Depth-𝑑 Threshold Circuits vs. Depth-(𝑑+1) AND-OR Trees

For any n ∈ ℕ and d = o (loglog( n )), we prove that there is a Boolean function F on n bits and a value γ = 2 −Θ( d ) such that F can be computed by a uniform depth-( d + 1) AC 0 circuit with O ( n ) wires, but F cannot be computed by any depth- d TC 0 circuit with n 1 + γ wires. This bound matches the current state-of-the-art lower bounds for computing explicit functions by threshold circuits of depth d > 2, which were previously known only for functions outside AC 0 such as the parity function. Furthermore, in our result, the AC 0 circuit computing F is a monotone *read-once formula* (i.e., an AND-OR tree), and the lower bound holds even in the average-case setting with respect to advantage n −γ . At a high level, our proof strategy combines two prominent approaches in circuit complexity from the last decade: The celebrated *random projections* method of Håstad, Rossman, Servedio, and Tan (J. ACM 2017), which was previously used to show a tight average-case depth hierarchy for AC 0 ; and the line of works analyzing the effect of *random restrictions* on threshold circuits. We show that under a modified version of Håstad, Rossman, Servedio, and Tan’s projection procedure, any depth- d threshold circuit with n 1 + γ wires simplifies to a near-trivial function, whereas an appropriately parameterized AND-OR tree of depth d + 1 maintains structure.