Stability of Boolean function classes with respect to clones of linear functions

TL;DR

This paper characterizes classes of Boolean functions that remain stable when composed with linear idempotent functions, refining previous results and connecting to broader theories of function minors and stable classes.

Contribution

It provides a complete description of Boolean function classes stable under compositions with linear idempotent clones, advancing the understanding of their structure.

Findings

Countably many linearly definable classes of Boolean functions

Complete classification of these classes

Connections to function minors and stable classes

Abstract

We consider classes of Boolean functions stable under compositions both from the right and from the left with clones. Motivated by the question how many properties of Boolean functions can be defined by means of linear equations, we focus on stability under compositions with the clone of linear idempotent functions. It follows from a result by Sparks that there are countably many such linearly definable classes of Boolean functions. In this paper, we refine this result by completely describing these classes. This work is tightly related with the theory of function minors, stable classes, clonoids, and hereditary classes, topics that have been widely investigated in recent years by several authors including Maurice Pouzet and his coauthors.