File size: 2,225 Bytes
41c2fe3 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 |
---
license: mit
tags:
- formal-verification
- coq
- threshold-logic
- neuromorphic
- functionally-complete
---
# tiny-NOR-verified
Formally verified NOR gate. Single threshold neuron computing negated disjunction with 100% accuracy.
## Architecture
| Component | Value |
|-----------|-------|
| Inputs | 2 |
| Outputs | 1 |
| Neurons | 1 |
| Parameters | 3 |
| Weights | [-1, -1] |
| Bias | 0 |
| Activation | Heaviside step |
## Key Properties
- 100% accuracy (4/4 inputs correct)
- Coq-proven correctness
- Single threshold neuron
- Integer weights
- Commutative: NOR(x,y) = NOR(y,x)
- Functionally complete (can build any Boolean function)
- Self-dual: NOR(x,x) = NOT(x)
## Usage
```python
import torch
from safetensors.torch import load_file
weights = load_file('nor.safetensors')
def nor_gate(x, y):
# Heaviside: weighted_sum + bias >= 0
inputs = torch.tensor([float(x), float(y)])
weighted_sum = (inputs * weights['weight']).sum() + weights['bias']
return int(weighted_sum >= 0)
# Test
print(nor_gate(0, 0)) # 1
print(nor_gate(0, 1)) # 0
print(nor_gate(1, 0)) # 0
print(nor_gate(1, 1)) # 0
```
## Verification
**Coq Theorem**:
```coq
Theorem nor_correct : forall x y, nor_circuit x y = negb (orb x y).
```
Proven axiom-free with properties:
- Commutativity
- Self-duality (NOR(x,x) = NOT(x))
- Functional completeness
- Identity with false gives NOT
- Absorption with true gives false
Full proof: [coq-circuits/Boolean/NOR.v](https://github.com/CharlesCNorton/coq-circuits/blob/main/coq/Boolean/NOR.v)
## Circuit Operation
Input combination produces weighted sum:
- (0,0): 0*(-1) + 0*(-1) + 0 = 0 >= 0 → 1
- (0,1): 0*(-1) + 1*(-1) + 0 = -1 < 0 → 0
- (1,0): 1*(-1) + 0*(-1) + 0 = -1 < 0 → 0
- (1,1): 1*(-1) + 1*(-1) + 0 = -2 < 0 → 0
Fires only when both inputs are false.
## Functional Completeness
NOR is functionally complete - any Boolean function can be built from NOR gates alone. This makes it particularly important for circuit composition.
## Citation
```bibtex
@software{tiny_nor_prover_2025,
title={tiny-NOR-verified: Formally Verified NOR Gate},
author={Norton, Charles},
url={https://huggingface.co/phanerozoic/tiny-NOR-verified},
year={2025}
}
```
|