|
|
---
|
|
|
license: mit
|
|
|
tags:
|
|
|
- formal-verification
|
|
|
- coq
|
|
|
- threshold-logic
|
|
|
- neuromorphic
|
|
|
- modular-arithmetic
|
|
|
---
|
|
|
|
|
|
# tiny-mod6-verified
|
|
|
|
|
|
Formally verified MOD-6 circuit. Single-layer threshold network computing modulo-6 arithmetic with 100% accuracy.
|
|
|
|
|
|
## Architecture
|
|
|
|
|
|
| Component | Value |
|
|
|
|-----------|-------|
|
|
|
| Inputs | 8 |
|
|
|
| Outputs | 1 (per residue class) |
|
|
|
| Neurons | 6 (one per residue 0-5) |
|
|
|
| Parameters | 54 (6 × 9) |
|
|
|
| Weights | [1, 1, 1, 1, 1, -5, 1, 1] |
|
|
|
| Bias | 0 |
|
|
|
| Activation | Heaviside step |
|
|
|
|
|
|
## Key Properties
|
|
|
|
|
|
- 100% accuracy (256/256 inputs correct)
|
|
|
- Coq-proven correctness
|
|
|
- Algebraic weight pattern: resets every 6 positions
|
|
|
- Computes Hamming weight mod 6
|
|
|
- Compatible with neuromorphic hardware
|
|
|
|
|
|
## Algebraic Pattern
|
|
|
|
|
|
MOD-6 uses the pattern with reset at position 6:
|
|
|
- Positions 1-5: weight = 1
|
|
|
- Position 6: weight = 1-6 = -5
|
|
|
- Positions 7-8: weight = 1
|
|
|
|
|
|
This creates a cumulative sum that cycles mod 6.
|
|
|
|
|
|
## Usage
|
|
|
|
|
|
```python
|
|
|
import torch
|
|
|
from safetensors.torch import load_file
|
|
|
|
|
|
weights = load_file('mod6.safetensors')
|
|
|
|
|
|
def mod6_circuit(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 weighted_sum.item()
|
|
|
|
|
|
# Test
|
|
|
print(mod6_circuit([1,1,1,1,1,1,0,0])) # 6 mod 6 = 0
|
|
|
print(mod6_circuit([1,1,1,1,1,1,1,0])) # 7 mod 6 = 1
|
|
|
```
|
|
|
|
|
|
## Verification
|
|
|
|
|
|
**Coq Theorem**:
|
|
|
```coq
|
|
|
Theorem mod6_correct_residue_0 : forall x0 x1 x2 x3 x4 x5 x6 x7,
|
|
|
mod6_is_zero [x0; x1; x2; x3; x4; x5; x6; x7] =
|
|
|
Z.eqb ((Z.of_nat (hamming_weight [x0; x1; x2; x3; x4; x5; x6; x7])) mod 6) 0.
|
|
|
```
|
|
|
|
|
|
Proven axiom-free using algebraic weight patterns.
|
|
|
|
|
|
Full proof: [coq-circuits/Modular/Mod6.v](https://github.com/CharlesCNorton/coq-circuits/blob/main/coq/Modular/Mod6.v)
|
|
|
|
|
|
## Residue Distribution
|
|
|
|
|
|
For 8-bit inputs (256 total):
|
|
|
- Residue 0: 29 inputs
|
|
|
- Residue 1: 16 inputs
|
|
|
- Residue 2: 29 inputs
|
|
|
- Residue 3: 56 inputs
|
|
|
- Residue 4: 70 inputs
|
|
|
- Residue 5: 56 inputs
|
|
|
|
|
|
## Citation
|
|
|
|
|
|
```bibtex
|
|
|
@software{tiny_mod6_verified_2025,
|
|
|
title={tiny-mod6-verified: Formally Verified MOD-6 Circuit},
|
|
|
author={Norton, Charles},
|
|
|
url={https://huggingface.co/phanerozoic/tiny-mod6-verified},
|
|
|
year={2025}
|
|
|
}
|
|
|
```
|
|
|
|
