threshold-popcount3

3-bit population count (Hamming weight). Counts the number of 1-bits in a 3-bit input, producing a 2-bit output (0-3).

Circuit

    x0      x1      x2
     │       │       │
     └───┬───┴───┬───┘
         │       │
    ┌────┴────┐  │
    │ Layer 1 │  │
    │ atleast1│  │  (sum >= 1)
    │ atleast2│  │  (sum >= 2)
    │ atleast3├──┘  (sum >= 3)
    └────┬────┘
         │
    ┌────┴────┐
    │ Layer 2 │
    │  XOR    │     out1 = atleast1 XOR atleast2
    │  pass   │     out0 = atleast2 XOR atleast3
    └────┬────┘
         │
         ▼
   [out1, out0]

Function

popcount3(x0, x1, x2) -> (out1, out0)

where output = 2*out1 + out0 = number of 1-bits in input

Truth Table

x0	x1	x2	Count	out1	out0
0	0	0	0	0	0
0	0	1	1	0	1
0	1	0	1	0	1
0	1	1	2	1	0
1	0	0	1	0	1
1	0	1	2	1	0
1	1	0	2	1	0
1	1	1	3	1	1

Mechanism

The circuit uses threshold gates to detect "at least k" conditions, then XOR gates to convert to binary:

Layer 1 - Threshold Detection:

Gate	Weights	Bias	Fires when
atleast1	[1,1,1]	-1	sum >= 1
atleast2	[1,1,1]	-2	sum >= 2
atleast3	[1,1,1]	-3	sum >= 3

Layer 2 - Binary Encoding:

The key insight: binary output bits can be computed from threshold outputs:

• out1 (2's place) = atleast2 XOR atleast3 = (sum >= 2) XOR (sum >= 3)

  • True when sum is exactly 2 or 3
  • Actually: out1 = atleast2 (since atleast3 implies atleast2)
  • Simplified: out1 = atleast2

• out0 (1's place) = atleast1 XOR atleast2 XOR atleast3

  • True when sum is 1 or 3 (odd from {1,2,3} perspective)
  • Simplified: out0 = parity of threshold outputs

Actually, simpler encoding:

• out1 = atleast2
• out0 = atleast1 XOR atleast2

Architecture

Layer	Components	Neurons
1	atleast1, atleast2, atleast3	3
2-3	XOR for out0	3

Total: 6 neurons

Alternative simpler design:

• out1 = atleast2 (direct wire, 0 extra neurons)
• out0 = atleast1 XOR atleast2 (3 neurons for XOR)

Using direct threshold for out1: 4 neurons total

Parameters

Inputs	3
Outputs	2
Neurons	6
Layers	3
Parameters	21
Magnitude	22

Usage

from safetensors.torch import load_file
import torch

w = load_file('model.safetensors')

def popcount3(x0, x1, x2):
    inp = torch.tensor([float(x0), float(x1), float(x2)])

    # Layer 1: Threshold detection
    at1 = int((inp @ w['atleast1.weight'].T + w['atleast1.bias'] >= 0).item())
    at2 = int((inp @ w['atleast2.weight'].T + w['atleast2.bias'] >= 0).item())

    # out1 = atleast2 (sum >= 2 means bit 1 is set)
    out1 = at2

    # out0 = atleast1 XOR atleast2
    l1 = torch.tensor([float(at1), float(at2)])
    or_out = int((l1 @ w['xor.or.weight'].T + w['xor.or.bias'] >= 0).item())
    nand_out = int((l1 @ w['xor.nand.weight'].T + w['xor.nand.bias'] >= 0).item())
    l2 = torch.tensor([float(or_out), float(nand_out)])
    out0 = int((l2 @ w['xor.and.weight'].T + w['xor.and.bias'] >= 0).item())

    return out1, out0

# Examples
print(popcount3(0, 0, 0))  # (0, 0) = 0
print(popcount3(1, 0, 0))  # (0, 1) = 1
print(popcount3(1, 1, 0))  # (1, 0) = 2
print(popcount3(1, 1, 1))  # (1, 1) = 3

Applications

• Hamming distance calculation
• Set cardinality in bit vectors
• Error weight computation
• Branch prediction (counting set bits)
• Cryptographic operations

Files

threshold-popcount3/
├── model.safetensors
├── model.py
├── create_safetensors.py
├── config.json
└── README.md

License

MIT