2026-07-26T16:58:46Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:8863 2024-03-06T10:42:53Z ddc:004 open_access

Small Normalized Boolean Circuits for Semi-disjoint Bilinear Forms Require Logarithmic Conjunction-depth Lingas, Andrzej Boolean circuits semi-disjoint bilinear form Boolean vector convolution Boolean matrix product 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Andrzej Lingas 2018 Is Part Of LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CCC.2018.26 urn:nbn:de:0030-drops-88630 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.26 eng https://creativecommons.org/licenses/by/3.0/legalcode