Gödel Machine Skill

"A Gödel Machine can rewrite any part of itself, including the learning algorithm, provided it can first prove that the rewrite is beneficial."
— Jürgen Schmidhuber

Overview

The Gödel Machine is a self-improving system that:

1. Contains a formal proof system (e.g., Lean4, Coq)
2. Has a utility function defining "better"
3. Can rewrite any part of itself if it proves the rewrite improves utility
4. The proof constraint prevents reckless self-modification

Core Architecture

┌─────────────────────────────────────────────────────┐
│                  GÖDEL MACHINE                      │
├─────────────────────────────────────────────────────┤
│  ┌─────────────┐    ┌─────────────┐                │
│  │   Policy    │───▶│   Prover    │                │
│  │  (current)  │    │  (verifier) │                │
│  └─────────────┘    └──────┬──────┘                │
│         ▲                   │                       │
│         │            ┌──────▼──────┐                │
│         │            │  Candidate  │                │
│         │            │   Policy    │                │
│         │            └──────┬──────┘                │
│         │                   │                       │
│  ┌──────┴──────┐     ┌──────▼──────┐               │
│  │   Rewrite   │◀────│  Utility    │               │
│  │   if proof  │     │   Check     │               │
│  └─────────────┘     └─────────────┘               │
└─────────────────────────────────────────────────────┘

Darwin Gödel Machine (DGM)

Combines evolutionary search with formal proofs:

class DarwinGodelMachine:
    """
    DGM: Open-ended evolution of self-improving agents.
    
    Archive of agents, LLM-based mutation, fitness evaluation,
    keep if novel and beneficial.
    """
    
    def __init__(self, initial_agent: Agent, prover: TheoremProver):
        self.archive = [initial_agent]
        self.prover = prover
        self.generation = 0
    
    def evolve_step(self) -> Agent:
        # Sample parent from archive (fitness-proportionate)
        parent = self.sample_archive()
        
        # LLM-based mutation
        child = self.llm_mutate(parent)
        
        # Evaluate on benchmarks
        fitness = self.evaluate(child)
        
        # Optionally: verify improvement formally
        if self.prover.can_prove(f"utility({child}) > utility({parent})"):
            child.proven = True
        
        # Add if novel and good
        if self.is_novel(child) and fitness > 0:
            self.archive.append(child)
        
        return child
    
    def llm_mutate(self, agent: Agent) -> Agent:
        """Use LLM to generate improved version."""
        prompt = f"""
        Current agent code:
        {agent.code}
        
        Current fitness: {agent.fitness}
        
        Suggest an improvement to make this agent better.
        Return only the improved code.
        """
        
        new_code = self.llm.generate(prompt)
        return Agent(code=new_code, generation=self.generation + 1)

GF(3) Triads

# Self-Improvement Triads
kolmogorov-compression (-1) ⊗ cognitive-superposition (0) ⊗ godel-machine (+1) = 0 ✓
proofgeneral-narya (-1) ⊗ self-evolving-agent (0) ⊗ godel-machine (+1) = 0 ✓
sheaf-cohomology (-1) ⊗ epistemic-arbitrage (0) ⊗ godel-machine (+1) = 0 ✓

Integration with Interaction Entropy

module GodelMachine
  def self.attempt_improvement(current_policy, seed)
    gen = SplitMixTernary::Generator.new(seed)
    color = gen.next_color
    
    # Generate candidate via color-guided mutation
    candidate = mutate(current_policy, color)
    
    # Attempt proof
    proof = attempt_prove(candidate, current_policy)
    
    if proof[:success]
      {
        improved: true,
        new_policy: candidate,
        proof: proof[:theorem],
        trit: 1  # Generator role
      }
    else
      { improved: false, reason: proof[:failure_reason] }
    end
  end
end

Key Properties

1. Halting Problem: Cannot prove all beneficial rewrites (incompleteness)
2. Safety: Only rewrites with proofs are applied
3. Bootstrapping: Initial prover must be trustworthy
4. Asymptotic Optimality: Converges to optimal policy (given enough time)

References

1. Schmidhuber, J. (2003). "Gödel Machines: Self-Referential Universal Problem Solvers."
2. Zhang, J. et al. (2025). "Darwin Gödel Machine: Open-Ended Evolution of Self-Improving Agents."
3. Schmidhuber, J. (2007). "New Millennium AI and the Convergence of History."