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Neural networks powered research of semigroups

Project description

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Neural Semigroups

The project is abandoned.

If you want to reproduce results from the paper, please use this notebook.

Here we try to model Cayley tables of semigroups using neural networks.

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 amusement. As a result, one can create a puzzle from a full Cayley table of a semigroup but there may be many distinct solutions.

How to Install

The best way to install this package is to use pip:

pip install neural-semigroups

How to use

The simplest way to get started is to use Google Colaboratory.

To look at more experimental results for different semigroups cardinalities, you can use Kaggle:

There is also an experimental notebook contributed by Žarko Bulić.

How to Contribute

Pull requests are welcome. To start:

git clone
cd neural-semigroups
# activate python virtual environment with Python 3.6+
pip install -U pip
pip install -U setuptools wheel poetry
poetry install
# recommended but not necessary
pre-commit install

To check the code quality before creating a pull request, one might run the script It locally does nearly the same as the CI pipeline after the PR is created.

Reporting issues or problems with the software

Questions and bug reports are welcome on the tracker.

More documentation

More documentation can be found here.

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