Two function algebras defining functions in [formula omitted] boolean circuits

Abstract

We describe the functions computed by boolean circuits in NCk by means of functions algebra for k≥1 in the spirit of implicit computational complexity. The whole hierarchy defines NC. In other words, we give a recursion-theoretic characterization of the complexity classes NCk for k≥1 without reference to a machine model, nor explicit bounds in the recursion schema. Actually, we give two equivalent descriptions of the classes NCk, k≥1. One is based on a tree structure à la Leivant, the other is based on words. This latter puts into light the role of computation of pointers in circuit complexity. We show that transducers are a key concept for pointer evaluation.