plurigrid 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:
- Contains a formal proof system (e.g., Lean4, Coq)
- Has a utility function defining "better"
- Can rewrite any part of itself if it proves the rewrite improves utility
- 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)
Key Properties
- Halting Problem: Cannot prove all beneficial rewrites (incompleteness)
- Safety: Only rewrites with proofs are applied
- Bootstrapping: Initial prover must be trustworthy
- 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
- Schmidhuber, J. (2003). "Gödel Machines: Self-Referential Universal Problem Solvers."
- Zhang, J. et al. (2025). "Darwin Gödel Machine: Open-Ended Evolution of Self-Improving Agents."
- Schmidhuber, J. (2007). "New Millennium AI and the Convergence of History."