Depth-Three Circuits for Inner Product and Majority Functions

We consider the complexity of depth-three Boolean circuits with limited bottom fan-in that compute some explicit functions. This is one of the simplest circuit classes for which we cannot derive tight bounds on the complexity for many functions. A Σ₃^k-circuit is a depth-three OR ∘ AND ∘ OR circuit in which each bottom gate has fan-in at most k. First, we investigate the complexity of Σ₃^k-circuits computing the inner product mod two function IP_n on n pairs of variables for small values of k. We give an explicit construction of a Σ²₃-circuit of size smaller than 2^{0.952n} for IP_n as well as a Σ³₃-circuit of size smaller than 2^{0.692n}. These improve the known upper bounds of 2^{n-o(n)} for Σ₃²-circuits and 3^{n/2} ∼ 2^{0.792n} for Σ₃³-circuits by Golovnev, Kulikov and Williams (ITCS 2021), and also the upper bound of 2^{(0.965…)n} for Σ₃²-circuits shown in a recent concurrent work by Göös, Guan and Mosnoi (MFCS 2023). Second, we investigate the complexity of the majority function MAJ_n aiming for exploring the effect of negations. Currently, the smallest known depth-three circuit for MAJ_n is a monotone circuit. A Σ₃^{(+k,-𝓁)}-circuit is a Σ₃-circuit in which each bottom gate has at most k positive literals and 𝓁 negative literals as its input. We show that, for k ≤ 2, the minimum size of a Σ₃^{(+k,-∞)}-circuit for MAJ_n is essentially equal to the minimum size of a monotone Σ₃^k-circuit for MAJ_n. In sharp contrast, we also show that, for k = 3,4 and 5, there exists a Σ₃^{(+k, -𝓁)}-circuit computing MAJ_n (for an appropriately chosen 𝓁) that is smaller than the smallest known monotone Σ₃^k-circuit for MAJ_n. Our results suggest that negations may help to speed up the computation of the majority function even for depth-three circuits. All these constructions rely on efficient circuits or formulas on a small number of variables that we found through a computer search.