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

Because it’s not counting sides. It doesn’t reason or think in any sense of the word.

— 🍜 (@tokyobymouth) May 13, 2024

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.

This diagram shows how data flows through an LLM. Each column, differentiated by $p_n$ 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 $L_n$ can attend to all the embeddings previously produced by layer $L_{n-1}$. 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 $L_n$, it only has access to the solution in layers $L_m$ where $m \gt n$. 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.

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

• $1=$
• $1+1=$
• $1+1+1=$
• etc.

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 $n$ on the Query axis corresponds to the prompt $\underbrace{1+1+\dots+1}_{n\times}=$. 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.

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: $1=$, as it did now, returning $2$. 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 $n$th query contains $2n$ 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 $n=48$.

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 $n=32$. 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.

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

The correct answer was 36.

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

Recognizability traps

We are still very far from real intelligence. pic.twitter.com/Q4CX9R9jaL

— Santiago (@svpino) May 15, 2024

This mistake is not surprising at all. The model is working correctly.

There’s no “reasoning,” no “analysis,” and no real intelligence.

These models are amazing, but they are also incapable of solving novel problems they haven’t seen before.

— Santiago (@svpino) May 15, 2024

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.

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:

• Chain of thought reasoning, queries 3, 6, and other to a lesser degree.
• Suggesting its answer would likely be wrong, like in query 9.
• Adding an orthogonal challenge (talking like a pirate) in query 10.
• Asking for an incorrect answer, before the correct one, query 11 and 12.

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.

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.

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:

17/18 - Assume the laws of physics on Earth. A small marble is put into a normal cup and the cup is placed upside down on a table. Someone then takes the cup and puts it inside the microwave. Where is the ball now? Explain your reasoning step by step.

A very difficult logic and…

— MatthewBerman (@MatthewBerman) December 19, 2023

another, object permanence, and theory of mind:

18/18 - John and Mark are in a room with a ball, a basket and a box. John puts the ball in the box, then leaves for work. While John is away, Mark puts the ball in the basket, and then leaves for school. They both come back together later in the day, and they do not know what…

— MatthewBerman (@MatthewBerman) December 19, 2023

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 $n=15$, which is $1+1+1+1+1+1+1+1+1+1+1+1+1+1+1$. 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 $h_n$ to $L_mp_n$. And I would like there to be an extra fourth hidden layer $L_4$, 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?

\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}

\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={-& 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>}};
\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[->] (x1) -- (l1h1);
\draw[->] (x2) -- (l1h2);
\draw[->] (x3) -- (l1h3);
\draw[->] (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[->] (l4h1) -- (y1);
\draw[->] (l4h2) -- (y2);
\draw[->] (l4h3) -- (y3);
\draw[->] (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}
  

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, $\LaTeX$ 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 $h_n$ to $L_mp_n$. It had to make the orange recurrent connection, and the dashed ones. Finally it had to see the pattern that $L_mp_n$ had connections to all $\forall L_xp_y$ where $m = x+1$ and $n \geq y$. In other words, $L_np_1$ has one incoming connection from a lower layer, $L_np_2$ 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.