Research map / Large language models / Reasoning and chain of thought
Reasoning by Superposition: A Theoretical Perspective on Chain of Continuous Thought
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Proves a two-layer transformer with D steps of continuous thought solves directed graph reachability via parallel BFS in superposition, and validates the construction with GPT-2-style experiments on ProsQA.
Problem
Theoretical understanding of why chain-of-continuous-thought (Coconut) outperforms discrete CoT on graph reachability; discrete CoT for constant-depth transformers needs O(n^2) steps.
Approach
Each continuous thought is a normalized superposition of all vertices reachable within c steps, enabling implicit parallel BFS in a two-layer transformer; discrete CoT collapses this to a single path. The construction works with sinusoidal and RoPE positional encodings and emerges from gradient-based training.
Key results
- Coconut reaches near-perfect accuracy on ProsQA with a 2-layer transformer trained from scratch, while CoT and No CoT solve only about 75%.
- A 12-layer CoT model improves only to about 83% accuracy on ProsQA.
- Coconut-BFS (uniform frontier supervision) also achieves near-perfect accuracy, matching original Coconut.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use continuous CoT instead of discrete CoT | GPT-2 style decoder · ProsQA | accuracy: about 75\% -> near-perfect | Helped |
| use 12-layer discrete CoT instead of 2-layer continuous CoT | GPT-2 style decoder · ProsQA | accuracy: near-perfect -> 83% | Hurt |
| draw each intermediate supervision token uniformly from frontier nodes instead of using the CoT solution node (ablation) | coconut · ProsQA | accuracy | No clear effect |
Surprising findings
- Superposition-like frontier encoding emerges from training on only the optimal path, without explicit supervision for multiple paths.
- A 2-layer continuous-CoT model outperforms a 12-layer discrete-CoT model.
Limitations
- Theoretical construction is for a specific two-layer architecture with orthonormal token embeddings and a fixed input format.
- Experiments are limited to a synthetic ProsQA subset trained from scratch.
Experimental setup
Datasets: ProsQA
Benchmarks: ProsQA
Metrics: accuracy attention score inner product similarity
Baselines: CoT No CoT Coconut-BFS
Builds on: Coconut ProsQA sinusoidal positional encoding RoPE
Closest papers by meaning
- Emergence of Superposition: Unveiling the Training Dynamics of Chain of Continuous Thought (ICLR 2026 · similarity 0.95)
- Continuous Chain of Thought Enables Parallel Exploration and Reasoning (ICLR 2026 · similarity 0.91)
- Metastable Dynamics of Chain-of-Thought Reasoning: Provable Benefits of Search, RL and Distillation (ICML 2025 · similarity 0.91)
- Reasoning about Reasoning: BAPO Bounds on Chain-of-Thought Token Complexity in LLMs (ICML 2026 · similarity 0.90)
- Mind the Gap: Bridging Thought Leap for Improved Chain-of-Thought Tuning (NeurIPS 2025 · similarity 0.90)
- DAG-Math: Graph-Guided Mathematical Reasoning in LLMs (ICLR 2026 · similarity 0.90)
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