Kolmogorov Compression

Kolmogorov Compression Skill

"The Kolmogorov complexity of x is the length of the shortest program that outputs x." — Andrey Kolmogorov

Overview

Kolmogorov complexity K(x) = length of shortest program P where P() = x.

Intelligence = Compression: Finding short descriptions of data.

Core Concept

K(x) = min { |P| : U(P) = x }

Where:
  U = Universal Turing Machine
  P = program (binary string)
  |P| = length of P

Properties:
  - K(x) ≤ |x| + O(1)  (trivial: print x)
  - K(x) is uncomputable (halting problem)
  - K(x|y) = conditional complexity given y

The KoLMogorov-Test (2025)

Use LLMs to approximate Kolmogorov complexity:

class KolmogorovCompressor:
    """
    Approximate K(x) via code generation.
    """
    
    def __init__(self, llm):
        self.llm = llm
    
    def compress(self, data: str) -> str:
        """Generate shortest program that outputs data."""
        prompt = f"""
        Generate the shortest Python program that prints exactly:
        {data[:100]}...
        
        The program must output EXACTLY this string.
        Make it as SHORT as possible.
        """
        
        program = self.llm.generate(prompt)
        return self.extract_code(program)
    
    def complexity(self, data: str) -> int:
        """Estimate K(data)."""
        program = self.compress(data)
        return len(program.encode())
    
    def intelligence_score(self, model, data: str) -> float:
        """
        KoLMogorov-Test score.
        
        Higher = better compression = more intelligent.
        """
        program = model.compress(data)
        ratio = len(program) / len(data)
        return 1 - ratio  # Higher = better

Connection to Theorem Proving

For proof P of theorem T:
  K(T) ≈ min |P| over all proofs P

Short proofs = Simple theorems
Long proofs = Complex theorems (but still provable)

Gödel: Some true statements have K(T) = ∞ (unprovable)

End-of-Skill Interface

Integration with Sutskever's Thesis

Sutskever's Insight:
  Compression = Prediction = Understanding = Intelligence

If you can compress x to K(x) bits:
  - You understand x's structure
  - You can predict x from the program
  - You have a model of x

GF(3) Triads

kolmogorov-compression (-1) ⊗ cognitive-superposition (0) ⊗ godel-machine (+1) = 0 ✓
kolmogorov-compression (-1) ⊗ turing-chemputer (0) ⊗ dna-origami (+1) = 0 ✓
kolmogorov-compression (-1) ⊗ solomonoff-induction (0) ⊗ information-capacity (+1) = 0 ✓

As Validator (-1), kolmogorov-compression:

• Measures true complexity (validates claims)
• Filters noise from signal
• Provides lower bound on description

References

1. Kolmogorov, A.N. (1965). "Three approaches to the quantitative definition of information."
2. Solomonoff, R.J. (1964). "A formal theory of inductive inference."
3. Li, M. & Vitányi, P. (2008). An Introduction to Kolmogorov Complexity and Its Applications.
4. Fan et al. (2025). "The KoLMogorov-Test: Compression-Based Intelligence Evaluation."

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