threshold-xnor

The equality detector. Fires when both inputs match - both 0 or both 1. Like XOR, this function is not linearly separable and requires two layers.

Circuit

      x       y
      │       │
      ├───┬───┤
      │   │   │
      ▼   │   ▼
  ┌──────┐│┌──────┐
  │ NOR  │││ AND  │   Layer 1
  │w:-1,-1││w:1,1 │
  │b: 0  │││b: -2 │
  └──────┘│└──────┘
      │   │   │
      └───┼───┘
          ▼
      ┌──────┐
      │  OR  │         Layer 2
      │w: 1,1│
      │b: -1 │
      └──────┘
          │
          ▼
      XNOR(x,y)

Mechanism

XNOR catches equality via two cases:

• NOR fires when both inputs are 0 (both quiet)
• AND fires when both inputs are 1 (both active)
• OR combines: at least one case holds

x	y	NOR	AND	OR(NOR,AND)
0	0	1	0	1
0	1	0	0	0
1	0	0	0	0
1	1	0	1	1

XNOR = NOT(XOR). Same non-linear structure, opposite meaning.

Parameters

Layer	Weights	Bias
NOR	[-1, -1]	0
AND	[1, 1]	-2
OR	[1, 1]	-1
Total		9

Properties

• Commutative: XNOR(x,y) = XNOR(y,x)
• Reflexive: XNOR(x,x) = 1
• Equality: x = y iff XNOR(x,y) = 1

Usage

from safetensors.torch import load_file
import torch

w = load_file('model.safetensors')

def xnor_gate(x, y):
    inp = torch.tensor([float(x), float(y)])

    nor_out = int((inp * w['layer1.neuron1.weight']).sum() + w['layer1.neuron1.bias'] >= 0)
    and_out = int((inp * w['layer1.neuron2.weight']).sum() + w['layer1.neuron2.bias'] >= 0)

    l1 = torch.tensor([float(nor_out), float(and_out)])
    return int((l1 * w['layer2.weight']).sum() + w['layer2.bias'] >= 0)

Files

threshold-xnor/
├── model.safetensors
├── model.py
├── config.json
└── README.md

License

MIT