---
license: mit
tags:
- pytorch
- safetensors
- threshold-logic
- neuromorphic
---

# threshold-implies

Material implication: x → y. The only two-input Boolean function with asymmetric weights.

## Circuit

```
    x   y
    │   │
    └─┬─┘
      ▼
  ┌───────┐
  │w: -1,1│
  │ b:  0 │
  └───────┘
      │
      ▼
    x → y
```

## Mechanism

The antecedent x has weight -1 (inhibitory), the consequent y has weight +1 (excitatory):

| x | y | sum | output | meaning |
|---|---|-----|--------|---------|
| 0 | 0 | 0 | 1 | false → false |
| 0 | 1 | +1 | 1 | false → true |
| 1 | 0 | -1 | 0 | true → false ✗ |
| 1 | 1 | 0 | 1 | true → true |

The only failure: asserting a true antecedent with a false consequent. This is the only thing implication forbids.

## Equivalent Forms

- x → y = ¬x ∨ y
- x → y = ¬(x ∧ ¬y)

The weights [-1, +1] directly implement ¬x + y.

## Parameters

| | |
|---|---|
| Weights | [-1, +1] |
| Bias | 0 |
| Total | 3 parameters |

## Optimality

Exhaustive enumeration of all 25 weight configurations at magnitudes 0-2 confirms this circuit is **already at minimum magnitude (2)**. There is exactly one valid configuration at magnitude 2, and no valid configurations exist below it.

## Properties

- Linearly separable (unlike XOR)
- Not commutative: (x → y) ≠ (y → x)
- Reflexive: x → x = 1
- Ex falso quodlibet: 0 → y = 1

## Usage

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

w = load_file('model.safetensors')

def implies_gate(x, y):
    inputs = torch.tensor([float(x), float(y)])
    return int((inputs * w['weight']).sum() + w['bias'] >= 0)
```

## Files

```
threshold-implies/
├── model.safetensors
├── model.py
├── config.json
└── README.md
```

## License

MIT