metadata
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
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