Transformers Linearly Represent Highly Structured World Models

The Tower of Hanoi is a simple planning puzzle that in prior work has proven challenging for large reasoning models (LRMs). Current models solve the standard formulation of the puzzle, but still struggle with the flat-to-flat variant (where initial and goal states are not restricted to have all rings on a single peg). This paper presents an in-depth study of how both small, in-house Transformers and large, third-party LRMs solve this task. To understand the failures mechanistically, we first train small Transformers from scratch on precomputed solution traces. Using a variety of interpretability techniques, we show that these Transformers develop an emergent world model: a linearly decodable, geometrically faithful representation of the puzzle's state space (the Sierpinski triangle), that is causally involved in solving the puzzles. Second, we return to the large LLMs and apply our techniques to two frontier reasoning models, Qwen3.6-27B and DeepSeek-R1-Distill-Qwen-32B, that attempt to solve the task through extended chain-of-thought. Surprisingly, we find that both models encode the Sierpinski world model near-perfectly at the end of the prompt, and yet fail at the majority of tasks when there are more than 3 rings. We locate the source of this failure in the decaying representation of the world model. We probe for the representation at different stages during planning, and establish causality by showing that performance can be improved by injecting the prompt-time representation at inference. The failure of the models is thus one of maintenance of the required representations, not their absence, and performance is at least partially recoverable. These results thus reframe the reported collapse in performance from prior work: current Large Reasoning Models build a world model, and then lose it.

Existing expressivity results for transformers typically rely on hardmax attention, high precision, and other architectural modifications that disconnect them from the models used in practice. We bridge this gap by analyzing standard transformer decoders with softmax attention and rounding of activations and attention weights, while allowing depth and width to grow logarithmically with the context length. As an intermediate step, we construct hardmax transformers with ternary activations and well-separated attention scores that simulate Turing machines using Chain-of-Thought (CoT). This lets us convert the constructions to equivalent softmax transformers without the unrealistic parameter magnitudes or activation precision that prior approaches would require. Using the same technique, we analyze a recently proposed summarized CoT paradigm and show that it simulates Turing machines more efficiently, with model size scaling logarithmically in a space bound rather than a time bound. We empirically test predictions made by our results on a Sudoku reasoning task and find better alignment with learnability than for prior high-precision results. Our code is available at https://github.com/moritzbroe/transformer-expressivity.

We introduce the Lattice Deduction Transformer (LDT), a recurrent transformer that approximates logically sound deduction by projecting its latent state through a lattice between forward passes. We train on-policy in a process that mirrors deduction in a search-based constraint solver and supervise training via a domain-agnostic, abstract-interpretation-based approximation of the set of solution candidates. An $800$K-parameter LDT achieves $100\%$ accuracy on Sudoku-Extreme and Snowflake Sudoku, at a fraction of the training cost of prior small recurrent reasoners, while remaining empirically sound: the model returns a correct answer or abstains. A $1.8$M-parameter variant reaches $99.9\%$ accuracy on Maze-Hard. Frontier LLMs score $0\%$ on all three benchmarks.