# RBF++: Quantifying and Optimizing Reasoning Boundaries across Measurable and Unmeasurable Capabilities for Chain-of-Thought Reasoning

arXiv: https://arxiv.org/abs/2505.13307 · HTML: https://mlsanity.com/paper/2505.13307/

## Key facts

- Cited by 4 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 25 of its 64 references that are in the map.
- First posted on arXiv 19 May 2025.

## 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, 25 of 25)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [A Survey of Large Language Models](https://mlsanity.com/paper/2303.18223/) (2023)
- [Code Llama: Open Foundation Models for Code](https://mlsanity.com/paper/2308.12950/) (2023)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Gemini: A Family of Highly Capable Multimodal Models](https://mlsanity.com/paper/2312.11805/) (2023)
- [DeepSeek-V3 Technical Report](https://mlsanity.com/paper/2412.19437/) (2024)
- [Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models](https://mlsanity.com/paper/2503.09567/) (2025)
- [ART: Automatic multi-step reasoning and tool-use for large language models](https://mlsanity.com/paper/2303.09014/) (2023)
- [Rethinking the Bounds of LLM Reasoning: Are Multi-Agent Discussions the Key?](https://mlsanity.com/paper/2402.18272/) (2024)
- [The Impact of Reasoning Step Length on Large Language Models](https://mlsanity.com/paper/2401.04925/) (2024)
- [OpenMathInstruct-1: A 1.8 Million Math Instruction Tuning Dataset](https://mlsanity.com/paper/2402.10176/) (2024)
- [Large Language Models Meet NLP: A Survey](https://mlsanity.com/paper/2405.12819/) (2024)
- [How Far Are We on the Decision-Making of LLMs? Evaluating LLMs' Gaming Ability in Multi-Agent Environments](https://mlsanity.com/paper/2403.11807/) (2024)
- [How to think step-by-step: A mechanistic understanding of chain-of-thought reasoning](https://mlsanity.com/paper/2402.18312/) (2024)
- [A Preliminary Evaluation of ChatGPT for Zero-shot Dialogue Understanding](https://mlsanity.com/paper/2304.04256/) (2023)
- [Code Prompting: a Neural Symbolic Method for Complex Reasoning in Large Language Models](https://mlsanity.com/paper/2305.18507/) (2023)
- [Enhancing Chain of Thought Prompting in Large Language Models via Reasoning Patterns](https://mlsanity.com/paper/2404.14812/) (2024)
- [A Theory for Length Generalization in Learning to Reason](https://mlsanity.com/paper/2404.00560/) (2024)
- [Evaluating Mathematical Reasoning Across Large Language Models: A Fine-Grained Approach](https://mlsanity.com/paper/2503.10573/) (2025)
- [Electronic Circuit Principles of Large Language Models](https://mlsanity.com/paper/2502.03325/) (2025)

## Built on by (most cited first, 4 of 4)

- [Aware First, Think Less: Dynamic Boundary Self-Awareness Drives Extreme Reasoning Efficiency in Large Language Models](https://mlsanity.com/paper/2508.11582/) (2025)
- [Do Coding Agents Understand Least-Privilege Authorization?](https://mlsanity.com/paper/2605.14859/) (2026)
- [Stop Before You Fail: Operational Capability Boundaries for Mitigating Unproductive Reasoning in Large Reasoning Models](https://mlsanity.com/paper/2509.24711/) (2025)
- [FantasyVLN: Unified Multimodal Chain-of-Thought Reasoning for Vision-Language Navigation](https://mlsanity.com/paper/2601.13976/) (2026)

## Abstract (arXiv)

Chain-of-Thought (CoT) reasoning has proven effective in enhancing large language models (LLMs) on complex tasks, spurring research into its underlying mechanisms. However, two primary challenges remain for real-world applications: (1) the lack of quantitative metrics and actionable guidelines for evaluating and optimizing measurable boundaries of CoT capability, and (2) the absence of methods to assess boundaries of unmeasurable CoT capability, such as multimodal perception. To address these gaps, we introduce the Reasoning Boundary Framework++ (RBF++). To tackle the first challenge, we define the reasoning boundary (RB) as the maximum limit of CoT performance. We also propose a combination law for RBs, enabling quantitative analysis and offering actionable guidance across various CoT tasks. For the second challenge, particularly in multimodal scenarios, we introduce a constant assumption, which replaces unmeasurable RBs with scenario-specific constants. Additionally, we propose the reasoning boundary division mechanism, which divides unmeasurable RBs into two sub-boundaries, facilitating the quantification and optimization of both unmeasurable domain knowledge and multimodal perception capabilities. Extensive experiments involving 38 models across 13 tasks validate the feasibility of our framework in cross-modal settings. Additionally, we evaluate 10 CoT strategies, offer insights into optimization and decay from two complementary perspectives, and expand evaluation benchmarks for measuring RBs in LLM reasoning. We hope this work advances the understanding of RBs and optimization strategies in LLMs. Code and data are available at https://github.com/LightChen233/reasoning-boundary.