Not-trained-Neural-Networks-Notes

A comprehensive collection of notes, implementations, and examples of neural networks that don't rely on traditional gradient-based training methods.

Contents

• Algebraic Neural Networks
• Uncomputable Neural Networks
• Theory and Mathematical Foundations
• Implementations
• Examples and Use Cases

Algebraic Neural Networks

Algebraic Neural Networks (ANNs) represent a paradigm shift from traditional neural networks by utilizing algebraic structures and operations instead of gradient-based optimization. These networks leverage:

• Algebraic Group Theory: Using group operations for network transformations
• Polynomial Algebras: Networks based on polynomial computations
• Geometric Algebra: Incorporating geometric algebraic structures
• Fixed Algebraic Transformations: Pre-defined algebraic operations

Key Features

1. No Training Required: Networks are constructed using algebraic principles
2. Deterministic Behavior: Outputs are fully determined by algebraic rules
3. Mathematical Rigor: Based on well-established algebraic foundations
4. Interpretability: Clear mathematical interpretation of operations

Uncomputable Neural Networks

Uncomputable Neural Networks extend the paradigm of non-trained networks by incorporating theoretical concepts from computability theory. These networks explore computational boundaries by simulating uncomputable functions and operations:

• Halting Oracle Layers: Simulate access to halting oracles for program termination decisions
• Kolmogorov Complexity Layers: Approximate uncomputable complexity measures using compression heuristics
• Busy Beaver Layers: Utilize the uncomputable Busy Beaver function values and approximations
• Non-Recursive Layers: Operate on computably enumerable but non-computable sets

Key Features

1. Theoretical Foundations: Based on computability theory and hypercomputation concepts
2. Bounded Approximations: Practical implementations of theoretically uncomputable functions
3. Deterministic Simulation: Consistent behavior through fixed-seed randomness and heuristics
4. Educational Value: Demonstrates limits and possibilities of computation

Getting Started

git clone https://github.com/ewdlop/Not-trained-Neural-Networks-Notes.git
cd Not-trained-Neural-Networks-Notes

# Install dependencies
pip install numpy matplotlib

# Quick demo
python demo.py

# Run main implementation
python algebraic_neural_network.py

# Run comprehensive tests
python test_comprehensive.py

Quick Demo

python demo.py

This runs a simple demonstration showing how algebraic neural networks process data without any training.

Examples

# Polynomial-based networks
python examples/polynomial_network.py

# Group theory networks
python examples/group_theory_network.py

# Geometric algebra networks
python examples/geometric_algebra_network.py

# Uncomputable neural networks
python examples/uncomputable_networks.py

Structure

├── README.md                          # This file
├── demo.py                            # Quick demonstration script
├── algebraic_neural_network.py        # Main implementation
├── test_comprehensive.py              # Test suite
├── theory/                            # Theoretical background
│   ├── algebraic_foundations.md       # Mathematical foundations
│   ├── uncomputable_networks.md       # Uncomputable neural networks theory
│   └── examples.md                    # Worked examples
└── examples/                          # Practical examples
    ├── polynomial_network.py          # Polynomial-based network
    ├── group_theory_network.py        # Group theory implementation
    ├── geometric_algebra_network.py   # Geometric algebra network
    └── uncomputable_networks.py       # Uncomputable neural networks

Testing

Run the comprehensive test suite to verify all components:

python test_comprehensive.py

This tests:

• Basic functionality of all layer types (algebraic and uncomputable)
• Network composition and data flow
• Deterministic behavior (same input → same output)
• Mathematical properties of algebraic operations
• Uncomputable layer approximations and bounds
• Edge cases and boundary conditions