∇Continuous Function
FoundationsChecking saved investigationReading browser-local route memory before showing a continuation.

Foundation Lab

Chain-of-Thought Prompting

Dramatically improves reasoning performance—math, logic, coding

Concept 75 of 100Scaling & AlignmentPhase 11
Open DemoAll Foundations
#75CoTScaling & Alignment
key equationP(answer | question, reasoning)

Selected Foundation Object

Keep the equation fixed; move through the evidence.

Concept 75 of 100CoTScaling & Alignment / Phase 11: Frontier research & scaling
Current question

CoT gives the model "scratchpad" space for intermediate computation

P(answer | question, reasoning)
PredictionCommit before the demo.

Ask what should change when the equation is manipulated, then let the visualization test that expectation.

EvidenceCompare local witness and source.

Use the runnable panel, the key equation, and canonical papers as separate forms of evidence for the same object.

InvariantName what survives notation changes.

The useful learning product is the reusable mechanism you can carry into another model, paper, or engineering tradeoff.

Next moveContinue through the atlas.

Use prerequisites, dependents, and semantic links to repair the next gap without leaving the object behind.

Open demoInspect equationBrowse atlasNext foundation

Why It Matters for Modern Models

  • Dramatically improves reasoning performance—math, logic, coding
  • Emergent capability: only works well in large models (>100B parameters)
  • Foundation for o1-style reasoning: explicit intermediate computation steps

What Tutorials Skip

What is still poorly explained in textbooks and papers:

  • CoT gives the model "scratchpad" space for intermediate computation
  • The reasoning doesn't need to be human-readable—it just needs to help the model
  • Self-consistency leverages diversity: different reasoning paths vote on the answer

Interactive Visualization

Core Math (Optional Deep Dive)

If you want intuition first, start with the key equation and the visualization. Come back here for the full walkthrough.

Key Equation
P(answer∣question,reasoning)P(answer | question, reasoning)

Chain-of-Thought: Prompt model to show reasoning steps:

P(answer∣question)→P(answer∣question,reasoning)P(answer | question) \to P(answer | question, reasoning)

Self-Consistency: Sample multiple reasoning paths, majority vote:

answer∗=arg⁡max⁡a∑i=1N1[ai=a]answer^* = \arg\max_a \sum_{i=1}^N \mathbf{1}[a_i = a]

where ai∼P(a∣q,ri)a_i \sim P(a | q, r_i) with reasoning path rir_i.

Decomposition: Break complex problem into sub-problems:

P(y∣x)=∏t=1TP(yt∣y<t,x)P(y | x) = \prod_{t=1}^T P(y_t | y_{<t}, x)

Canonical Papers

Chain-of-Thought Prompting Elicits Reasoning in Large Language Models

Wei et al.2022NeurIPS
Read paper →

Self-Consistency Improves Chain of Thought Reasoning

Wang et al.2023ICLR
Read paper →

Connections

Prerequisites

📝ICL🧠Test-Time

Enables

🎯Deliberative🌲MCTS-LLM

Next Moves

Choose the next question to carry this object forward.