Systematic Minimization Technique for Majority-Majority Digital Combinational Circuits

Monotone Circuits for the Majority Function

We present a simple randomized construction of size O(n3) and depth 5:3logn+O(1) monotone circuits for the majority function on n variables. This result can be viewed as a reduction in the size and a partial derandomization of Valiant’s construction of an O(n5:3) monotone formula, [15]. On the other hand, compared with the deterministic monotone circuit obtained from the sorting network of Ajta...

Inversion Minimization in Majority-Inverter Graphs

In this paper, we present new optimization techniques for the recently introduced Majority-Inverter Graph (MIG). Our optimizations exploit intrinsic algebraic properties of MIGs and aim at rewriting the complemented edges of the graph without changing its shape. Two exact algorithms are proposed to minimize the number of complemented edges in the graph. The former is a dynamic programming metho...

Majority is incompressible by AC[p] circuits

We consider C-compression games, a hybrid model between computational and communication complexity. A C-compression game for a function f : {0, 1} → {0, 1} is a two-party communication game, where the first party Alice knows the entire input x but is restricted to use strategies computed by C-circuits, while the second party Bob initially has no information about the input, but is computational...

Typed semigroups, majority logic, and threshold circuits

Separating Ac from Depth-2 Majority Circuits∗

We construct a function in AC that cannot be computed by a depth-2 majority circuit of size less than exp(Θ(n1/5)). This solves an open problem due to Krause and Pudlák (1997) and matches Allender’s classic result (1989) that AC can be efficiently simulated by depth-3 majority circuits. To obtain our result, we develop a novel technique for proving lower bounds on communication complexity. This...