neural-semigroups 0.2.1

Neural networks powered research of semigroups

Neural Semigroups

Here we try to learn Cayley tables of semigroups using neural networks. The supposed workflow:

• install the package
• get the data
• build a model
• print a model testing report

More documentation can be found here.

Motivation

This work was inspired by a sudoku solver. A solved Sudoku puzzle is nothing more than a Cayley table of a quasigroup from 9 items with some well-known additional properties. So, one can imagine a puzzle made from a Cayley table of any other magma, e. g. a semigroup, by hiding part of its cells.

There are two major differences between sudoku and puzzles based on semigroups:

1. it's easy to take a glance on a table to understand whether it is a sudoku or not. That's why it was possible to encode numbers in a table cells as colour intensities. Sudoku is a picture, and a semigroup is not. It's difficult to check a Cayley table's associativity with a naked eye;

2. sudoku puzzles are solved by humans for fun and thus catalogued. When solving a sudoku one knows for sure that there is a unique solution. On the contrary, nobody guesses values in a partially filled Cayley table of a semigroup as a form of amuzement. As a result, one can create a puzzle from a full Cayley table of a semigroup but there may be many distinct solutions.

Package Installation

First, get Python 3.8

Then run several commands in a terminal:

git clone [e-mail]:inpefess/neural-semigroups.git
cd neural-semigroups
python -m venv venv
source ./venv/bin/activate
pip install -U pip poetry
poetry install

This could take some time (mostly downloading torch).

After that you can go to the project's directory and source ./venv/bin/activate to start a virtual environment there.

Getting Data

This package uses data included into a smallsemi package for GAP system. One can download it from the GAP page.

You can get the data by running a script from a scripts folder:

cd scripts
./download_smallsemi.sh

Training a Model

Here are several examples of commands to train a model depending on semigroup's cardinality:

python train_denoising_autoencoder.py --cardinality 4 --epochs 100 \
--learning_rate 0.1 --batch_size 32 --train_size 10 --validation_size 10

python train_denoising_autoencoder.py --cardinality 5 --epochs 100 \
--learning_rate 0.01 --batch_size 256 --train_size 100 --validation_size 100

python train_denoising_autoencoder.py --cardinality 6 --epochs 100 \
--learning_rate 0.001 --batch_size 2048 --train_size 1000 --validation_size 100

python train_denoising_autoencoder.py --cardinality 7 --epochs 100 \
--learning_rate 0.001 --batch_size 2048 --train_size 1000 --validation_size 100

Printing a Testing Report

One can print a model quality report using the following command:

python test_model.py --cardinality 4