---
license: mit
tags:
- pytorch
- safetensors
- threshold-logic
- neuromorphic
- functionally-complete
---

# threshold-nand

The universal gate. Any Boolean function can be constructed from NAND alone.

## Circuit

```
    x   y
    │   │
    └─┬─┘
      ▼
  ┌────────┐
  │w: -1,-1│
  │ b: +1  │
  └────────┘
      │
      ▼
  NAND(x,y)
```

## Mechanism

Negative weights mean inputs *subtract* from the sum. The positive bias starts us above threshold, and inputs pull us down:

| x | y | sum | output |
|---|---|-----|--------|
| 0 | 0 | +1 | 1 |
| 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | -1 | 0 |

Only when both inputs are active do we fall below threshold. This is AND with inverted output.

## Parameters

| | |
|---|---|
| Weights | [-1, -1] |
| Bias | +1 |
| Total | 3 parameters |

## Optimality

Exhaustive enumeration of all 25 weight configurations at magnitudes 0-3 confirms this circuit is **already at minimum magnitude (3)**. There is exactly one valid configuration at magnitude 3, and no valid configurations exist below it.

## Functional Completeness

NAND can build any Boolean function:

- NOT(x) = NAND(x, x)
- AND(x,y) = NOT(NAND(x,y)) = NAND(NAND(x,y), NAND(x,y))
- OR(x,y) = NAND(NOT(x), NOT(y))

This is why NAND gates dominate digital logic fabrication.

## Usage

```python
from safetensors.torch import load_file
import torch

w = load_file('model.safetensors')

def nand_gate(x, y):
    inputs = torch.tensor([float(x), float(y)])
    return int((inputs * w['weight']).sum() + w['bias'] >= 0)
```

## Files

```
threshold-nand/
├── model.safetensors
├── model.py
├── config.json
└── README.md
```

## License

MIT