tiny-Minority-verified

Formally verified minority gate for 8-bit inputs. Single threshold neuron computing minority function with 100% accuracy.

Architecture

Component	Value
Inputs	8
Outputs	1
Neurons	1
Parameters	9
Weights	[-1, -1, -1, -1, -1, -1, -1, -1]
Bias	3
Activation	Heaviside step

Key Properties

• 100% accuracy (256/256 inputs correct)
• Coq-proven correctness
• Single threshold neuron
• Integer weights
• Fires when ≤3 of 8 inputs are true
• Complement of majority (inverted weights)

Usage

import torch
from safetensors.torch import load_file

weights = load_file('minority.safetensors')

def minority_gate(bits):
    # bits: list of 8 binary values
    inputs = torch.tensor([float(b) for b in bits])
    weighted_sum = (inputs * weights['weight']).sum() + weights['bias']
    return int(weighted_sum >= 0)

# Test
print(minority_gate([0,0,0,0,0,0,0,0]))  # 1 (minority)
print(minority_gate([1,1,1,0,0,0,0,0]))  # 1 (3/8, minority)
print(minority_gate([1,1,1,1,0,0,0,0]))  # 0 (4/8, not minority)
print(minority_gate([1,1,1,1,1,1,1,1]))  # 0 (no minority)

Verification

Coq Theorem:

Theorem minority_correct : forall x0 x1 x2 x3 x4 x5 x6 x7,
  minority_circuit [x0; x1; x2; x3; x4; x5; x6; x7] =
  minority_spec [x0; x1; x2; x3; x4; x5; x6; x7].

Proven axiom-free via three methods:

1. Exhaustive: Verified on all 256 inputs
2. Universal: Quantified proof over all boolean combinations
3. Algebraic: Characterized via hamming weight ≤ 3

Algebraic characterization:

Theorem minority_hamming_weight (xs : list bool) :
  length xs = 8 ->
  minority_circuit xs = true <-> hamming_weight xs <= 3.

Full proof: coq-circuits/Threshold/Minority.v

Circuit Operation

Input with k true bits produces weighted sum: -k + 3

• k ≤ 3: weighted_sum ≥ 0 → output 1 (minority)
• k > 3: weighted_sum < 0 → output 0 (not minority)

Applications

• Inverted voting systems
• Fault detection (low activation)
• Sparse pattern detection
• Neuromorphic hardware

Citation

@software{tiny_minority_prover_2025,
  title={tiny-Minority-verified: Formally Verified Minority Gate},
  author={Norton, Charles},
  url={https://huggingface.co/phanerozoic/tiny-Minority-verified},
  year={2025}
}