Circuits with Arbitrary Gates

Abstract

We consider boolean circuits computing n-operators f : {0, 1} n → {0, 1} n . As gates we allow arbitrary boolean functions; neither fanin nor fanout of gates is restricted. An operator is linear if it computes n linear forms, that is, computes a matrix-vector product A x over GF (2).We prove the existence of n-operators requiring about n 2 wires in any circuit, and linear n-operators requiring about n 2 / log n wires in depth-2 circuits, if either all output gates or all gates on the middle layer are linear.

An Adaptive Step Toward the Multiphase Conjecture

“…It also suggests that attacking the Multiphase conjecture for (general) dynamic data strucutres via the the NOF Game, should exploit the fact that data strucutres induce highly restricted NOF protocols. First, note that Conjecture 1.1 in particular implies the following special case: Circuits with arbitrary gates As mentioned in the introduction, a long-standing open problem in circuit complexity is whether non-linear gates can significantly (polynomially) reduce the number of wires of circuits computing linear operators [JS10]. We consider Valiant's depth-2 circuit model [Val77] with arbitrary gates, and its generalizations to arbitrary depths.…”