threshold-xor-mag7

Minimum-magnitude threshold circuit for XOR. Magnitude 7 is proven optimal via exhaustive Coq computation.

Key Results

Magnitude	Valid XOR Circuits
0-6	0 (proven impossible)
7	6 (all included here)

The 6 Solutions

All solutions use weights from {-1, 0, 1} only.

Solution	Neuron 1	Neuron 2	Output	Structure
1	[-1,1] b=0	[1,-1] b=0	[-1,-1] b=1	Symmetric opposites
2	[1,-1] b=0	[-1,1] b=0	[-1,-1] b=1	Symmetric opposites
3	[-1,1] b=0	[-1,1] b=-1	[-1,1] b=0	Same weights
4	[1,-1] b=0	[1,-1] b=-1	[-1,1] b=0	Same weights
5	[-1,1] b=-1	[-1,1] b=0	[1,-1] b=0	Same weights
6	[1,-1] b=-1	[1,-1] b=0	[1,-1] b=0	Same weights

Solutions 1-2 form one structural family (symmetric opposites in layer 1). Solutions 3-6 form another family (identical weights in layer 1, differing biases).

Improvement Over Original

The original XOR circuit uses OR + NAND + AND with magnitude 10:

Original (magnitude 10):
  OR:   [1, 1], b=-1   → magnitude 3
  NAND: [-1,-1], b=1   → magnitude 3
  AND:  [1, 1], b=-2   → magnitude 4  ← costly -2 bias

Optimized (magnitude 7):
  n1:   [-1, 1], b=0   → magnitude 2  ← zero bias
  n2:   [1, -1], b=0   → magnitude 2  ← zero bias
  out:  [-1,-1], b=1   → magnitude 3

30% magnitude reduction (10 → 7).

Coq Proof

optimality_proof.v contains a machine-verified proof that:

1. No configuration with magnitude ≤ 6 computes XOR
2. At least one configuration with magnitude 7 computes XOR

The proof uses exhaustive computation via native_compute/vm_compute.

Theorem mag_6 : any_xor (configs_at_mag 6) = false.
Proof. native_compute. reflexivity. Qed.

Theorem mag_7 : any_xor (configs_at_mag 7) = true.
Proof. native_compute. reflexivity. Qed.

Benefits of Lower Magnitude

• Fewer bits to represent weights
• Less energy for multiply-accumulate
• Sparser (2 zero biases vs 0 in original)
• Simpler hardware implementation

Files

threshold-xor-mag7/
├── solution1.safetensors   # Symmetric family
├── solution2.safetensors   # Symmetric family
├── solution3.safetensors   # Same-weights family
├── solution4.safetensors   # Same-weights family
├── solution5.safetensors   # Same-weights family
├── solution6.safetensors   # Same-weights family
├── optimality_proof.v      # Coq proof of optimality
├── model.py                # Python implementation
├── config.json             # Metadata
└── README.md

Usage

from safetensors.torch import load_file
import torch

w = load_file('solution1.safetensors')

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

    h1 = int((inp * w['layer1.neuron1.weight']).sum() + w['layer1.neuron1.bias'] >= 0)
    h2 = int((inp * w['layer1.neuron2.weight']).sum() + w['layer1.neuron2.bias'] >= 0)

    hid = torch.tensor([float(h1), float(h2)])
    return int((hid * w['layer2.weight']).sum() + w['layer2.bias'] >= 0)

# Test
assert xor_gate(0, 0) == 0
assert xor_gate(0, 1) == 1
assert xor_gate(1, 0) == 1
assert xor_gate(1, 1) == 0

Compile the Proof

coqc optimality_proof.v

License

MIT