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Linearly definable classes of Boolean functions

Couceiro, Miguel; Lehtonen, Erkko

Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), (2020), 39–46

In this paper we address the question "How many properties of Boolean functions can be defined by means of linear equations?" 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 and stable classes, a topic that has been widely investigated in recent years by several authors including Maurice Pouzet.