On Circuit Complexity of Parity and Majority Functions in Antichain Basis

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Contribution

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Abstract

We study the circuit complexity of boolean functions in a certain infinite basis. The basis consists of all functions that take value $1$ on antichains over the boolean cube. We prove that the circuit complexity of the parity function and the majority function of $n$ variables in this basis is $\lfloor \frac{n+1}{2} \rfloor$ and $\left\lfloor \frac{n}{2} \right \rfloor +1$ respectively. We show that the asymptotic of the maximum complexity of $n$-variable boolean functions in this basis equals $n.$