A degree-decreasing Lemma for (MOD q, MOD p) circuits

Abstract

Consider a (MODq,MODp) circuit, where the inputs of the bottom MODp gates are degree-d polynomials of the input variables (p, q are different primes). Using our main tool — the Degree Decreasing Lemma — we show that this circuit can be converted to a (MODq, MODp) circuit with linear polynomials on the input-level with the price of increasing the size of the circuit. This result has numerous consequences: for the Constant Degree Hypothesis of Barrington, Straubing and Thérien [3], and generalizing the lower bound results of Yan and Parberry [21], Krause and Waack [13] and Krause and Pudlák [12]. Perhaps the most important application is an exponential lower bound for the size of (MODq, MODp) circuits computing the n fan-in AND, where the input of each MODp gate at the bottom is an arbitrary integer valued function of cn variables (c < 1) plus an arbitrary linear function of n input variables.We believe that the Degree Decreasing Lemma becomes a standard tool in modular circuit theory.