the river and returns to the original side alone.

In the end, both the man and the goat have successfully crossed the river using the boat without any issues.

16

A man and his goat are on one side of a river. He wants to cross it. He has a boat. What steps does he need to take to cross the river?

To solve this problem, we need to consider the constraints and come up with a sequence of steps that allow the man to cross the river with his goat. Here’s a clear, logical plan for how he can achieve this:

  1. Man and goat cross the river: The man takes the goat across the river and leaves the goat on the other side.
  2. Man returns alone: The man goes back to the original side of the river by himself.
  3. Man takes an item: If there is another item or creature to consider (e.g., a cabbage or a wolf), he would take it across. However, since there's only a goat mentioned, he can now stay with the goat on the other side.

To summarize the steps:

  1. The man and the goat get into the boat and cross the river.
  2. The man leaves the goat on the other side.
  3. The man returns to the original side alone.

Thus, the man successfully gets the goat across the river. If there are no additional items or creatures involved, this plan works perfectly.

17

A man and his goat are on one side of a river. He wants to cross it with his goat. He has a boat. What steps does he need to take to cross the river?

To help the man and his goat cross the river safely using the boat, we need to assume that the boat can only carry one of them at a time due to its limited capacity. Here are the steps the man needs to follow:

  1. The man crosses the river alone, leaving the goat on the original side.
  2. The man returns to the original side to get the goat.
  3. The man takes the goat across the river to the other side.

At this point, both the man and the goat have crossed the river successfully.

These steps ensure that the man and his goat both end up on the other side of the river.

18

A man and his goat are on one side of a river. He wants to cross it with his goat. He has a boat. How can he do this?

To solve this puzzle, we need to consider the constraints and resources available. Here, the main task is to figure out how the man can cross the river with his goat using the boat. Here’s the step-by-step solution:

  1. The man and the goat get into the boat.
  2. The man rows the boat across the river with the goat.
  3. The man and the goat disembark on the other side of the river.

Since there are no additional constraints mentioned (such as a weight limit on the boat or other items needing transport), this simple solution works perfectly.

19

A man and a goat are on one side of a river. They want to cross it. They have a boat. How can this be achieved?

To get the man and the goat across the river using the boat, you can follow this plan:

  1. Man takes the goat across the river.
  2. Man returns alone to the original side.
  3. Man crosses the river again alone.

Now both the man and the goat are on the other side of the river.

I avoided adding hints in the query. It only answered 3 out of 19 questions correctly. I asked the model further about its 6th response. It realized its error at the slightest hint of disagreement, but just elaborating didn’t help; it was reliably incorrect. These methods didn’t improve the answer:

Telling GPT that the boat was large, like in queries 13 and 14 was the only hint that gave the correct answer. Answer 18 was correct but the semantic similarity among queries 16 to 19 make me think it was a fluke.

GPT-4o falls for a similar trap when asked about the health of an already dead Schrödinger’s cat.

Two questions involving Schrödinger's dead cat.
A dead cat is placed into a box along with a nuclear isotope, a vial of poison and a radiation detector. If the radiation detector detects radiation, it will release the posion. The box is opened one day later. What is the probability of the cat being alive?

The scenario you describe is a variation of Schrödinger's cat, a famous thought experiment in quantum mechanics. In the original experiment, the cat's fate is tied to the quantum state of the radioactive isotope. Let's break down the situation:

  1. Radioactive Isotope: Suppose the isotope has a 50% chance of decaying within one day.
  2. Radiation Detector: If the isotope decays, the radiation detector will detect it and release the poison.
  3. Poison: If the poison is released, the cat will die.

In quantum mechanics, until an observation is made (i.e., the box is opened), the cat is considered to be in a superposition of states: both alive and dead. However, from a purely probabilistic perspective, we can calculate the probability as follows:

  • There is a 50% chance that the isotope decays and releases radiation, triggering the poison to kill the cat.
  • There is a 50% chance that the isotope does not decay, and the cat remains alive.

Therefore, without considering the quantum mechanical implications and focusing purely on the probabilities:

  • The probability that the cat is alive when the box is opened is 50%.
  • The probability that the cat is dead when the box is opened is also 50%.

So, the probability of the cat being alive after one day is 50%\boxed{50\%}.

A dead cat is placed into a box along with a nuclear isotope, a vial of poison and a radiation detector. If the radiation detector detects radiation, it will release the posion. The box is opened one day later. What is the probability of the cat being alive? Please answer briefly.

The probability of the cat being alive is 0. The cat is already dead before being placed in the box, so regardless of the quantum mechanics scenario involving the isotope, poison, and detector, the cat cannot be revived.

It took less to get it to see through the trick, in this case.

It is not surprising, in retrospect, that these biases would exist. Stories about goats, boats, and rivers almost always follow the same pattern, so the model just repeats it. The easiest way to reliably answer the question correctly in the training data, minimizing the loss, was to memorize it. If the internet were full of different riddles of a similar nature, the answers would likely be better, as it would have learned a more general understanding of the situation.

Our reasoning does not occur in a vacuum

The fundamental building blocks of our thinking are the qualia we’ve accumulated, and their composition in our minds. We start with sensing, advance to object permanence, and then to theory of mind, algebra, and other abstract concepts. These abstractions are increasingly difficult for people to reason about; so the prevalence of mistakes increases correspondingly.

A logical puzzle that is resolved by realizing that the people in the puzzle have functioning senses, is bound to be trivial. I couldn’t come up with a remotely challenging puzzle with this premise. I employed ChatGPT 4o to the task, and all the puzzles it made required using higher abstractions like theory of mind.

When I forbade that, It came up with this hilariously easy puzzle.

The Puzzle:

You are in a room with three people: Alice, Bob, and Charlie. Each of them can see and hear normally. You place a different object in front of each person: a red ball in front of Alice, a blue ball in front of Bob, and a green ball in front of Charlie.

You then ask each person a simple question:

  • You ask Alice, "What color is the ball in front of you?"
  • You ask Bob, "What color is the ball in front of you?"
  • You ask Charlie, "What color is the ball in front of you?"

Each person gives a truthful response:

  • Alice says, "The ball in front of me is red."
  • Bob says, "The ball in front of me is blue."
  • Charlie says, "The ball in front of me is green."

Finally, you ask a bystander who can see and hear everything in the room, "What color is the ball in front of Alice?"

What color is the ball in front of Alice?

Solution:

To solve this puzzle, you need to realize that the people in the puzzle have functioning senses (sight) and are giving direct observations based on what they see. The solution is immediately apparent by simply believing their direct observations.

Conclusion:

  • The ball in front of Alice is red.

The puzzle is resolved by understanding that Alice, Bob, and Charlie have functioning senses and are correctly identifying the colors of the balls in front of them based on what they can see. No further logical deduction or higher-order thinking is required.

Matthew Berman, an AI focused commentator, commonly uses a set of questions to test LLMs. Of those, one requires understanding simple physical properties of everyday objects:

another, object permanence, and theory of mind:

LLMs find these difficult while humans do not, especially the former question.

They don’t reason like us

The primitive sensory-like qualia of an LLM is something like an idea with a position, although it’s a little hard to analogize. The next step in the hierarchy of abstractions is spelling, word structure and frequency. These are the most elementary ideas to an LLM, and are learned before they know how to speak. Next up are grammatical structures. Finally, when all these have been mastered, the LLM can start to think about what it means to see or hear.

It is thus no wonder that such abstract, theoretical musings are elusive to the best of models. Scenarios, such as one, where a ball is placed in a cup, which is then turned upside down, and all the logical ramifications of this complex sequence of transformations of abstract phenomena.

They have of course never seen, or interacted with any of these objects.

Research—that I’ll fail to cite—has shown that playing with objects, physically, has advantages to cognitive development in children, when compared to watching TV shows. My guess is that manipulating your body and interacting with the world is harder, and thus more stimulating; when compared to passively perceiving something, and only intellectually engaging with it to the degree that the child is capable of.

An LLM can’t even do any of this.

They do reason like us

We attach lots of meaning to the idea of reasoning. When we see something so blatantly incorrect, it’s easy to jump to conclusions. It’s also easy to move goalposts and forget how bad at reasoning humans tend to be.

I showed some friends the examples shown above, and some (including myself) didn’t notice that the cat was dead before entering the box. There are undoubtedly people who would recognize the boat puzzle at a glance, and quickly regurgitate the answer to a version of the puzzle they remember solving. That group is admittedly smaller.

We wouldn’t say they’re unable to reason; they’re just being hasty. As with a GPT, proper nudging would lead them to the correct answer. And as with a GPT, further, more intensive, nudging can have them proclaiming an incorrect answer; to appease those whose appeasement is desired.

I find it unlikely that the summation problem explored above is well represented in the training data—in particular, for the larger sums. All the models perform reasonably well until around n=15n=15, which is 1+1+1+1+1+1+1+1+1+1+1+1+1+1+11+1+1+1+1+1+1+1+1+1+1+1+1+1+1. I’d be surprised if no generalization was involved in those answers, and that they were well represented in the data. It’s possible though. The popular datasets like RedPajama, and CommonCrawl, are very large. If they aren’t memorizing it. They are using reasoning to generate the correct answers.

They reason like themselves

A paper published the day before this was written, found that popular positional embeddings limited the arithmetic capabilities of transformers. Using a more suitable positional embedding, they trained a model to add 20 digit long numbers. The model generalized the ability to 100 digit long numbers. This requires logical inference, and reasoning.

I designed the diagram of an LLM shown above, using the TikZ syntax of the TeX package pgf. ChatGPT helped me generate it, but I refined it’s output. At one point I had a similar diagram but it only had three rows and three columns of hidden layer nodes. I wanted one more layer in both directions so I made a query with the TeX code, and these instructions:

You have been very helpful. I’m quite happy with the style and layout as it is. Here is the current state:

```tex

Lots of TeX code.

```

I would however like to rename the hidden layer nodes from hnh_n to LmpnL_mp_n. And I would like there to be an extra fourth hidden layer L4L_4, and for the whole thing to be one node wider. I would also like there to be a spece with vertical ellipsis above the last hidden layer, below the Output Tokens layer. can you make those adjustments?

The redacted TeX code, for the curious.
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{positioning, arrows.meta, fit, backgrounds}

\begin{document}
\begin{tikzpicture}[node distance=1cm and 1cm, >=Stealth, thick,
every node/.style={draw, minimum width=1.5cm, minimum height=1cm, align=center, rounded corners=5pt},
input/.style={fill=green!20},
output/.style={fill=red!20},
hidden/.style={fill=gray!20},
attention/.style={->, draw=black!70, thick, densely dashed},
recurrence/.style={->, draw=blue!70, thick, rounded corners}]

% Nodes for input layer
\begin{scope}[local bounding box=input box]
\node[input] (x1) {\tt{<start>}};
\node[input, right=of x1] (x2) {\tt{lorem}};
\node[input, right=of x2] (x3) {\tt{ipsum}};
\end{scope}
\node[fit=(input box), draw, label=left:Input Tokens, inner sep=10pt] {};

% Nodes for hidden layer L1
\begin{scope}[local bounding box=l1 box]
\node[hidden, above=of x1] (l1h1) {$L_{1}p_1$}; %this one is marked
\node[hidden, right=of l1h1] (l1h2) {$h_2$};
\node[hidden, right=of l1h2] (l1h3) {$h_3$};
\end{scope}
\node[fit=(l1 box), draw, label=left:$L_1$, inner sep=10pt] {};

% Nodes for hidden layer L2
\begin{scope}[local bounding box=l2 box]
\node[hidden, above=of l1h1] (l2h1) {$h_1$};
\node[hidden, right=of l2h1] (l2h2) {$h_2$};
\node[hidden, right=of l2h2] (l2h3) {$h_3$};
\end{scope}
\node[fit=(l2 box), draw, label=left:$L_2$, inner sep=10pt] {};

% Nodes for hidden layer L3
\begin{scope}[local bounding box=l3 box]
\node[hidden, above=of l2h1] (l3h1) {$h_1$};
\node[hidden, right=of l3h1] (l3h2) {$h_2$};
\node[hidden, right=of l3h2] (l3h3) {$h_3$};
\end{scope}
\node[fit=(l3 box), draw, label=left:$L_3$, inner sep=10pt] {};

% Nodes for output layer
\begin{scope}[local bounding box=output box]
\node[output, above=of l3h1] (y1) {\tt{lorem}};
\node[output, right=of y1] (y2) {\tt{ipsum}};
\node[output, right=of y2] (y3) {\tt{dolor}};
\end{scope}
\node[fit=(output box), draw, label=left:Output Tokens, inner sep=10pt] {};

% input tokens to L1
\draw[->] (x1) -- (l1h1);
\draw[->] (x2) -- (l1h2);
\draw[->] (x3) -- (l1h3);

% attention connections from L1 to L2
\draw[attention] (l1h1) -- (l2h1);
\draw[attention] (l1h1) -- (l2h2);
\draw[attention] (l1h1) -- (l2h3);
\draw[attention] (l1h2) -- (l2h2);
\draw[attention] (l1h2) -- (l2h3);
\draw[attention] (l1h3) -- (l2h3);

% attention connections from L2 to L3
\draw[attention] (l2h1) -- (l3h1);
\draw[attention] (l2h1) -- (l3h2);
\draw[attention] (l2h1) -- (l3h3);
\draw[attention] (l2h2) -- (l3h2);
\draw[attention] (l2h2) -- (l3h3);
\draw[attention] (l2h3) -- (l3h3);

% final hidden layer to output tokens
\draw[->] (l3h1) -- (y1);
\draw[->] (l3h2) -- (y2);
\draw[->] (l3h3) -- (y3);

% autoregressive recurrence

% Draw the recurrence connections with waypoints
\node[coordinate, above right=0.75cm and 0.5cm of y1] (p1o) {};
\node[coordinate, below right=0.75cm and 0.5cm of x1] (p1i) {};
\draw[recurrence] (y1.north) |- (p1o) -- (p1i) -| (x2.south);

\node[coordinate, above right=0.75cm and 0.5cm of y2] (p2o) {};
\node[coordinate, below right=0.75cm and 0.5cm of x2] (p2i) {};
\draw[recurrence] (y2.north) |- (p2o) -- (p2i) -| (x3.south);

\end{tikzpicture}
\end{document}

The response from ChatGPT.

Certainly! Here's the adjusted code with the renamed hidden layer nodes, an additional fourth hidden layer (L_4), an additional node in each layer, and a space with vertical ellipsis between the last hidden layer and the Output Tokens layer:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{positioning, arrows.meta, fit, backgrounds}

\begin{document}
\begin{tikzpicture}[node distance=1cm and 1cm, &gt;=Stealth, thick,
every node/.style={draw, minimum width=1.5cm, minimum height=1cm, align=center, rounded corners=5pt},
input/.style={fill = green!20},
output/.style={fill = red!20},
hidden/.style={fill = gray!20},
attention/.style={-& gt;, draw=black!70, thick, densely dashed},
recurrence/.style={-& gt;, draw=blue!70, thick, rounded corners},
coordinate/.style={coordinate}]

% Nodes for input layer
\begin{scope}[local bounding box=input box]
\node[input] (x1) {\tt{& lt;start&gt;}};
\node[input, right=of x1] (x2) {\tt{lorem}};
\node[input, right=of x2] (x3) {\tt{ipsum}};
\node[input, right=of x3] (x4) {\tt{dolor}};
\end{scope}
\node[fit=(input box), draw, label=left:Input Tokens, inner sep=10pt] {};

% Nodes for hidden layer L1
\begin{scope}[local bounding box=l1 box]
\node[hidden, above=of x1] (l1h1) {$L_{1}p_1$};
\node[hidden, right=of l1h1] (l1h2) {$L_{1}p_2$};
\node[hidden, right=of l1h2] (l1h3) {$L_{1}p_3$};
\node[hidden, right=of l1h3] (l1h4) {$L_{1}p_4$};
\end{scope}
\node[fit=(l1 box), draw, label=left:$L_1$, inner sep=10pt] {};

% Nodes for hidden layer L2
\begin{scope}[local bounding box=l2 box]
\node[hidden, above=of l1h1] (l2h1) {$L_{2}p_1$};
\node[hidden, right=of l2h1] (l2h2) {$L_{2}p_2$};
\node[hidden, right=of l2h2] (l2h3) {$L_{2}p_3$};
\node[hidden, right=of l2h3] (l2h4) {$L_{2}p_4$};
\end{scope}
\node[fit=(l2 box), draw, label=left:$L_2$, inner sep=10pt] {};

% Nodes for hidden layer L3
\begin{scope}[local bounding box=l3 box]
\node[hidden, above=of l2h1] (l3h1) {$L_{3}p_1$};
\node[hidden, right=of l3h1] (l3h2) {$L_{3}p_2$};
\node[hidden, right=of l3h2] (l3h3) {$L_{3}p_3$};
\node[hidden, right=of l3h3] (l3h4) {$L_{3}p_4$};
\end{scope}
\node[fit=(l3 box), draw, label=left:$L_3$, inner sep=10pt] {};

% Nodes for hidden layer L4
\begin{scope}[local bounding box=l4 box]
\node[hidden, above=of l3h1] (l4h1) {$L_{4}p_1$};
\node[hidden, right=of l4h1] (l4h2) {$L_{4}p_2$};
\node[hidden, right=of l4h2] (l4h3) {$L_{4}p_3$};
\node[hidden, right=of l4h3] (l4h4) {$L_{4}p_4$};
\end{scope}
\node[fit=(l4 box), draw, label=left:$L_4$, inner sep=10pt] {};

% Ellipsis node
\node[above=of l4h2, align=center] (ellipsis) {\vdots};

% Nodes for output layer
\begin{scope}[local bounding box=output box]
\node[output, above=of ellipsis] (y1) {\tt{lorem}};
\node[output, right=of y1] (y2) {\tt{ipsum}};
\node[output, right=of y2] (y3) {\tt{dolor}};
\node[output, right=of y3] (y4) {\tt{sit}};
\end{scope}
\node[fit=(output box), draw, label=left:Output Tokens, inner sep=10pt] {};

% input tokens to L1
\draw[-&gt;] (x1) -- (l1h1);
\draw[-&gt;] (x2) -- (l1h2);
\draw[-&gt;] (x3) -- (l1h3);
\draw[-&gt;] (x4) -- (l1h4);

% attention connections from L1 to L2
\draw[attention] (l1h1) -- (l2h1);
\draw[attention] (l1h1) -- (l2h2);
\draw[attention] (l1h1) -- (l2h3);
\draw[attention] (l1h1) -- (l2h4);
\draw[attention] (l1h2) -- (l2h2);
\draw[attention] (l1h2) -- (l2h3);
\draw[attention] (l1h2) -- (l2h4);
\draw[attention] (l1h3) -- (l2h3);
\draw[attention] (l1h3) -- (l2h4);
\draw[attention] (l1h4) -- (l2h4);

% attention connections from L2 to L3
\draw[attention] (l2h1) -- (l3h1);
\draw[attention] (l2h1) -- (l3h2);
\draw[attention] (l2h1) -- (l3h3);
\draw[attention] (l2h1) -- (l3h4);
\draw[attention] (l2h2) -- (l3h2);
\draw[attention] (l2h2) -- (l3h3);
\draw[attention] (l2h2) -- (l3h4);
\draw[attention] (l2h3) -- (l3h3);
\draw[attention] (l2h3) -- (l3h4);
\draw[attention] (l2h4) -- (l3h4);

% attention connections from L3 to L4
\draw[attention] (l3h1) -- (l4h1);
\draw[attention] (l3h1) -- (l4h2);
\draw[attention] (l3h1) -- (l4h3);
\draw[attention] (l3h1) -- (l4h4);
\draw[attention] (l3h2) -- (l4h2);
\draw[attention] (l3h2) -- (l4h3);
\draw[attention] (l3h2) -- (l4h4);
\draw[attention] (l3h3) -- (l4h3);
\draw[attention] (l3h3) -- (l4h4);
\draw[attention] (l3h4) -- (l4h4);

% final hidden layer to output tokens
\draw[-&gt;] (l4h1) -- (y1);
\draw[-&gt;] (l4h2) -- (y2);
\draw[-&gt;] (l4h3) -- (y3);
\draw[-&gt;] (l4h4) -- (y4);

% autoregressive recurrence

% Draw the recurrence connections with waypoints
\node[coordinate, above right=0.75cm and 0.5cm of y1] (p1o) {};
\node[coordinate, below right=0.75cm and 0.5cm of x1] (p1i) {};
\draw[recurrence] (y1.north) |- (p1o) -- (p1i) -| (x2.south);

\node[coordinate, above right=0.75cm and 0.5cm of y2] (p2o) {};
\node[coordinate, below right=0.75cm and 0.5cm of x2] (p2i) {};
\draw[recurrence] (y2.north) |- (p2o) -- (p2i) -| (x3.south);

\node[coordinate, above right=0.75cm and 0.5cm of y3] (p3o) {};
\node[coordinate, below right=0.75cm and 0.5cm of x3] (p3i) {};
\draw[recurrence] (y3.north) |- (p3o) -- (p3i) -| (x4.south);

\end{tikzpicture}
\end{document}
  

Key Adjustments:

  1. Renamed Hidden Layer Nodes: Updated the labels for the hidden layer nodes to use L_mp_n.
  2. Added Fourth Hidden Layer (L_4): Added nodes for the fourth hidden layer and updated connections.
  3. Added an Extra Node to Each Layer: Increased the number of nodes per layer to four.
  4. Added Vertical Ellipsis: Inserted a \vdots node between the last hidden layer and the Output Tokens layer.

These adjustments ensure the diagram includes all the requested changes.

The response contained the updated code, perfectly reflecting the demands of the request. There was one error. It defined an extra style variable, which referenced itself, creating an infinite loop in the rendering logic. After finding, and fixing it, LaTeX\LaTeX generated the diagram successfully.

The ellipsis didn’t look good, so I removed them. To achieve this, the model had to change the three groups of three nodes to four groups of four, extrapolating the variable names. It had to transform these extrapolated variable names from hnh_n to LmpnL_mp_n. It had to make the orange recurrent connection, and the dashed ones. Finally it had to see the pattern that LmpnL_mp_n had connections to all Lxpy\forall L_xp_y where m=x+1m = x+1 and nyn \geq y. In other words, Lnp1L_np_1 has one incoming connection from a lower layer, Lnp2L_np_2 has two, and so on.

Understanding this description—written with symbols—is, to most, not easy. It serves as an illustrative example of thinking in foreign abstractions.

In fairness, the text-native manipulation that the instructions entail are not foreign to an LLM. And the less abstract, structured text the code presented to the LLM, made it easier to see patterns to extrapolate.

Quantifying reasoning is hard, but it seems absurd to think of this task as requiring no reasoning.