Geodesic
On the structure of one class of perfect $\Pi$-partitions
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1499-1518

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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 $\Pi$-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.
Keywords: boolean functions, $\pi$-schemes, normalized formulas, lower bounds on the complexity, formula representation. 
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     author = {K. L. Rychkov},
     title = {On the structure of one class of perfect $\Pi$-partitions},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
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K. L. Rychkov. On the structure of one class of perfect $\Pi$-partitions. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1499-1518. http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a40/