Compression Progress Skill: Curiosity-Driven Learning

Status: ✅ Production Ready Trit: +1 (PLUS - generator) Color: #D82626 (Red) Principle: Learning = Compression improvement Frame: Compressor improvement rate as reward signal

Overview

Compression Progress measures the derivative of compression ability over time. When a learner compresses data better than before, that improvement is intrinsic reward—the formal theory of curiosity and creativity.

1. Compressor C(t): Current world model
2. Compression ratio: |C(data)| / |data|
3. Progress: C(t) - C(t-1) improvement
4. Reward: Proportional to progress, not absolute compression

Core Formula

r(t) = |C(t-1)(data)| - |C(t)(data)|

Curiosity reward = compression improvement rate
Boredom = zero progress (already compressed or incompressible)

def compression_progress(compressor_old, compressor_new, data) -> float:
    """Intrinsic reward from model improvement."""
    old_bits = len(compressor_old.compress(data))
    new_bits = len(compressor_new.compress(data))
    return old_bits - new_bits  # positive = learned something

Key Concepts

1. Curiosity as Compression Gradient

class CuriousAgent:
    def __init__(self):
        self.world_model = Compressor()
        self.history = []
    
    def intrinsic_reward(self, observation) -> float:
        old_len = self.world_model.compressed_length(observation)
        self.world_model.update(observation)
        new_len = self.world_model.compressed_length(observation)
        return old_len - new_len  # curiosity signal
    
    def should_explore(self, state) -> bool:
        """Explore where compression progress is expected."""
        return self.expected_progress(state) > self.threshold

2. Creativity as Compression Search

def generate_interesting(compressor) -> Data:
    """Generate data that maximizes expected compression progress."""
    candidates = sample_latent_space()
    return max(candidates, 
               key=lambda x: expected_progress(compressor, x))

3. Optimal Curriculum via Progress

def select_next_task(tasks, compressor) -> Task:
    """Choose task with maximum learning potential."""
    progress_estimates = [
        estimate_compression_progress(compressor, task)
        for task in tasks
    ]
    # Not too easy (zero progress), not too hard (negative/zero)
    return tasks[argmax(progress_estimates)]

Commands

# Measure compression progress
just compression-progress before.model after.model data/

# Generate curiosity curriculum
just curiosity-curriculum tasks.json

# Visualize learning trajectory
just compression-trajectory log.json

Integration with GF(3) Triads

yoneda-directed (-1) ⊗ cognitive-superposition (0) ⊗ compression-progress (+1) = 0 ✓  [Riehl-Schmidhuber]
kolmogorov-compression (-1) ⊗ turing-chemputer (0) ⊗ compression-progress (+1) = 0 ✓  [Formal Learning]

Related Skills

• kolmogorov-compression (-1): Absolute complexity baseline
• godel-machine (+1): Self-improvement via provable progress
• cognitive-superposition (0): Multi-hypothesis compression

Skill Name: compression-progress Type: Curiosity Generator Trit: +1 (PLUS) Color: #D82626 (Red)

Scientific Skill Interleaving

This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:

Graph Theory

• networkx [○] via bicomodule
  • Universal graph hub

Bibliography References

• general: 734 citations in bib.duckdb

Cat# Integration

This skill maps to Cat# = Comod(P) as a bicomodule in the equipment structure:

Trit: 0 (ERGODIC)
Home: Prof
Poly Op: ⊗
Kan Role: Adj
Color: #26D826

GF(3) Naturality

The skill participates in triads satisfying:

(-1) + (0) + (+1) ≡ 0 (mod 3)

This ensures compositional coherence in the Cat# equipment structure.