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

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