Depth-reduction for composites Shiteng Chen * Tsinghua University Periklis A. Papakonstantinou † Rutgers University May 27, 2016 Abstract We obtain a new depth-reduction construction, which implies a super-exponential improve- ment in the depth lower bound separating NEXP from non-uniform ACC. In particular, we show that every circuit with AND, OR, NOT, and MOD m gates, m ∈ Z + , of polynomial size and depth d can be reduced to a depth-2, SYM ◦ AND, circuit of size 2 (log n) O(d) . This is an exponential size improvement over the traditional Yao-Beigel-Tarui, which has size blowup 2 (log n) 2 O(d) . Therefore, depth-reduction for composite m matches the size of the Allender-Hertrampf construction for primes from 1989. One immediate implication of depth reduction is an improvement of the depth from o(log log n) to o(log n/ log log n), in Williams’ program for ACC circuit lower bounds against NEXP. This is just short of O(log n/ log log n) and thus pushes William’s program to the NC 1 barrier, since NC 1 is contained in ACC of depth O(log n/ log log n). A second, but non-immediate, implication regards the strengthening of the ACC lower bound in the Chattopadhyay-Santhanam interactive compression setting. Keywords composite modulus, depth-reduction, circuit lower bound, Williams’ program, inter- active compression 1 Introduction The development of computational complexity is vastly a history of conjectures, and gaps between these conjectures and what is actually proved. One such story regards the power of MOD m gates in small-depth boolean circuits that also have AND, OR, NOT gates. A MOD m gate outputs 1 if and only if the number of 1s in its input is a multiple of m. What is known for prime m = p stands in sharp contrast to what is known for composite m ∈ Z + . In a sense we settle the question about depth-reduction of ACC circuits. 1 Depth-reduction is an algorithm compressing a low-depth ACC circuit (highly parallel algorithm) of depth d into * IIIS, Tsinghua University, shitengchen@gmail.com † MSIS, Business School, Rutgers University, periklis.research@gmail.com. Part of this work done when both authors were at Tsinghua University. 1 In the literature ACC 0 denotes the class of boolean functions computable by polynomial size {AND, OR, NOT, MODm} families of circuits of constant depth and m ∈ Z + . We will be referring to ACC 0 both as the class of boolean functions and the circuits characterizing it. Since, we consider circuits of different depth d and size s we will commonly refer to such circuits as ACC circuits (or ACCm for a fixed modulus) of depth d and size s. 1 ISSN 1433-8092