emic

Epsilon Machine Inference & Characterization

A Python framework for constructing and analyzing epsilon-machines based on computational mechanics.

📚 Documentation | 🚀 Getting Started

What is an Epsilon-Machine?

An epsilon-machine (ε-machine) is the minimal, optimal predictor of a stochastic process. Introduced by James Crutchfield and collaborators, ε-machines capture the intrinsic computational structure hidden in sequential data.

Key concepts:

• Causal states: Equivalence classes of histories that yield identical predictions
• Statistical complexity (Cμ): The entropy of the causal state distribution — a measure of structural complexity
• Entropy rate (hμ): The irreducible randomness in the process

ε-machines reveal the emic structure of a process — the computational organization that exists within the system itself, not imposed from outside.

Features

• 🔮 Inference: Reconstruct ε-machines using multiple algorithms (CSSR, CSM, BSI, Spectral, NSD)
• 📊 Analysis: Compute complexity measures (Cμ, hμ, excess entropy E, crypticity χ)
• 🎲 Sources: Built-in stochastic process generators (Golden Mean, Even Process, Biased Coin, Periodic) with noise transforms (BitFlipNoise)
• 🔗 Pipeline: Composable >> operator for source → inference → analysis workflows
• 🧪 Experiments: CLI and framework for reproducible algorithm benchmarking
• 📈 Visualization: State diagram rendering with Graphviz
• 📝 Export: LaTeX tables, TikZ diagrams, DOT, Mermaid, and JSON formats
• 🧩 Extensible: Protocol-based architecture for custom algorithms and sources

Installation

pip install emic

Or install from source with uv:

git clone https://github.com/johnazariah/emic.git
cd emic
uv sync --dev

Quick Start

from emic.sources import GoldenMeanSource, TakeN
from emic.inference import CSSR, CSSRConfig
from emic.analysis import analyze

# Generate data from the Golden Mean process (no consecutive 1s)
source = GoldenMeanSource(p=0.5, _seed=42)
data = TakeN(10_000)(source)

# Infer the epsilon-machine using CSSR
config = CSSRConfig(max_history=5, significance=0.001)
result = CSSR(config).infer(data)

# Analyze the inferred machine
summary = analyze(result.machine)
print(f"States: {len(result.machine.states)}")
print(f"Statistical Complexity: Cμ = {summary.statistical_complexity:.4f}")
print(f"Entropy Rate: hμ = {summary.entropy_rate:.4f}")

Pipeline Composition

Chain operations using the >> operator:

from emic.sources import GoldenMeanSource, TakeN
from emic.inference import CSSR, CSSRConfig
from emic.analysis import analyze

# Compose source and transforms
source = GoldenMeanSource(p=0.5, _seed=42)
data = source >> TakeN(10_000)

# Infer and analyze
config = CSSRConfig(max_history=5, significance=0.001)
result = CSSR(config).infer(data)
summary = analyze(result.machine)

print(summary)

Built-in Sources

Experiments

Run reproducible experiments to evaluate algorithm performance:

# Run all experiments with parallel execution
emic-experiment --all --parallel 4

# Quick mode for development
emic-experiment --quick

# List available experiments
emic-experiment --list

Algorithm Accuracy (January 2026)

See the Experiments Guide for full details.

Project Status

✅ Core implementation complete — The framework is functional with:

• Multiple inference algorithms: CSSR, CSM, BSI, Spectral, NSD
• Full analysis suite (Cμ, hμ, excess entropy E, crypticity χ)
• Synthetic and empirical data sources with noise transforms
• Pipeline composition
• 429 tests with 82%+ coverage
• Deep dive documentation: CSSR, Spectral Learning, Complexity Measures, Working with Real Data

📚 Full documentation available

Testing

All 429 tests are catalogued in the Testing Register, each with a plain English intent and classified by kind:

Test categories:

• Unit tests — Types, analysis measures, all 5 inference algorithms, sources, transforms, output formats
• Golden tests — Algorithms verified against analytically known ε-machines (Golden Mean, Even Process, Biased Coin, Periodic)
• Integration tests — Pipeline composition from source through inference to analysis
• Machine invariant tests — Every algorithm's output validated for stochastic correctness (transition sums ≤ 1.0)

Pre-commit hooks enforce that the testing register is updated whenever tests change.

Etymology

The name emic works on multiple levels:

1. Acronym: Epsilon Machine Inference & Characterization
2. Linguistic: In linguistics/anthropology, emic refers to analysis from within the system — understanding structure on its own terms. This resonates with computational mechanics: ε-machines reveal the intrinsic structure of a process.
3. Phonetic: Pronounced "EE-mik" or "EH-mic" — a nod to "ε-machine"

References

• Crutchfield, J.P. (1994). "The Calculus of Emergence: Computation, Dynamics, and Induction". Physica D.
• Shalizi, C.R. & Crutchfield, J.P. (2001). "Computational Mechanics: Pattern and Prediction, Structure and Simplicity". Journal of Statistical Physics.
• Crutchfield, J.P. & Young, K. (1989). "Inferring Statistical Complexity". Physical Review Letters.

Contributing

Contributions are welcome! See the Contributing Guide for details.

License

MIT License — see LICENSE for details.

Author

John Azariah (@johnazariah)