Discrepancy and the Power of Bottom Fan-in in Depth-three Circuits

Abstract

We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty 'Number on the Forehead' model. Our method is based on the notion of voting polynomial degree of functions and extends the Degree-Discrepancy Lemma in the recent work of Sherstov [24]. Using this we prove that depth three circuits consisting of a MAJORITY gate at the output, gates computing arbitrary symmetric function at the second layer and arbitrary gates of bounded fan-in at the base layer i.e. circuits of type MAJ • SYMM • ANY O(1) cannot simulate the circuit class AC 0 in sub-exponential size. Further, even if the fan-in of the bottom ANY gates are increased to o(log log n), such circuits cannot simulate AC 0 in quasi-polynomial size. This is in contrast to the classical result of Yao and Beigel-Tarui that shows that such circuits, having only MAJORITY gates, can simulate the class ACC 0 in quasi-polynomial size when the bottom fan-in is increased to poly-logarithmic size.In the second part, we simplify the arguments in the breakthrough work of Bourgain [7] for obtaining exponentially small upper bounds on the correlation between the boolean function MOD q and functions represented by polynomials of small degree over Z m , when m, q ≥ 2 are co-prime integers. Our calculation also shows similarity with techniques used to estimate discrepancy of functions in the multiparty communication setting. This results in a slight improvement of the estimates of [7,14]. It is known that such estimates imply that circuits of type MAJ • MOD m • AND log n cannot compute the MOD q function in sub-exponential size. It remains a major open question to determine if such circuits can simulate ACC 0 in polynomial size when the bottom fan-in is increased to poly-logarithmic size.

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“…In the final section of this paper, we revisit several other multiparty generalizations [6,18,7] of the pattern matrix method. By applying our techniques in these other settings, we are able to obtain similar exponential separations for smaller k, by functions as simple as constant-depth circuits.…”

“…Originally formulated in [28,27] for the two-party model, the pattern matrix method has been adapted to the multiparty model by several authors [6,18,7,10,11]. The first adaptation of the method to the multiparty model gave improved lower bounds for the multiparty disjointness function [18,7].…”

“…The proof of Theorem 1.1 comes next, in Section 5. In the final section of the paper, we explore some implications of this work in the light of other multiparty papers [6,18,7].…”

“…The communication lower bounds in Theorems 3.6 and 3.7 were improved to optimal in [27] using matrix-analytic techniques. Unlike the combinatorial argument above, however, the matrix-analytic proof is not known to extend to the multiparty model and is not used in the follow-up multiparty papers [6,18,7,10,11] …”

“…In this section, we revisit recent multiparty analyses of the disjointness function [6,18,7]. We will see that the program of the previous sections applies here essentially unchanged.…”