# Reasoning about Reasoning: BAPO Bounds on Chain-of-Thought Token Complexity in LLMs

arXiv: https://arxiv.org/abs/2602.02909 · HTML: https://mlsanity.com/paper/2602.02909/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 22 of its 32 references that are in the map.
- First posted on arXiv 2 Feb 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Stop Overthinking: A Survey on Efficient Reasoning for Large Language Models](https://mlsanity.com/paper/2503.16419/) (2025)
- [Chain of Draft: Thinking Faster by Writing Less](https://mlsanity.com/paper/2502.18600/) (2025)
- [Towards Revealing the Mystery behind Chain of Thought: A Theoretical Perspective](https://mlsanity.com/paper/2305.15408/) (2023)
- [O1-Pruner: Length-Harmonizing Fine-Tuning for O1-Like Reasoning Pruning](https://mlsanity.com/paper/2501.12570/) (2025)
- [Chain of Thought Empowers Transformers to Solve Inherently Serial Problems](https://mlsanity.com/paper/2402.12875/) (2024)
- [TokenSkip: Controllable Chain-of-Thought Compression in LLMs](https://mlsanity.com/paper/2502.12067/) (2025)
- [Token-Budget-Aware LLM Reasoning](https://mlsanity.com/paper/2412.18547/) (2024)
- [When More is Less: Understanding Chain-of-Thought Length in LLMs](https://mlsanity.com/paper/2502.07266/) (2025)
- [The Impact of Reasoning Step Length on Large Language Models](https://mlsanity.com/paper/2401.04925/) (2024)
- [Concise Thoughts: Impact of Output Length on LLM Reasoning and Cost](https://mlsanity.com/paper/2407.19825/) (2024)
- [How Well do LLMs Compress Their Own Chain-of-Thought? A Token Complexity Approach](https://mlsanity.com/paper/2503.01141/) (2025)
- [ShorterBetter: Guiding Reasoning Models to Find Optimal Inference Length for Efficient Reasoning](https://mlsanity.com/paper/2504.21370/) (2025)
- [Can Language Models Learn to Skip Steps?](https://mlsanity.com/paper/2411.01855/) (2024)
- [How Far Can Transformers Reason? The Globality Barrier and Inductive Scratchpad](https://mlsanity.com/paper/2406.06467/) (2024)
- [Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers](https://mlsanity.com/paper/2502.02393/) (2025)
- [Metastable Dynamics of Chain-of-Thought Reasoning: Provable Benefits of Search, RL and Distillation](https://mlsanity.com/paper/2502.01694/) (2025)
- [What Characterizes Effective Reasoning? Revisiting Length, Review, and Structure of CoT](https://mlsanity.com/paper/2509.19284/) (2025)
- [The Price of a Second Thought: On the Evaluation of Reasoning Efficiency in Large Language Models](https://mlsanity.com/paper/2505.22017/) (2025)
- [Lost in Transmission: When and Why LLMs Fail to Reason Globally](https://mlsanity.com/paper/2505.08140/) (2025)
- [Re-FORC: Adaptive Reward Prediction for Efficient Chain-of-Thought Reasoning](https://mlsanity.com/paper/2511.02130/) (2025)

## Built on by (most cited first, 1 of 1)

- [Scaling of Capability and Efficiency at Inference Time in Large Reasoning Models](https://mlsanity.com/paper/2609.27166/) (2026)

## Abstract (arXiv)

Inference-time scaling via chain-of-thought (CoT) reasoning is a major driver of state-of-the-art LLM performance, but it comes with substantial latency and compute costs. We address a fundamental theoretical question: how many reasoning tokens are required to solve a problem as input size grows? By extending the bounded attention prefix oracle (BAPO) model--an abstraction of LLMs that quantifies the information flow required to solve a task--we prove lower bounds on the CoT tokens required for three canonical BAPO-hard tasks: binary majority, triplet matching, and graph reachability. We show that each requires Omega(n) reasoning tokens when the input size is n. We complement these results with matching or near-matching upper bounds via explicit constructions. Finally, our experiments with frontier reasoning models show approximately linear reasoning token scaling on these tasks and failures when constrained to smaller reasoning budgets, consistent with our theoretical lower bounds. Together, our results identify fundamental bottlenecks in inference-time compute through CoT and offer a principled tool for analyzing optimal reasoning length.