Abstract:
Modular gates are known to be immune for the random restriction techniques of Ajtai; Furst, Saxe, Sipser; and Yao and Hastad. We demonstrate here a random clustering technique which overcomes this difficulty and is capable to prove generalizations of several known modular circuit lower bounds of Barrington, Straubing, Therien; Krause and Pudlak; and others, characterizing symmetric functions computable by small (MOD_p, AND_t, MOD_m) circuits. Applying a degree-decreasing technique together with random restriction methods for the AND gates at the bottom level, we also prove a hard special case of the Constant Degree Hypothesis of Barrington, Straubing, Therien, and some other related lower bounds for certain (MOD_p, MOD_m, AND) circuits. Most of the previous lower bounds on circuits with modular gates used special definitions of the modular gates (i.e., the gate outputs one if the sum of its inputs is divisible by m, or is not divisible by m), and were not valid for more general MOD_m gates. Our methods are applicable -- and our lower bounds are valid -- for the most general modular gates as well.