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

# tiny-AND-verified

Formally verified AND gate. Single threshold neuron computing conjunction with 100% accuracy.

## Architecture

| Component | Value |
|-----------|-------|
| Inputs | 2 |
| Outputs | 1 |
| Neurons | 1 |
| Parameters | 3 |
| Weights | [1, 1] |
| Bias | -2 |
| Activation | Heaviside step |

## Key Properties

- 100% accuracy (4/4 inputs correct)
- Coq-proven correctness
- Single threshold neuron
- Integer weights
- Commutative: AND(x,y) = AND(y,x)
- Associative: AND(x,AND(y,z)) = AND(AND(x,y),z)
- Idempotent: AND(x,x) = x

## Usage

```python
import torch
from safetensors.torch import load_file

weights = load_file('and.safetensors')

def and_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(and_gate(0, 0))  # 0
print(and_gate(0, 1))  # 0
print(and_gate(1, 0))  # 0
print(and_gate(1, 1))  # 1
```

## Verification

**Coq Theorem**:
```coq
Theorem and_correct : forall x y, and_circuit x y = andb x y.
```

Proven axiom-free with properties:
- Commutativity
- Associativity
- Identity (AND with true)
- Absorption (AND with false)
- Idempotence

Full proof: [coq-circuits/Boolean/AND.v](https://github.com/CharlesCNorton/coq-circuits/blob/main/coq/Boolean/AND.v)

## Circuit Operation

Input combination produces weighted sum:
- (0,0): 0*1 + 0*1 - 2 = -2 < 0 → 0
- (0,1): 0*1 + 1*1 - 2 = -1 < 0 → 0
- (1,0): 1*1 + 0*1 - 2 = -1 < 0 → 0
- (1,1): 1*1 + 1*1 - 2 = 0 >= 0 → 1

Requires both inputs to reach threshold.

## Citation

```bibtex
@software{tiny_and_prover_2025,
  title={tiny-AND-verified: Formally Verified AND Gate},
  author={Norton, Charles},
  url={https://huggingface.co/phanerozoic/tiny-AND-verified},
  year={2025}
}
```