Two function algebras defining functions in NCkNCk boolean circuits

We describe the functions computed by boolean circuits in NCkNCk by means of functions algebra for k≥1k≥1 in the spirit of implicit computational complexity. The whole hierarchy defines NCNC. In other words, we give a recursion-theoretic characterization of the complexity classes NCkNCk for k≥1k≥1 without reference to a machine model, nor explicit bounds in the recursion schema. Actually, we give two equivalent descriptions of the classes NCkNCk, k≥1k≥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.