Abstract:

By means of a modification of the method proposed by S. V. Yablonsky for constructing an economical (hypothetically minimal) normalized formula ($\Pi$-circuit) that calculates a given linear Boolean function, a whole class of similar formulas was constructed -- the class of so-called optimal perfect normalized formulas. Presumably it is the class of all minimal normalized formulas that compute this function. To prove this conjecture, we consider extending this class to the class of perfect normalized formulas that also compute the same function. It is established that a normalized formula is perfect if and only if it has a perfect representation on the own rectangle of the specified
function.