Polynomial programs and the Razborov-Smolensky method

Representations of boolean functions as polynomials (over rings) have been used to establish lower bounds in complexity theory. Such representations were used to great effect by Smolensky, who established that MOD q \notin AC^0[MOD p] (for distinct primes p, q) by representing AC^0[MOD p] functions as low-degree multilinear polynomials over fields of characteristic p. Another tool which has yielded insight into small-depth circuit complexity classes is the program-over-monoids model of computation, which has provided characterizations of circuit complexity classes such as AC^0 and NC^1. We introduce a new model of computation, the polynomial program, which naturally unifies both the polynomial (over ring) model of computation and the program-over-monoids model of computation. Our motivation is to extend Smolensky's result to prove AC^0[MOD m] lower bounds, where m is a composite with at least two distinct prime factors. In this paper, we first study the basic properties of this new model of computation and its relationship to previous work. Then, we show that Smolensky's proof of the ``hardness'' of certain functions for the polynomial model of computation has an analog in the polynomial program model. We also prove a dichotomy theorem on finite groups, essentially showing that a finite group G either is so ``complicated'' that every boolean function has a degree one polynomial program over G, or is too ``simple'' in that the function MOD p can be computed by a low-degree polynomial program over G for at most one prime p. The property of nilpotence gives the dividing line. This dichotomy theorem rigorously demonstrates limitations of the Razborov-Smolensky method for polynomial programs over finite groups