Lower Bounds for (MOD p - MOD m) Circuits

Modular gates are known to be immune for the random restriction techniques of Ajtai Ajt83], Furst, Saxe, Sipser FSS84], Yao Yao85] and H astad H as86]. We demonstrate here a random clustering technique which overcomes this diiculty and is capable to prove generalizations of several known modular circuit lower bounds of Barrington, Straubing, Th erien BST90], Krause and Pudll ak KP94], 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, Th erien BST90], and 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 deeni-tions 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.