---
license: mit
tags:
- formal-verification
- coq
- threshold-logic
- neuromorphic
- multi-layer
- equivalence
---

# tiny-BiImplies-verified

Formally verified biconditional gate (if and only if). Two-layer threshold network computing logical equivalence with 100% accuracy.

## Architecture

| Component | Value |
|-----------|-------|
| Inputs | 2 |
| Outputs | 1 |
| Neurons | 3 (2 hidden, 1 output) |
| Layers | 2 |
| Parameters | 9 |
| **Layer 1, Neuron 1 (NOR)** | |
| Weights | [-1, -1] |
| Bias | 0 |
| **Layer 1, Neuron 2 (AND)** | |
| Weights | [1, 1] |
| Bias | -2 |
| **Layer 2 (OR)** | |
| Weights | [1, 1] |
| Bias | -1 |
| Activation | Heaviside step (all layers) |

## Key Properties

- 100% accuracy (4/4 inputs correct)
- Coq-proven correctness
- Minimal 2-layer architecture
- Integer weights
- Equivalence relation (reflexive, symmetric, transitive)
- Identical to XNOR

## Usage

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

weights = load_file('biimplies.safetensors')

def biimplies_gate(x, y):
    inputs = torch.tensor([float(x), float(y)])

    # Layer 1: NOR and AND
    nor_sum = (inputs * weights['layer1.neuron1.weight']).sum() + weights['layer1.neuron1.bias']
    nor_out = int(nor_sum >= 0)

    and_sum = (inputs * weights['layer1.neuron2.weight']).sum() + weights['layer1.neuron2.bias']
    and_out = int(and_sum >= 0)

    # Layer 2: OR
    layer1_outs = torch.tensor([float(nor_out), float(and_out)])
    or_sum = (layer1_outs * weights['layer2.weight']).sum() + weights['layer2.bias']
    return int(or_sum >= 0)

# Test
print(biimplies_gate(0, 0))  # 1 (both false, equivalent)
print(biimplies_gate(0, 1))  # 0 (different, not equivalent)
print(biimplies_gate(1, 0))  # 0 (different, not equivalent)
print(biimplies_gate(1, 1))  # 1 (both true, equivalent)
```

## Verification

**Coq Theorem**:
```coq
Theorem biimplies_correct : forall x y, biimplies_circuit x y = eqb x y.
```

Proven axiom-free with properties:
- Reflexivity (x ↔ x)
- Symmetry (x ↔ y → y ↔ x)
- Transitivity (x ↔ y ∧ y ↔ z → x ↔ z)
- Full equivalence relation

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

## Circuit Operation

BiImplies outputs true when inputs are equal.

BiImplies(x,y) = OR(NOR(x,y), AND(x,y)) = (x ↔ y)

## Citation

```bibtex
@software{tiny_biimplies_prover_2025,
  title={tiny-BiImplies-verified: Formally Verified Biconditional Gate},
  author={Norton, Charles},
  url={https://huggingface.co/phanerozoic/tiny-BiImplies-verified},
  year={2025}
}
```