|
|
--- |
|
|
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} |
|
|
} |
|
|
``` |
|
|
|
