godel-machine

name	godel-machine
description	Schmidhuber''s Gödel Machine: Self-improving systems that prove their
version	1.0.0

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: 
        
        Suggest an improvement to make this agent better.
        Return only the improved code.
        """
        
        new_code = .llm.generate(prompt)
         Agent(code=new_code, generation=.generation + )

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)

---

End-of-Skill Interface

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

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 ✓

r2con Speaker Resources

Speaker	Relevance	Repository/Talk
cryptax	Malware evolution/mutation	droidlysis
unixfreaxjp	Self-modifying malware	r2con malware analysis
cmatthewbrooks	Binary mutation analysis	malchive

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."

SDF Interleaving

This skill connects to Software Design for Flexibility (Hanson & Sussman, 2021):

Primary Chapter: 10. Adventure Game Example

Concepts: autonomous agent, game, synthesis

GF(3) Balanced Triad

godel-machine (+) + SDF.Ch10 (+) + [balancer] (+) = 0

Skill Trit: 1 (PLUS - generation)

Secondary Chapters

• Ch5: Evaluation
• Ch7: Propagators

Connection Pattern

Adventure games synthesize techniques. This skill integrates multiple patterns.