SIGACT News complexity theory column 32
ACM SIGACT News
Computational Complexity
Depth Reduction for Circuits with a Single Layer of Modular Counting Gates
CSR '09 Proceedings of the Fourth International Computer Science Symposium in Russia on Computer Science - Theory and Applications
Some properties of MODmcircuits computing simple functions
CIAC'03 Proceedings of the 5th Italian conference on Algorithms and complexity
Size-energy tradeoffs for unate circuits computing symmetric Boolean functions
Theoretical Computer Science
On the correlation between parity and modular polynomials
MFCS'06 Proceedings of the 31st international conference on Mathematical Foundations of Computer Science
Nonuniform ACC Circuit Lower Bounds
Journal of the ACM (JACM)
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Modular gates are known to be immune for the random restriction techniques of Ajtai (1983), Furst, Saxe, and Sipser (1984), Yao (1985), and Hå stad (1986). We demonstrate here a random clustering technique which overcomes this difficulty and is capable of proving generalizations of several known modular circuit lower bounds of Barrington, Straubing, and Th{érien (1990), Krause and Pudl{ák (1994), and others, characterizing symmetric functions computable by small (MODp, ANDt, MODm) 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, and Th{érien (1990) and other related lower bounds for certain (MODp, MODm, 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 MODm gates. Our methods are applicable, and our lower bounds are valid for the most general modular gates as well.