Abstract

We investigate the computational power of depth two 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. (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. (ii) For all integers r, all problems computable by depth two fAND;OR;NOTg-- circuits of polynomial size have threshold--MOD r circuits with polynomially many edges. (iii) There is a problem computable by depth 3 fAND;OR;NOTg--circuits of linear size and constant bottom fan--in which for all r needs threshold--MOD r circuits with exponentially many nodes. (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 e...

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