We consider normalized Boolean circuits that use binary operations of disjunction and conjunction, and unary negation, with the restriction that negation can be only applied to input variables. We derive a lower bound trade-off between the size of normalized Boolean circuits computing Boolean semi-disjoint bilinear forms and their conjunction-depth (i.e., the maximum number of and-gates on a directed path to an output gate). In particular, we show that any normalized Boolean circuit of at most epsilon log n conjunction-depth computing the n-dimensional Boolean vector convolution has Omega(n^{2-4 epsilon}) and-gates. Analogously, any normalized Boolean circuit of at most epsilon log n conjunction-depth computing the n x n Boolean matrix product has Omega(n^{3-4 epsilon}) and-gates. We complete our lower-bound trade-offs with upper-bound trade-offs of similar form yielded by the known fast algebraic algorithms.
@InProceedings{lingas:LIPIcs.CCC.2018.26, author = {Lingas, Andrzej}, title = {{Small Normalized Boolean Circuits for Semi-disjoint Bilinear Forms Require Logarithmic Conjunction-depth}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {26:1--26:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.26}, URN = {urn:nbn:de:0030-drops-88630}, doi = {10.4230/LIPIcs.CCC.2018.26}, annote = {Keywords: Boolean circuits, semi-disjoint bilinear form, Boolean vector convolution, Boolean matrix product} }
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