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

# threshold-and4

A 4-of-4 threshold gate. All four inputs must be active to reach the firing threshold.

## Circuit

```
  x1  x2  x3  x4
   │   │   │   │
   └───┴───┴───┘
         │
         ▼
    ┌─────────┐
    │w: 1,1,1,1│
    │ b: -4   │
    └─────────┘
         │
         ▼
   AND4(x1,x2,x3,x4)
```

## Mechanism

Each input contributes +1 to the sum. The bias of -4 means exactly four contributions are required to reach zero:

| x1 | x2 | x3 | x4 | sum | output |
|----|----|----|-----|-----|--------|
| 0 | 0 | 0 | 0 | -4 | 0 |
| 0 | 0 | 0 | 1 | -3 | 0 |
| ... | ... | ... | ... | ... | 0 |
| 1 | 1 | 1 | 0 | -1 | 0 |
| 1 | 1 | 1 | 1 | 0 | 1 |

Only the all-ones input reaches the threshold.

## Parameters

| | |
|---|---|
| Weights | [1, 1, 1, 1] |
| Bias | -4 |
| Magnitude | 8 |
| Total | 5 parameters |

## Optimality

This circuit is at minimum magnitude (8). The pattern generalizes: n-input AND requires weights all 1 and bias -n, giving magnitude 2n.

| Gate | Weights | Bias | Magnitude |
|------|---------|------|-----------|
| AND2 | [1, 1] | -2 | 4 |
| AND3 | [1, 1, 1] | -3 | 6 |
| AND4 | [1, 1, 1, 1] | -4 | 8 |
| ANDn | [1, ..., 1] | -n | 2n |

## Properties

- Linearly separable (single neuron suffices)
- Commutative, associative, idempotent
- De Morgan dual: AND4(x1,x2,x3,x4) = NOT(OR4(NOT(x1), NOT(x2), NOT(x3), NOT(x4)))

## Usage

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

w = load_file('model.safetensors')

def and4_gate(x1, x2, x3, x4):
    inputs = torch.tensor([float(x1), float(x2), float(x3), float(x4)])
    return int((inputs * w['weight']).sum() + w['bias'] >= 0)

# Test
assert and4_gate(1, 1, 1, 1) == 1
assert and4_gate(1, 1, 1, 0) == 0
assert and4_gate(0, 0, 0, 0) == 0
```

## Files

```
threshold-and4/
├── model.safetensors
├── model.py
├── config.json
├── create_safetensors.py
└── README.md
```

## License

MIT