On the computational power of depth-2 circuits with threshold and modulo gates

Abstract

We investigate the computational power of depth-2 circuits consisting of MOD r gates at the bottom and a threshold gate with arbitrary weights at the top (for short, threshold-MOD r circuits) and circuits with two levels of MOD gates ( MOD p-MOD q circuits). In particular, we will show the following results: 1. (i) For all prime numbers p and integers q, r, it holds that if p divides r but not q then all threshold-MOD q circuits for MOD r have exponentially many nodes. 2. (ii) For all integers r, all problems computable by depth-2 AND,OR,NOT circuits of polynomial size have threshold-MOD r circuits with polynomially many edges. 3. (iii) There is a problem computable by depth 3 AND,OR,NOT circuits of linear size and constant bottom fan-in which for all r needs threshold-MOD r circuits with exponentially many nodes. 4. (iv) For p, r different primes, and q ⩾ 2, k positive integers, where r does not divide q, every MOD p k -MOD q circuit for MOD r has exponentially many nodes. Results (i) and (iii) imply the first known exponential lower bounds on the number of nodes of threshold-MOD r circuits, r ≠ 2. They are based on a new method for estimating the minimum length of threshold realizations over predefined function bases, which, in contrast to previous related techniques (Goldmann et al., 1992; Bruck and Smolensky, 1990; Kailath et al., 1991; Goldmann, 1993; Grolmusz, 1993) works even if the weight of the realization is allowed to be unbounded, and if the bases are allowed to be nonorthogonal. The special importance of result (iii) consists of the fact that the known spectral-theoretically based lower bound methods for threshold-XOR circuits (Bruck and Smolensky, 1990; Kailath et al., 1991) can provably not be applied to AC 0 functions. Thus, by (ii), result (iii) is sharp. It gives a partial negative answer to the open question whether there exist simulations of AC 0 -circuits by small depth threshold circuits which are more efficient than that given by Yao's important result that ACC functions have depth-3 threshold circuits of quasipolynomial weight (Yao, 1990). Finally we observe that our method works also for MOD p-MOD q circuits, if p is a power of a prime ((iv) above); see (Barrington et al., 1990; Krause and Waack, 1991; Yan and Parberry, 1994) for related results. A preliminary version of this paper appeared in (Krause and Pudlák, 1993).