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--- |
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license: mit |
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tags: |
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- formal-verification |
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- coq |
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- threshold-logic |
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- neuromorphic |
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- modular-arithmetic |
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--- |
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# tiny-mod4-verified |
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Formally verified MOD-4 circuit. Single-layer threshold network computing modulo-4 arithmetic with 100% accuracy. |
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## Architecture |
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| Component | Value | |
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|-----------|-------| |
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| Inputs | 8 | |
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| Outputs | 1 (per residue class) | |
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| Neurons | 4 (one per residue 0-3) | |
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| Parameters | 36 (4 × 9) | |
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| Weights | [1, 1, 1, -3, 1, 1, 1, -3] | |
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| Bias | 0 | |
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| Activation | Heaviside step | |
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## Key Properties |
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- 100% accuracy (256/256 inputs correct) |
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- Coq-proven correctness |
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- Algebraic weight pattern: (1, 1, 1, 1-m) repeating |
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- Computes Hamming weight mod 4 |
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- Compatible with neuromorphic hardware |
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## Algebraic Pattern |
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MOD-4 uses the repeating pattern `[1, 1, 1, -3]`: |
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- Positions 1-3: weight = 1 |
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- Position 4: weight = 1-4 = -3 |
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- Positions 5-7: weight = 1 |
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- Position 8: weight = 1-4 = -3 |
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This creates a cumulative sum that cycles mod 4. |
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## Usage |
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```python |
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import torch |
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from safetensors.torch import load_file |
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weights = load_file('mod4.safetensors') |
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def mod4_circuit(bits): |
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# bits: list of 8 binary values |
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inputs = torch.tensor([float(b) for b in bits]) |
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weighted_sum = (inputs * weights['weight']).sum() + weights['bias'] |
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# Output represents cumulative sum mod 4 |
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return weighted_sum.item() |
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# Test |
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print(mod4_circuit([1,0,0,0,0,0,0,0])) # 1 mod 4 = 1 |
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print(mod4_circuit([1,1,1,1,0,0,0,0])) # 4 mod 4 = 0 |
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print(mod4_circuit([1,1,1,1,1,0,0,0])) # 5 mod 4 = 1 |
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``` |
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## Verification |
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**Coq Theorem**: |
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```coq |
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Theorem mod4_correct_residue_0 : forall x0 x1 x2 x3 x4 x5 x6 x7, |
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mod4_is_zero [x0; x1; x2; x3; x4; x5; x6; x7] = |
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Z.eqb ((Z.of_nat (hamming_weight [x0; x1; x2; x3; x4; x5; x6; x7])) mod 4) 0. |
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``` |
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Proven axiom-free using: |
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1. **Algebraic correctness**: Weight pattern proven to maintain mod-4 invariant |
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2. **Universal quantification**: Verified for all 8-bit inputs |
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3. **Parametric instantiation**: Instantiates `mod_m_weights_8` with m=4 |
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Full proof: [coq-circuits/Modular/Mod4.v](https://github.com/CharlesCNorton/coq-circuits/blob/main/coq/Modular/Mod4.v) |
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## Residue Distribution |
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For 8-bit inputs (256 total): |
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- Residue 0: 72 inputs |
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- Residue 1: 64 inputs |
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- Residue 2: 56 inputs |
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- Residue 3: 64 inputs |
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## Citation |
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```bibtex |
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@software{tiny_mod4_verified_2025, |
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title={tiny-mod4-verified: Formally Verified MOD-4 Circuit}, |
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author={Norton, Charles}, |
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url={https://huggingface.co/phanerozoic/tiny-mod4-verified}, |
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year={2025} |
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} |
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``` |
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