threshold-symmetric-s23

Symmetric function S_2^3: outputs 1 if and only if exactly 2 of 3 inputs are 1.

Function

S_2^3(x2, x1, x0) = 1 iff (x2 + x1 + x0) = 2

Symmetric functions are invariant under input permutation.

Truth Table

x2	x1	x0	sum	y
0	0	0	0	0
0	0	1	1	0
0	1	0	1	0
0	1	1	2	1
1	0	0	1	0
1	0	1	2	1
1	1	0	2	1
1	1	1	3	0

Architecture

x2   x1   x0
 │    │    │
 └────┴────┘
      │
 ┌────┴────┐
 │         │
 ▼         ▼
┌─────┐  ┌─────┐
│>=2  │  │<=2  │   Layer 1
└─────┘  └─────┘
   │        │
   └───┬────┘
       │
       ▼
    ┌─────┐
    │ AND │        Layer 2
    └─────┘
       │
       ▼
       y

• at_least_2: w=[1,1,1], b=-2
• at_most_2: w=[-1,-1,-1], b=2
• y = AND(at_least_2, at_most_2)

Parameters

Inputs	3
Outputs	1
Neurons	3
Layers	2
Parameters	11
Magnitude	14

Symmetric Functions

The general symmetric function S_k^n outputs 1 iff exactly k of n inputs are 1. Special cases:

• S_0^n = NOR (no inputs set)
• S_n^n = AND (all inputs set)
• S_1^n to S_{n-1}^n = "exactly k" functions

Usage

from safetensors.torch import load_file
import torch

w = load_file('model.safetensors')

def s23(x2, x1, x0):
    inp = torch.tensor([float(x2), float(x1), float(x0)])
    al2 = int((inp @ w['at_least_2.weight'].T + w['at_least_2.bias'] >= 0).item())
    am2 = int((inp @ w['at_most_2.weight'].T + w['at_most_2.bias'] >= 0).item())
    l1 = torch.tensor([float(al2), float(am2)])
    return int((l1 @ w['y.weight'].T + w['y.bias'] >= 0).item())

# s23(1, 1, 0) = 1  # exactly 2 of 3
# s23(1, 1, 1) = 0  # 3 of 3, not exactly 2

License

MIT