# 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](#algebraic-neural-networks)
- [Uncomputable Neural Networks](#uncomputable-neural-networks)
- [Theory and Mathematical Foundations](#theory-and-mathematical-foundations)
- [Implementations](#implementations)
- [Examples and Use Cases](#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

```bash
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
```bash
python demo.py
```
This runs a simple demonstration showing how algebraic neural networks process data without any training.

### Examples
```bash
# 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:

```bash
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