threshold-thermometertobinary

Converts 7-bit thermometer code to 3-bit binary. The inverse of BinaryToThermometer.

Circuit

  t₀  t₁  t₂  t₃  t₄  t₅  t₆
   │   │   │   │   │   │   │
   │   │   │   │   │   │   │
   ├───┴───┴───┴───┴───┴───┘
   │   │       │       │
   │   ├───────┼───────┼──────────► b₂ (direct from t₃)
   │   │       │       │
   │   │   ┌───┴───┐   │
   │   │   │t₁∧¬t₃ │   │
   │   │   └───┬───┘   │
   │   │       │       │
   │   │       ├───OR──┴──────────► b₁
   │   │       │   │
   │   │       │   t₅
   │   │       │
   ├───┴───┐ ┌─┴─┐ ┌───┐
   │t₀∧¬t₁│ │...│ │t₆ │
   └───┬───┘ └─┬─┘ └─┬─┘
       │       │     │
       └───────┴─OR──┴────────────► b₀

Conversion Table

Thermometer	Value	Binary
0000000	0	000
1000000	1	001
1100000	2	010
1110000	3	011
1111000	4	100
1111100	5	101
1111110	6	110
1111111	7	111

Mechanism

For valid thermometer (monotonic ones then zeros), tᵢ = 1 iff value > i.

b₂ (bit 2): Directly equals t₃.

• t₃ = 1 iff value ≥ 4, which is exactly when b₂ = 1

b₁ (bit 1): Fires for values {2, 3, 6, 7}.

b₁ = (t₁ AND NOT(t₃)) OR t₅

• t₁ AND NOT(t₃) catches values 2, 3 (≥2 but <4)
• t₅ catches values 6, 7 (≥6)

b₀ (bit 0): Fires for odd values {1, 3, 5, 7}.

b₀ = (t₀ AND NOT(t₁)) OR (t₂ AND NOT(t₃)) OR (t₄ AND NOT(t₅)) OR t₆

Each AND-NOT term detects a transition from 1 to 0 at an odd position.

The Transition Detection Insight

In valid thermometer code, the value equals the position of the last 1. The formula detects "where does the thermometer stop?"

Value	Last 1 at	Detected by
1	t₀	t₀ AND NOT(t₁)
3	t₂	t₂ AND NOT(t₃)
5	t₄	t₄ AND NOT(t₅)
7	t₆	t₆ (no t₇ to check)

Architecture

Output	Neurons	Parameters
b₂	1	8
b₁	2	11
b₀	4	29
Total	7	48

Layers: 2

Asymmetry Note

BinaryToThermometer is single-layer (7 parallel thresholds). ThermometerToBinary requires 2 layers because extracting individual bits from a sum needs non-trivial logic.

Usage

from safetensors.torch import load_file
import torch

w = load_file('model.safetensors')

def therm_to_binary(therm):
    """therm: 7-element list"""
    # See model.py for full implementation
    pass

# Thermometer for 5 -> binary 101
therm = [1, 1, 1, 1, 1, 0, 0]
b2, b1, b0 = therm_to_binary(therm)
print(b2, b1, b0)  # 1, 0, 1

Files

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

License

MIT