threshold-xnor-mag7

Minimum-magnitude threshold circuit for XNOR (equivalence). Magnitude 7 is optimal via exhaustive enumeration of all 224,143 configurations.

Circuit Diagram

        x1      x2
        │       │
        │       │
   ┌────┴───────┴────┐
   │                 │
   ▼                 ▼
┌─────┐           ┌─────┐
│ N1  │           │ N2  │    Layer 1
│[-1,-1]          │[-1,-1]
│ b=0 │           │ b=1 │
└──┬──┘           └──┬──┘
   │                 │
   │    ┌────────────┘
   │    │
   ▼    ▼
  ┌──────┐
  │ OUT  │               Layer 2
  │[1,-1]│
  │ b=0  │
  └──┬───┘
     │
     ▼
   XNOR(x1,x2)

Key Results

Magnitude	Valid XNOR Circuits
0-6	0 (exhaustively verified)
7	2 (both included here)

Truth Table

x1	x2	N1	N2	OUT
0	0	1	1	1
0	1	0	1	0
1	0	0	1	0
1	1	0	0	1

The 2 Solutions

Both solutions use weights from {-1, 0, 1} only. They are related by neuron permutation symmetry (swap N1 ↔ N2).

Solution	Neuron 1	Neuron 2	Output
1	[-1,-1] b=0	[-1,-1] b=1	[1,-1] b=0
2	[-1,-1] b=1	[-1,-1] b=0	[-1,1] b=0

Interpretation:

• N1 (b=0): fires when x1 + x2 = 0 (both zero) — NOR
• N2 (b=1): fires when x1 + x2 ≤ 1 (at most one) — NAND
• OUT: fires when N1=1 OR N2=0 — equivalence detector

Comparison with XOR

Property	XOR-mag7	XNOR-mag7
Optimal magnitude	7	7
Solutions at optimal	6	2
Solution families	2	1
Zero biases	2	3

XNOR has fewer solutions because OR at layer 2 creates tighter constraints than XOR's AND.

Improvement Over Original

Original XNOR (magnitude 9):
  NOR:  [-1,-1], b=0   → magnitude 2
  AND:  [1, 1], b=-2   → magnitude 4  ← costly -2 bias
  OR:   [1, 1], b=-1   → magnitude 3

Optimized (magnitude 7, solution 1):
  N1:   [-1,-1], b=0   → magnitude 2
  N2:   [-1,-1], b=1   → magnitude 3
  OUT:  [1,-1],  b=0   → magnitude 2  ← zero bias

22% magnitude reduction (9 → 7).

Benefits of Lower Magnitude

• Fewer bits to represent weights (all weights ∈ {-1, 0, 1})
• Less energy for multiply-accumulate
• 3 zero biases (vs 1 in original)
• Simpler hardware implementation

Usage

from safetensors.torch import load_file
import torch

w = load_file('solution1.safetensors')

def xnor_gate(x1, x2):
    inp = torch.tensor([float(x1), float(x2)])

    n1 = int((inp * w['layer1.neuron1.weight']).sum() + w['layer1.neuron1.bias'] >= 0)
    n2 = int((inp * w['layer1.neuron2.weight']).sum() + w['layer1.neuron2.bias'] >= 0)

    hid = torch.tensor([float(n1), float(n2)])
    return int((hid * w['layer2.weight']).sum() + w['layer2.bias'] >= 0)

# Test
assert xnor_gate(0, 0) == 1  # same
assert xnor_gate(0, 1) == 0  # different
assert xnor_gate(1, 0) == 0  # different
assert xnor_gate(1, 1) == 1  # same

Files

threshold-xnor-mag7/
├── solution1.safetensors   # First solution
├── solution2.safetensors   # Second solution (neuron swap)
├── model.py                # Python implementation
├── config.json             # Metadata
├── create_safetensors.py   # Script to generate weights
└── README.md

License

MIT