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

# threshold-and3

3-input AND gate. Fires when all three inputs are active. The 3-of-3 threshold gate.

## Circuit

```
    a   b   c
    │   │   │
    └───┼───┘
        │
        ▼
   ┌─────────┐
   │ w: 1,1,1│
   │ b:  -3  │
   └─────────┘
        │
        ▼
   AND(a,b,c)
```

## The Unanimity Requirement

3-input AND is the smallest non-trivial unanimity gate:

| Inputs | Sum | Output |
|--------|-----|--------|
| 000 | -3 | 0 |
| 001 | -2 | 0 |
| 010 | -2 | 0 |
| 011 | -1 | 0 |
| 100 | -2 | 0 |
| 101 | -1 | 0 |
| 110 | -1 | 0 |
| **111** | **0** | **1** |

Only when all three contribute does the sum reach the threshold.

## Generalization from 2-input AND

| Gate | Weights | Bias | Threshold |
|------|---------|------|-----------|
| AND(a,b) | [1, 1] | -2 | 2 votes needed |
| **AND(a,b,c)** | [1, 1, 1] | -3 | 3 votes needed |
| AND(a,b,c,d) | [1, 1, 1, 1] | -4 | 4 votes needed |

The pattern: n inputs, all weight +1, bias -n.

## Single Point of Veto

Any single 0 input blocks the gate:

- a=0: max sum = 0 + 1 + 1 - 3 = -1 < 0
- b=0: max sum = 1 + 0 + 1 - 3 = -1 < 0
- c=0: max sum = 1 + 1 + 0 - 3 = -1 < 0

Each input has veto power. This is unanimous consent.

## Parameters

| Component | Value |
|-----------|-------|
| Weights | [1, 1, 1] |
| Bias | -3 |
| **Total** | **4 parameters** |

## Optimality

Exhaustive enumeration of all 1,289 weight configurations at magnitudes 0-6 confirms this circuit is **already at minimum magnitude (6)**. There is exactly one valid configuration at magnitude 6, and no valid configurations exist below it.

## Usage

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

w = load_file('model.safetensors')

def and3(a, b, c):
    inp = torch.tensor([float(a), float(b), float(c)])
    return int((inp * w['weight']).sum() + w['bias'] >= 0)

print(and3(1, 1, 1))  # 1
print(and3(1, 1, 0))  # 0
```

## Files

```
threshold-and3/
├── model.safetensors
├── model.py
├── config.json
└── README.md
```

## License

MIT