We consider the class of constant depth AND/OR circuits augmented with a layer of modular counting gates at the bottom layer, i.e ${AC}^0 circ {MOD}_m$ circuits. We show that the following holds for several types of gates $G$: by adding a gate of type $G$ at the output, it is possible to obtain an equivalent randomized depth 2 circuit of quasipolynomial size consisting of a gate of type $G$ at the output and a layer of modular counting gates, i.e $G circ {MOD}_m$ circuits. The types of gates $G$ we consider are modular counting gates and threshold-style gates. For all of these, strong lower bounds are known for (deterministic) $G circ {MOD}_m$ circuits.
@InProceedings{hansen:DagSemProc.08381.4, author = {Hansen, Kristoffer Arnsfelt}, title = {{Depth Reduction for Circuits with a Single Layer of Modular Counting Gates}}, booktitle = {Computational Complexity of Discrete Problems}, pages = {1--11}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8381}, editor = {Peter Bro Miltersen and R\"{u}diger Reischuk and Georg Schnitger and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08381.4}, URN = {urn:nbn:de:0030-drops-17824}, doi = {10.4230/DagSemProc.08381.4}, annote = {Keywords: Boolean Circuits, Randomized Polynomials, Fourier sums} }
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