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

# tiny-NAND-verified

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

## Architecture

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

## Key Properties

- 100% accuracy (4/4 inputs correct)
- Coq-proven correctness
- Single threshold neuron
- Integer weights
- Commutative: NAND(x,y) = NAND(y,x)
- Functionally complete (can build any Boolean function)
- Self-dual: NAND(x,x) = NOT(x)

## Usage

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

weights = load_file('nand.safetensors')

def nand_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(nand_gate(0, 0))  # 1
print(nand_gate(0, 1))  # 1
print(nand_gate(1, 0))  # 1
print(nand_gate(1, 1))  # 0
```

## Verification

**Coq Theorem**:
```coq
Theorem nand_correct : forall x y, nand_circuit x y = negb (andb x y).
```

Proven axiom-free with properties:
- Commutativity
- Self-duality (NAND(x,x) = NOT(x))
- Functional completeness
- Identity with true gives NOT
- Absorption with false gives true

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

## Circuit Operation

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

Fails to fire only when both inputs are true.

## Functional Completeness

NAND is functionally complete - any Boolean function can be built from NAND gates alone. This makes it particularly important for circuit composition.

## Citation

```bibtex
@software{tiny_nand_prover_2025,
  title={tiny-NAND-verified: Formally Verified NAND Gate},
  author={Norton, Charles},
  url={https://huggingface.co/phanerozoic/tiny-NAND-verified},
  year={2025}
}
```