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

# threshold-nand3

3-input NAND gate. Fires unless all three inputs are active. The universal gate extended.

## Circuit

```
    a   b   c
    │   │   │
    └───┼───┘
        │
        ▼
   ┌──────────┐
   │w: -1,-1,-1│
   │ b:  +2   │
   └──────────┘
        │
        ▼
  NAND(a,b,c)
```

## The Veto Detector

3-input NAND fires when at least one input is 0:

| Inputs | Sum | Output |
|--------|-----|--------|
| 000 | +2 | 1 |
| 001 | +1 | 1 |
| 010 | +1 | 1 |
| 011 | 0 | 1 |
| 100 | +1 | 1 |
| 101 | 0 | 1 |
| 110 | 0 | 1 |
| **111** | **-1** | **0** |

Only unanimous activation silences the gate.

## Negative Weights

Each input *subtracts* from the sum. The positive bias provides headroom:

```
sum = -a - b - c + 2 = 2 - HW
fires when 2 - HW >= 0
fires when HW <= 2
```

This is AtMost2 for 3 inputs.

## Functional Completeness

NAND is universal - any Boolean function can be built from NAND:

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

## Dual of 3-input NOR

| Gate | Weights | Bias | Fires when |
|------|---------|------|------------|
| **NAND3** | all -1 | +2 | HW ≤ 2 |
| NOR3 | all -1 | 0 | HW = 0 |

NAND is permissive (7 of 8 pass). NOR is restrictive (1 of 8 passes).

## Parameters

| Component | Value |
|-----------|-------|
| Weights | [-1, -1, -1] |
| Bias | +2 |
| **Total** | **4 parameters** |

## Optimality

Exhaustive enumeration of all 681 weight configurations at magnitudes 0-5 confirms this circuit is **already at minimum magnitude (5)**. There is exactly one valid configuration at magnitude 5, and no valid configurations exist below it.

## Usage

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

w = load_file('model.safetensors')

def nand3(a, b, c):
    inp = torch.tensor([float(a), float(b), float(c)])
    return int((inp * w['weight']).sum() + w['bias'] >= 0)

print(nand3(1, 1, 0))  # 1
print(nand3(1, 1, 1))  # 0
```

## Files

```
threshold-nand3/
├── model.safetensors
├── model.py
├── config.json
└── README.md
```

## License

MIT