The complexity of the parity function in unbounded fan-in, unbounded depth circuits

Abstract

Almost everything is known on the complexity of the parity function in fan-in 2 circuits over various bases. Also the minimal depth of polynomial-size, unbounded fan-in {∧, ∨, ⌝\\dl;} circuits for the parity function has been studied. Here the complexity without any depth restriction is considered. For the basis {∧, ∨, ⌝\\dl;} almost optimal bounds, and for the basis of NOR gates and the basis of all threshold functions optimal bounds on the number of gates are obtained. For the basis {∧, ∨, ⌝\\dl;} the minimal number of wires is determined. For threshold circuits an exponential gap between synchronous and asynchronous circuits is proved. The results not only answer open questions in complexity theory but also have implications for the real-life circuit design.