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Lower Bounds for (MOD p  MOD m) Circuits
 Proc. 39th IEEE FOCS
, 1998
"... Modular gates are known to be immune for the random restriction techniques of Ajtai (1983), Furst, Saxe, Sipser (1984), Yao (1985) and Hastad (1986). We demonstrate here a random clustering technique which overcomes this diculty and is capable to prove generalizations of several known modular circui ..."
Abstract

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Modular gates are known to be immune for the random restriction techniques of Ajtai (1983), Furst, Saxe, Sipser (1984), Yao (1985) and Hastad (1986). We demonstrate here a random clustering technique which overcomes this diculty and is capable to prove generalizations of several known modular circuit lower bounds of Barrington, Straubing, Therien (1990), Krause and Pudlak (1994), and others, characterizing symmetric functions computable by small (MOD p ; AND t ; MODm ) circuits. Applying a degreedecreasing 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 (1990), and other related lower bounds for certain (MOD p ; MODm ; AND) circuits. Most of the previous lower bounds on circuits with modular gates used special denitions 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 MODm gates. Our methods are applicable, and our lower bounds are valid, for the most general modular gates as well. 1