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

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