metadata
license: mit
tags:
- pytorch
- safetensors
- threshold-logic
- neuromorphic
- functionally-complete
threshold-nand
The universal gate. Any Boolean function can be constructed from NAND alone.
Circuit
x y
│ │
└─┬─┘
▼
┌────────┐
│w: -1,-1│
│ b: +1 │
└────────┘
│
▼
NAND(x,y)
Mechanism
Negative weights mean inputs subtract from the sum. The positive bias starts us above threshold, and inputs pull us down:
| x | y | sum | output |
|---|---|---|---|
| 0 | 0 | +1 | 1 |
| 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | -1 | 0 |
Only when both inputs are active do we fall below threshold. This is AND with inverted output.
Parameters
| Weights | [-1, -1] |
| Bias | +1 |
| Total | 3 parameters |
Optimality
Exhaustive enumeration of all 25 weight configurations at magnitudes 0-3 confirms this circuit is already at minimum magnitude (3). There is exactly one valid configuration at magnitude 3, and no valid configurations exist below it.
Functional Completeness
NAND can build any Boolean function:
- NOT(x) = NAND(x, x)
- AND(x,y) = NOT(NAND(x,y)) = NAND(NAND(x,y), NAND(x,y))
- OR(x,y) = NAND(NOT(x), NOT(y))
This is why NAND gates dominate digital logic fabrication.
Usage
from safetensors.torch import load_file
import torch
w = load_file('model.safetensors')
def nand_gate(x, y):
inputs = torch.tensor([float(x), float(y)])
return int((inputs * w['weight']).sum() + w['bias'] >= 0)
Files
threshold-nand/
├── model.safetensors
├── model.py
├── config.json
└── README.md
License
MIT