# Let's Think Dot by Dot: Hidden Computation in Transformer Language Models

arXiv: https://arxiv.org/abs/2404.15758 · HTML: https://mlsanity.com/paper/2404.15758/

## Key facts

- Cited by 191 papers in the map (data through 14 Oct 2026).
- Placed in Prompting and chain-of-thought reasoning, one of 17 directions in LLM reasoning.
- Builds on 14 of its 19 references that are in the map.
- First posted on arXiv 24 Apr 2024.

## Placed in

- LLM reasoning / [Prompting and chain-of-thought reasoning](https://mlsanity.com/lr/prompting-and-chain-of-thought-reasoning/)

## Builds on (most cited first, 14 of 14)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Challenging BIG-Bench Tasks and Whether Chain-of-Thought Can Solve Them](https://mlsanity.com/paper/2210.09261/) (2022)
- [Language Models Don't Always Say What They Think: Unfaithful Explanations in Chain-of-Thought Prompting](https://mlsanity.com/paper/2305.04388/) (2023)
- [Measuring Faithfulness in Chain-of-Thought Reasoning](https://mlsanity.com/paper/2307.13702/) (2023)
- [Designing and Interpreting Probes with Control Tasks](https://mlsanity.com/paper/1909.03368/) (2019)
- [Quiet-STaR: Language Models Can Teach Themselves to Think Before Speaking](https://mlsanity.com/paper/2403.09629/) (2024)
- [Towards Revealing the Mystery behind Chain of Thought: A Theoretical Perspective](https://mlsanity.com/paper/2305.15408/) (2023)
- [Think before you speak: Training Language Models With Pause Tokens](https://mlsanity.com/paper/2310.02226/) (2023)
- [The Parallelism Tradeoff: Limitations of Log-Precision Transformers](https://mlsanity.com/paper/2207.00729/) (2022)
- [Representational Strengths and Limitations of Transformers](https://mlsanity.com/paper/2306.02896/) (2023)
- [Future Lens: Anticipating Subsequent Tokens from a Single Hidden State](https://mlsanity.com/paper/2311.04897/) (2023)
- [Why are Sensitive Functions Hard for Transformers?](https://mlsanity.com/paper/2402.09963/) (2024)
- [A Logic for Expressing Log-Precision Transformers](https://mlsanity.com/paper/2210.02671/) (2022)
- [Do language models plan ahead for future tokens?](https://mlsanity.com/paper/2404.00859/) (2024)

## Built on by (most cited first, 30 of 191)

- [Training Large Language Models to Reason in a Continuous Latent Space](https://mlsanity.com/paper/2412.06769/) (2024)
- [Stop Overthinking: A Survey on Efficient Reasoning for Large Language Models](https://mlsanity.com/paper/2503.16419/) (2025)
- [RAGEN: Understanding Self-Evolution in LLM Agents via Multi-Turn Reinforcement Learning](https://mlsanity.com/paper/2504.20073/) (2025)
- [Inference Scaling Laws: An Empirical Analysis of Compute-Optimal Inference for Problem-Solving with Language Models](https://mlsanity.com/paper/2408.00724/) (2024)
- [CODI: Compressing Chain-of-Thought into Continuous Space via Self-Distillation](https://mlsanity.com/paper/2502.21074/) (2025)
- [A Survey on Test-Time Scaling in Large Language Models: What, How, Where, and How Well?](https://mlsanity.com/paper/2503.24235/) (2025)
- [Compressed Chain of Thought: Efficient Reasoning Through Dense Representations](https://mlsanity.com/paper/2412.13171/) (2024)
- [Thoughts Are All Over the Place: On the Underthinking of o1-Like LLMs](https://mlsanity.com/paper/2501.18585/) (2025)
- [Mixture-of-Recursions: Learning Dynamic Recursive Depths for Adaptive Token-Level Computation](https://mlsanity.com/paper/2507.10524/) (2025)
- [Efficient Reasoning Models: A Survey](https://mlsanity.com/paper/2504.10903/) (2025)
- [Token Assorted: Mixing Latent and Text Tokens for Improved Language Model Reasoning](https://mlsanity.com/paper/2502.03275/) (2025)
- [When Chain of Thought is Necessary, Language Models Struggle to Evade Monitors](https://mlsanity.com/paper/2507.05246/) (2025)
- [Relaxed Recursive Transformers: Effective Parameter Sharing with Layer-wise LoRA](https://mlsanity.com/paper/2410.20672/) (2024)
- [Reasoning by Superposition: A Theoretical Perspective on Chain of Continuous Thought](https://mlsanity.com/paper/2505.12514/) (2025)
- [Alice in Wonderland: Simple Tasks Showing Complete Reasoning Breakdown in State-Of-the-Art Large Language Models](https://mlsanity.com/paper/2406.02061/) (2024)
- [Understanding Transformer Reasoning Capabilities via Graph Algorithms](https://mlsanity.com/paper/2405.18512/) (2024)
- [Eliminating Position Bias of Language Models: A Mechanistic Approach](https://mlsanity.com/paper/2407.01100/) (2024)
- [LoopFormer: Elastic-Depth Looped Transformers for Latent Reasoning via Shortcut Modulation](https://mlsanity.com/paper/2602.11451/) (2026)
- [Training Strategies for Efficient Embodied Reasoning](https://mlsanity.com/paper/2505.08243/) (2025)
- [A Little Depth Goes a Long Way: The Expressive Power of Log-Depth Transformers](https://mlsanity.com/paper/2503.03961/) (2025)
- [Measuring Chain of Thought Faithfulness by Unlearning Reasoning Steps](https://mlsanity.com/paper/2502.14829/) (2025)
- [Thinking LLMs: General Instruction Following with Thought Generation](https://mlsanity.com/paper/2410.10630/) (2024)
- [PonderLM: Pretraining Language Models to Ponder in Continuous Space](https://mlsanity.com/paper/2505.20674/) (2025)
- [The Latent Space: Foundation, Evolution, Mechanism, Ability, and Outlook](https://mlsanity.com/paper/2604.02029/) (2026)
- [Beyond Semantics: The Unreasonable Effectiveness of Reasonless Intermediate Tokens](https://mlsanity.com/paper/2505.13775/) (2025)
- [Position: Stop Anthropomorphizing Intermediate Tokens as Reasoning/Thinking Traces!](https://mlsanity.com/paper/2504.09762/) (2025)
- [Soft Tokens, Hard Truths](https://mlsanity.com/paper/2509.19170/) (2025)
- [Enabling Agents to Communicate Entirely in Latent Space](https://mlsanity.com/paper/2511.09149/) (2025)
- [Break the Chain: Large Language Models Can be Shortcut Reasoners](https://mlsanity.com/paper/2406.06580/) (2024)
- [Hybrid Latent Reasoning via Reinforcement Learning](https://mlsanity.com/paper/2505.18454/) (2025)

## Abstract (arXiv)

Chain-of-thought responses from language models improve performance across most benchmarks. However, it remains unclear to what extent these performance gains can be attributed to human-like task decomposition or simply the greater computation that additional tokens allow. We show that transformers can use meaningless filler tokens (e.g., '......') in place of a chain of thought to solve two hard algorithmic tasks they could not solve when responding without intermediate tokens. However, we find empirically that learning to use filler tokens is difficult and requires specific, dense supervision to converge. We also provide a theoretical characterization of the class of problems where filler tokens are useful in terms of the quantifier depth of a first-order formula. For problems satisfying this characterization, chain-of-thought tokens need not provide information about the intermediate computational steps involved in multi-token computations. In summary, our results show that additional tokens can provide computational benefits independent of token choice. The fact that intermediate tokens can act as filler tokens raises concerns about large language models engaging in unauditable, hidden computations that are increasingly detached from the observed chain-of-thought tokens.