Research map / Large language models / Reasoning and chain of thought
Unlocking the Capabilities of Thought: A Reasoning Boundary Framework to Quantify and Optimize Chain-of-Thought
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Proposes a reasoning boundary framework (RBF) that quantifies the upper bound of chain-of-thought via a combination law over sub-task boundaries, and uses it to explain CoT strategies and guide optimization (MARP prompting), evaluated on BigGSM and other tasks.
Problem
Existing CoT analyses are qualitative; there is no quantitative metric for CoT upper bounds and little actionable guidance for optimizing CoT performance.
Approach
Defines a reasoning boundary as the maximum task difficulty at which accuracy hits a threshold, proposes a weighted-harmonic-mean combination law for boundaries of sub-capabilities (planning and calculation), partitions boundaries into completely feasible, partially feasible and completely infeasible regions, and uses these to explain and optimize CoT strategies (tool use, PoT, Complex CoT, Least-
Key results
- Combination law of reasoning boundaries (weighted harmonic mean) empirically matches accuracy boundaries at 90% and 10% thresholds for arithmetic, math reasoning, and multi-hop QA.
- Across 27 models, CFRB is reported as nearly zero for all open-source models except closed-source ones; o1 significantly enlarges CFRB, reportedly about three times other models.
- MARP prompting is reported to improve performance over vanilla CoT on BigGSM while reducing token consumption.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use Tool Usage instead of vanilla CoT | GPT-3.5-Turbo · mathematical reasoning | performance | Helped |
| use Program-of-Thought instead of Tool Usage | GPT-3.5-Turbo · mathematical reasoning | performance | Helped |
| use Minimum Acceptable Reasoning Paths prompting | GPT-3.5-Turbo · BigGSM | performance | Helped |
| use Minimum Acceptable Reasoning Paths prompting | GPT-3.5-Turbo · BigGSM | token consumption | Helped |
| increase Complex CoT reasoning steps (ablation) | GPT-3.5-Turbo · mathematical reasoning | performance | Mixed |
| use Complex CoT rather than Vanilla CoT on low-complexity problems (ablation) | GPT-3.5-Turbo · low-complexity problems | performance | Hurt |
| use Complex CoT rather than Vanilla CoT when reasoning steps exceed two (ablation) | GPT-3.5-Turbo · complex problems | performance | Helped |
| use Least-to-Most prompting instead of vanilla CoT (ablation) | GPT-3.5-Turbo · mathematical reasoning | performance | Helped |
| increase the number of reasoning paths combined with Self-Consistency (ablation) | GPT-3.5-Turbo · PFRB | accuracy | Helped |
| increase the number of reasoning paths combined with Self-Consistency (ablation) | GPT-3.5-Turbo · CIRB | accuracy | No clear effect |
| remove demonstrations from CoT prompting (ablation) | GPT-3.5-Turbo · CFRB vs other reasoning boundaries | correct-rationale improvement: 29.2% | Helped |
Surprising findings
- Complex CoT performance first rises then falls as the number of steps increases, because added planning pressure exceeds the model's planning boundary.
- Least-to-Most yields only limited gains because it increases global planning pressure by generating many sub-questions.
Limitations
- Does not model causal or complex dependencies among basic reasoning boundaries.
- Relies on manual annotation and fitted constants, and the combination law is approximate (k_i assumed near zero).
- Main experiments are on GPT-3.5-Turbo, and the LtM improvement is small.
Experimental setup
Datasets: HotpotQA MGSM GSM8K
Benchmarks: BigGSM HotpotQA MGSM
Metrics: accuracy token consumption
Baselines: Vanilla CoT Tool Usage Program-of-Thought Least-to-Most Complex CoT Self-Consistency
Builds on: Chain-of-Thought prompting Program-of-Thought Least-to-Most prompting Self-Consistency PAL
Closest papers by meaning
- The CoT Encyclopedia: Analyzing, Predicting, and Controlling how a Reasoning Model will Think (ICLR 2026 · similarity 0.93)
- Understanding Chain-of-Thought in LLMs through Information Theory (ICML 2025 · similarity 0.92)
- Causal Sufficiency and Necessity Improves Chain-of-Thought Reasoning (NeurIPS 2025 · similarity 0.92)
- To CoT or not to CoT? Chain-of-thought helps mainly on math and symbolic reasoning (ICLR 2025 · similarity 0.92)
- Rethinking Chain-of-Thought from the Perspective of Self-Training (ICML 2025 · similarity 0.92)
- SCOUT: Teaching Pre-trained Language Models to Enhance Reasoning via Flow Chain-of-Thought (NeurIPS 2025 · similarity 0.91)
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