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

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