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phanerozoic/
threshold-nand3

Name: threshold-nand3
Availability: InStock

functionally-complete

mit

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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

python

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

phanerozoic/threshold-nand3