On the structure of one class of perfect $\Pi$-partitions

Abstract

The concept of $\Pi$-partition is an
analogue of the concept of normalized formula (a formula in the basis $\{\vee,\wedge,^-\}$ in which negations are possible only over variables) and concept of $\{\vee,\wedge,^-\}$schema, just as these last two concepts are analogues of each other. At its core, a $\Pi$-partition is a kind of "imprint" of a formula in the Boolean function calculated by this formula and is considered as a representation of this formula. In order to describe the class of minimal normalized formulas that calculate linear Boolean functions, the structure of the $\Pi$-partitions representing these formulas has been clarified.