{"paper":{"title":"A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Christian Engels, Nutan Limaye, Srikanth Srinivasan, Suryajith Chillara","submitted_at":"2018-04-07T07:27:44Z","abstract_excerpt":"We study the size blow-up that is necessary to convert an algebraic circuit of product-depth $\\Delta+1$ to one of product-depth $\\Delta$ in the multilinear setting.\n  We show that for every positive $\\Delta = \\Delta(n) = o(\\log n/\\log \\log n),$ there is an explicit multilinear polynomial $P^{(\\Delta)}$ on $n$ variables that can be computed by a multilinear formula of product-depth $\\Delta+1$ and size $O(n)$, but not by any multilinear circuit of product-depth $\\Delta$ and size less than $\\exp(n^{\\Omega(1/\\Delta)})$. This result is tight up to the constant implicit in the double exponent for al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.02520","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}