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

TL;DR

RBF++ introduces a formal framework for quantifying and optimizing chain-of-thought reasoning by defining Reasoning Boundaries (RBs), a harmonic combination law to merge multiple RBs, constant assumptions for unmeasurable, and a boundary-division mechanism for finer granularity. The approach provides actionable metrics and prompting strategies (MARP, MARP++) and demonstrates across textual and multimodal tasks that RB-aware optimization improves CoT performance, with extensive benchmarks (BigGSM++, M3CoT, HotpotQA) and analysis across 38 models and 13 tasks. Key findings show distinct RB regimes (CFRB, PFRB, CIRB), the differential impact of tools and PoT in textual vs multimodal settings, and the value of boundary division and constants in unmeasurable domains. The work offers practical guidance for scaling reasoning capabilities in LLMs and provides publicly available code and data to facilitate further research and deployment in multimodal reasoning contexts.

Abstract

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.

Figures (16)

• Figure 1: Overview of the introduced concepts: (a) reasoning boundary (RB), (b) combination law for quantifying the upper bound of LLM capabilities in measurable scenarios; (c) constant assumption and (d) RB division mechanism for unmeasurable scenarios; (e) categories of RB for reasoning optimization guidance.
• Figure 2: Existence Verification for Reasoning Boundary. Figures (b, c) present evaluations performed on BigGSM, where the reasoning accuracy of each step is manually analyzed, without considering whether the final conclusions are correct.
• Figure 3: Combination law verification of RB on different tasks for RBF in textual modalities.
• Figure 4: Nature analysis at different reasoning boundaries on BigGSM with text-modal scenarios. For Fig. (c), all samples in CIRB are special value points, like $25000 \times 1000$. In fact, no real CIRB samples are obtained.
• Figure 5: The performance analysis of Least-to-Most prompting in textual modalities.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Definition 3