---
license: mit
tags:
- formal-verification
- coq
- threshold-logic
- neuromorphic
- minority
---

# 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

```python
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**:
```coq
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**:
```coq
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](https://github.com/CharlesCNorton/coq-circuits/blob/main/coq/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

```bibtex
@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}
}
```