Lower Bounds for the Size of Nondeterministic Circuits

TL;DR

This paper establishes a lower bound on the size of nondeterministic $U_2$-circuits computing the parity function, showing nondeterminism does not reduce circuit size for this problem, marking a significant theoretical advance.

Contribution

It proves the first nontrivial lower bound for nondeterministic circuit size for an explicit Boolean function, specifically for $U_2$-circuits computing parity.

Findings

Nondeterministic $U_2$-circuits require at least $3(n-1)$ size for parity

Nondeterminism does not reduce the size compared to deterministic circuits for parity

First nontrivial lower bound for nondeterministic circuit size for an explicit function

Abstract

Nondeterministic circuits are a nondeterministic computation model in circuit complexity theory. In this paper, we prove a $3(n-1)$ lower bound for the size of nondeterministic $U_2$-circuits computing the parity function. It is known that the minimum size of (deterministic) $U_2$-circuits computing the parity function exactly equals $3(n-1)$. Thus, our result means that nondeterministic computation is useless to compute the parity function by $U_2$-circuits and cannot reduce the size from $3(n-1)$. To the best of our knowledge, this is the first nontrivial lower bound for the size of nondeterministic circuits (including formulas, constant depth circuits, and so on) with unlimited nondeterminism for an explicit Boolean function. We also discuss an approach to proving lower bounds for the size of deterministic circuits via lower bounds for the size of nondeterministic restricted circuits.