threshold-exactly2outof8

Exactly-2-out-of-8 detector. Fires when precisely two inputs are active. The pair detector.

Circuit

  x₀ x₁ x₂ x₃ x₄ x₅ x₆ x₇
   │  │  │  │  │  │  │  │
   └──┴──┴──┴──┼──┴──┴──┴──┘
               │
       ┌───────┴───────┐
       ▼               ▼
  ┌─────────┐     ┌─────────┐
  │ AtLeast2│     │ AtMost2 │
  │  HW ≥ 2 │     │  HW ≤ 2 │
  └─────────┘     └─────────┘
       │               │
       └───────┬───────┘
               ▼
          ┌─────────┐
          │   AND   │
          └─────────┘
               │
               ▼
          (HW = 2?)

The Pair Detection Problem

Detecting exactly two active inputs is useful in:

• Error detection: Exactly 2 bit flips in a codeword
• Voting systems: Detecting ties in 4-person subcommittees
• Combinatorics: Enumerating pairs from a set

There are C(8,2) = 28 input patterns that fire this circuit.

Mechanism

Two threshold neurons work in concert:

AtLeast2 (lower bound):

sum = x₀ + x₁ + ... + x₇ - 2
fires when sum ≥ 0, i.e., HW ≥ 2

AtMost2 (upper bound):

sum = -x₀ - x₁ - ... - x₇ + 2
fires when sum ≥ 0, i.e., -HW + 2 ≥ 0, i.e., HW ≤ 2

The AND gate computes the intersection: HW ≥ 2 AND HW ≤ 2 = HW = 2.

Truth Table

HW	AtLeast2	AtMost2	Output
0	0	1	0
1	0	1	0
2	1	1	1
3	1	0	0
4+	1	0	0

Parameters

Component	Weights	Bias
AtLeast2	[+1, +1, +1, +1, +1, +1, +1, +1]	-2
AtMost2	[-1, -1, -1, -1, -1, -1, -1, -1]	+2
AND	[+1, +1]	-2

Total: 3 neurons, 21 parameters, 2 layers

Contrast with At-Least-2

Circuit	Condition	Fires on HW
2-out-of-8	HW ≥ 2	2, 3, 4, 5, 6, 7, 8
Exactly-2	HW = 2	2 only

The existing threshold-2outof8 fires on 247 of 256 inputs. This circuit fires on only 28.

Usage

from safetensors.torch import load_file
import torch

w = load_file('model.safetensors')

def exactly2(bits):
    inp = torch.tensor([float(b) for b in bits])
    atleast = int((inp * w['atleast.weight']).sum() + w['atleast.bias'] >= 0)
    atmost = int((inp * w['atmost.weight']).sum() + w['atmost.bias'] >= 0)
    comb = torch.tensor([float(atleast), float(atmost)])
    return int((comb * w['and.weight']).sum() + w['and.bias'] >= 0)

# Pair: positions 2 and 5 active
bits = [0, 0, 1, 0, 0, 1, 0, 0]
print(exactly2(bits))  # 1

# Three active - not a pair
bits = [1, 0, 1, 0, 0, 1, 0, 0]
print(exactly2(bits))  # 0

Files

threshold-exactly2outof8/
├── model.safetensors
├── model.py
├── config.json
└── README.md

License

MIT