# [2019.10.24] Associative Cayley Table Completion Task

Let us have a partially filled Cayley table of some magma about which
we know that it should be associative. The task sounds similar to
sudoku. There we also have only several cells known and need to
reconstruct the table with some properties of the binary operation
which it represents.

First, we can use tables filled not by exact values but with random
variables. Then we can estimate the distance between any two of them
using Kullback-Leibler divergence. A task of creating a puzzle from a
given complete Cayley table is hard per se. By 'puzzle' I mean a
table with several cells obscured, but so that one can fill them in a
precisely one way. These conditions characterize sudoku puzzles, for
example. To solve the task, we can consider hiding some cells as
random noise and build a denoising autoencoder. Namely, we have a
complete Cayley table as an input then we apply some noise, then we
encode and decode striving to get the undistorted original.