Symmetric Boolean functions

Abstract

The purpose of this chapter is to summarize what is known about symmetric Boolean functions, but will concentrate primarily on my own work. I will discuss several new types of symmetry, along with tools and analytical techniques for detecting and studying such symmetries. First, I will provide a comprehensive introduction to the mathematics of group theory, and show how this serves as a basis for the study of symmetric Boolean functions. I will extend these ideas to matrix-based symmetry, and show how matrix-based symmetry greatly expands the concept of permutation-based symmetry. The new types of symmetry discussed in this chapter will include conjugate symmetry, super-symmetry, exotic symmetry, and hypersymmetry. The names "super-symmetry," "exotic symmetry," and "hyper symmetry" have not appeared in print and will be defined in the chapter. These are all matrix-based symmetries. In addition I will discuss the existing symmetry types: anti-symmetry, hierarchical symmetry, and Kronecker symmetry and potential extensions to such.