On the Size of Depth-Three Boolean Circuits for Computing Multilinear Functions

This paper introduces and initiates a study of a new model of arithmetic circuits coupled with new complexity measures. The new model consists of multilinear circuits with arbitrary multilinear gates, rather than the standard multilinear circuits that use only addition and multiplication gates. In light of this generalization, the arity of gates becomes of crucial importance and is indeed one of our complexity measures. Our second complexity measure is the number of gates in the circuit, which (in our context) is significantly different from the number of wires in the circuit (which is typically used as a measure of size). Our main complexity measure, denoted \({\mathtt{AN}}(\cdot )\), is the maximum of these two measures (i.e., the maximum between the arity of the gates and the number of gates in the circuit). We also consider the depth of such circuits, focusing on depth-two and unbounded depth.

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