threshold-symmetric-s134

Symmetric function S_{1,3}^4: outputs 1 if exactly 1 or exactly 3 of 4 inputs are 1. Equivalent to 4-bit odd parity.

Function

S_{1,3}^4(x3, x2, x1, x0) = 1 iff sum ∈ {1, 3}

This is the "odd count" function on 4 bits.

Truth Table

x3	x2	x1	x0	sum	y
0	0	0	0	0	0
0	0	0	1	1	1
0	0	1	1	2	0
0	1	1	1	3	1
1	1	1	1	4	0

Output is 1 for odd sums (1, 3), 0 for even sums (0, 2, 4).

Architecture

    x3   x2   x1   x0
         │
    ┌────┴────┐
    │         │
┌───┴───┐ ┌───┴───┐
│ >=1   │ │ <=1   │ │ >=3   │ │ <=3   │   Layer 1
└───┬───┘ └───┬───┘ └───┬───┘ └───┬───┘
    │         │         │         │
    └────┬────┘         └────┬────┘
         │                   │
     ┌───┴───┐           ┌───┴───┐
     │ AND   │           │ AND   │         Layer 2
     │exact=1│           │exact=3│
     └───┬───┘           └───┬───┘
         │                   │
         └─────────┬─────────┘
                   │
               ┌───┴───┐
               │  OR   │               Layer 3
               └───┬───┘
                   │
                   y

Parameters

Inputs	4
Outputs	1
Neurons	7
Layers	3
Parameters	33
Magnitude	35

Relation to Parity

S_{1,3}^4 is equivalent to 4-bit parity (XOR of all bits). However, this implementation uses threshold counting rather than XOR decomposition, demonstrating an alternative approach.

Usage

from safetensors.torch import load_file
import torch

w = load_file('model.safetensors')

# Full implementation in model.py
# s134(0, 1, 1, 1) = 1  # sum=3, odd
# s134(1, 1, 0, 0) = 0  # sum=2, even

License

MIT