Boolean Functions and Circuits

Abstract

Many fundamental questions in complexity theory can be formulated as a language recognition problem, where by language L we understand a set of strings over a finite alphabet, say Σ = {0, 1}. The basic question is then to determine the computational resources (such as time, space, number of processors, energy consumption of a chip, etc.) needed in determining whether or not x ∈ L, for any string x ∈ Σ*. In turn, if for any n we let f _n denote the characteristic function of L ∩{0, 1} _n , then the previous question can be rephrased as a problem of determining the computational resources needed to compute the boolean function f _n (x), for arbitrary x, n.

Mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of conducting us to the knowledge of a mathematical law, ... H. Poincaré [Poi52]