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phanerozoic/
threshold-implies

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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

phanerozoic/threshold-implies