Definability by Constant-Depth Polynomial-Size Circuits

A function of boolean arguments is symmetric if its value depends solely on the number of l 's among its arguments. In the first part of this paper we partially characterize those symmetric functions that can be computed by constant-depth polynomial-size sequences of boolean circuits, and discuss the complete characterization. (We treat both uniform and non-uniform sequences of circuits.) Our results imply that these circuits can compute functions that are not definable in first-order logic. In the second part of the paper we generalize from circuits computing symmetric functions to circuits recognizing first-order structures. By imposing fairly natural restrictions we develop a circuit model with precisely the power of first-order logic: a class of structures is first-order definable if and only if it can be recognized by a constant-depth polynomial-time sequence of such circuits. © 1986 Academic Press, Inc.