Further human + AI + proof assistant work on Knuth's "Claude Cycles" problem

Orchestrate a human-AI-proof assistant collaboration to tackle complex theoretical computer science problems like Knuth's "Claude Cycles." Leverage AI for conjecture generation and formal proof assistants for rigorous verification, accelerating mathematical discovery.

advanced1 hour5 steps

The play

Formalize the Problem Definition
Review the problem (e.g., Knuth's 'Claude Cycles'), define its scope, known properties, and initial conjectures. Translate these into a formal language suitable for both AI processing and proof assistant input.
Generate Conjectures with AI
Utilize AI systems (LLMs or specialized search algorithms) to explore the problem space. Craft detailed prompts for LLMs to generate insights, hypotheses, potential solution paths, or counter-examples.
Human Review and Refinement
Human experts analyze the AI-generated conjectures and insights. Filter out irrelevant suggestions, refine promising hypotheses, and provide feedback to guide subsequent AI exploration, iterating on prompts or search parameters.
Rigorously Verify with Proof Assistants
Translate refined conjectures or potential proofs into the language of a formal proof assistant (e.g., Coq, Lean, Isabelle). Use the proof assistant to rigorously verify the correctness of the hypotheses, ensuring mathematical soundness and eliminating errors.
Iterate and Document Findings
Continuously iterate through problem formalization, AI exploration, human refinement, and formal verification. Document all findings, successful proofs, and unresolved challenges to advance research and build upon discoveries.

Starter code

curl -X POST "https://api.openai.com/v1/chat/completions" \
  -H "Content-Type: application/json" \
  -H "Authorization: Bearer $OPENAI_API_KEY" \
  -d '{
    "model": "gpt-4o",
    "messages": [
      {
        "role": "user",
        "content": "Given the definition of a \"Claude Cycle\" (a permutation p of {1,...,n} such that the sequence x_0=1, x_{i+1} = p(x_i) for i >= 0 eventually repeats, and the cycle length is maximized for a given n), explore properties of such permutations. Specifically, generate conjectures about the structure of permutations that yield maximal cycle lengths. Consider small values of n (e.g., n=1 to 10) and extrapolate. Also, suggest potential approaches for proving or disproving these conjectures."
      }
    ],
    "max_tokens": 500
  }'