Searching for symmetric permutations of binary patterns set with genetic algorithms

Symmetry is a powerful tool to reduce the freedom degrees of a problem. However, the applicability of the symmetry tool strongly depends on the possibility to calculate the symmetries of the system. General searching methods, e.g., ergodic searching, is unpractical for a high-dimensional system since the order of the symmetric group grows with n, where n is the dimension of the system. We propose an approach based on genetic algorithms to search for the symmetric permutations of a binary patterns set. Calculations for five kinds of dimensional pattern set are also given. Results show that the majority of symmetric permutations can be found within an acceptable time for a high-dimensional pattern set by the new approach, which makes it possible to study and design high-dimensional artificial neural networks by the method of symmetry.