Two function algebras defining functions in NCk boolean circuits
We describe the functions computed by boolean circuits in NC k 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 NC k 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 NC k , 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.