An algebra and a logic for NC/sup 1/

An algebra and a logic characterizing the complexity class NC/sup 1/, which consists of functions computed by uniform sequences of polynomial-size, log depth circuits, are presented. In both characterizations, NC/sup 1/ functions are regarded as functions from one class of finite relational structures to another. In the algebraic characterization, upward and downward tree recursion are applied to a class of simple functions. In the logical characterization, first-order logic is augmented by an operator for defining relations by primitive recursion. It is assumed that every structure has an underlying relation giving the binary representations of integers.<>