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