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---
license: mit
tags:
- pytorch
- safetensors
- threshold-logic
- neuromorphic
- multi-layer
---
# threshold-biimplies
The biconditional: x ↔ y ("if and only if"). Functionally identical to XNOR, but framed as the logical equivalence relation.
## Circuit
```
x y
│ │
├───┬───┤
│ │ │
▼ │ ▼
┌──────┐│┌──────┐
│ NOR │││ AND │ Layer 1
│w:-1,-1││w:1,1 │
│b: 0 │││b: -2 │
└──────┘│└──────┘
│ │ │
└───┼───┘
▼
┌──────┐
│ OR │ Layer 2
│w: 1,1│
│b: -1 │
└──────┘
│
▼
x ↔ y
```
## Mechanism
The biconditional tests whether x and y have the same truth value:
| x | y | NOR | AND | x ↔ y |
|---|---|-----|-----|-------|
| 0 | 0 | 1 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 |
NOR catches "both false," AND catches "both true," OR combines.
## Why Two Layers?
Unlike simple implication (x → y), the biconditional is not linearly separable. It requires detecting two diagonal cases - same problem as XOR.
Implication x → y can be computed with weights [-1, +1] because it fails only at (1,0). Biimplication fails at both (0,1) and (1,0) - these points cannot be separated from (0,0) and (1,1) by a single hyperplane.
## Parameters
| Layer | Weights | Bias |
|-------|---------|------|
| NOR | [-1, -1] | 0 |
| AND | [1, 1] | -2 |
| OR | [1, 1] | -1 |
| **Total** | | **9** |
## Properties
- Reflexive: x ↔ x = 1
- Symmetric: (x ↔ y) = (y ↔ x)
- Transitive: (x ↔ y) ∧ (y ↔ z) → (x ↔ z)
Full equivalence relation.
## Usage
```python
from safetensors.torch import load_file
import torch
w = load_file('model.safetensors')
def biimplies_gate(x, y):
inp = torch.tensor([float(x), float(y)])
nor_out = int((inp * w['layer1.neuron1.weight']).sum() + w['layer1.neuron1.bias'] >= 0)
and_out = int((inp * w['layer1.neuron2.weight']).sum() + w['layer1.neuron2.bias'] >= 0)
l1 = torch.tensor([float(nor_out), float(and_out)])
return int((l1 * w['layer2.weight']).sum() + w['layer2.bias'] >= 0)
```
## Files
```
threshold-biimplies/
├── model.safetensors
├── model.py
├── config.json
└── README.md
```
## License
MIT
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