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

Abstract

"We investigate the computational power of depth two circuits consisting of MODr -gates at the bottom and a threshold gate at the top (for short, threshold-MODr circuits) and circuits with two levels of MOD gates (MODp-MODq circuits.) In particular, we will show the following results For all prime numbers p and integers q,r it holds that if p divides r but not q then all threshold- MODq circuits for MOV have exponentially many nodes. For all integers r all problems computable by depth two {AND, OR, NOT] circuits of (quasi) polynomial size can be represented by threshold- MODr circuits with (quasi)polynormally many edges. There is a problem computable by depth three {AND,OR, ArOT}-circuits of linear size and constant bottom fan-in which for all r needs threshold- MODr circuits with exponentially many nodes. For p, r different primes, and q > 2, k positive integers, where p does not divide q, every MODp - MODq circuit for MODr has exponentially many nodes. Results (i) and (iii) imply the first known exponential lower bounds on the number of nodes of threshold- MODr circuits, r ≠ 2. They are based on a new lower bound method for the representation length of functions as threshold functions over predefined function bases, which, in contrast to previous related techniques [GHR92,BS90,KC>RS91,G93] works even if the edge weights are allowed to be unbounded and if the bases arc nonorthogonal. The spccial importance of result (iii) consists in the fact that the known spectral-Theoretically based lower bound methods for threshold- XOR circuits [BS90,KORS91] can provably not be applied to ACq-functions. Thus, by (ii), result (iii) is quite sharp and gives a partial (negative) answer to the (open) question whether there exist simulations of /iCo-circuits by small depth threshold circuits which are more efficient than that given by Yao's important result that ACC∗ C TC∗0.3 [Y90]. Finally we observe that our method works also for MODP-MODq circuits, if p is a power of a prime, see [KW91,YP94) for related results.