Classifications of Boolean functions and their closed sets

Abstract

We have continued and completed Post’s classification of two-valued functions and their closed sets. We use minimal bases of closed sets and two Mal 0 cev’s algebras instead of preserved relations and universal algebra. Then we use natural classifications of functions that are objects of the first Mal 0 cev’s algebra, and closed sets of functions that are objects of the second Mal 0 cev’s algebra. Classes of the classifications are disjoint. All these allow us to find more deep properties of functions and, in particular, to find fictitious closed sets that have gross volume and that give big gaps in Post’s classification. These allow also to get completeness of classification of closed sets. Post’s classification makes up 22% of the complete classification. Our class is a part of a closed set, the part is the rest of a closed set after removal all other closed sets in it. We call a closed set fictitious, if it becomes empty after removing all other closed sets containing in it. So fictitious closed sets are useless for classification of functions. But they take part in classification of closed sets since any classification must contain all objects of a theory. It turns out that non-fictitious closed sets have one-membered bases. These closed sets form level 1 in the multi-level classification of closed sets. Level i contains only closed sets without (i 1)-membered basis but with i-membered basis. We prove that levels 4 and above are empty. The classification by membered bases is well for many-valued functions, too.