Two function algebras defining functions in ${\mathsf{NC}}^k$ boolean circuits

Abstract

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