---
license: mit
tags:
- pytorch
- safetensors
- threshold-logic
- neuromorphic
- parity
---

# threshold-parity-tree

8-bit parity computation using balanced XOR tree structure for minimum depth.

## Circuit

```
x7 ─┬─[XOR]─┐
x6 ─┘       ├─[XOR]─┐
x5 ─┬─[XOR]─┘       │
x4 ─┘               ├─[XOR]──► parity
x3 ─┬─[XOR]─┐       │
x2 ─┘       ├─[XOR]─┘
x1 ─┬─[XOR]─┘
x0 ─┘
```

## Output

- **parity = 0**: Even number of 1s in input
- **parity = 1**: Odd number of 1s in input

## Tree vs Linear

```
Linear chain (depth 7):
x0─[XOR]─[XOR]─[XOR]─[XOR]─[XOR]─[XOR]─[XOR]─► parity
    x1    x2    x3    x4    x5    x6    x7

Tree structure (depth 3):
Layer 1: 4 parallel XORs
Layer 2: 2 parallel XORs
Layer 3: 1 final XOR
```

## Architecture

| Layer | XOR Gates | Function |
|-------|-----------|----------|
| 1 | 4 | Pairs: (0,1), (2,3), (4,5), (6,7) |
| 2 | 2 | Pairs: (01,23), (45,67) |
| 3 | 1 | Final: (0123,4567) |

**Total: 21 neurons (7 XORs × 3), 111 parameters, 3 layers**

## Scalability

For n-bit parity:
- Tree depth: log₂(n)
- XOR gates: n - 1
- Linear depth: n - 1

| Bits | Tree Depth | Linear Depth |
|------|------------|--------------|
| 8 | 3 | 7 |
| 16 | 4 | 15 |
| 32 | 5 | 31 |
| 64 | 6 | 63 |

## Usage

```python
from safetensors.torch import load_file

w = load_file('model.safetensors')

# Examples:
# 0x00 (00000000) → parity=0
# 0x01 (00000001) → parity=1
# 0x03 (00000011) → parity=0
# 0xFF (11111111) → parity=0
```

## Files

```
threshold-parity-tree/
├── model.safetensors
├── create_safetensors.py
├── config.json
└── README.md
```

## License

MIT