{"paper":{"title":"An average-case depth hierarchy theorem for Boolean circuits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Benjamin Rossman, Li-Yang Tan, Rocco A. Servedio","submitted_at":"2015-04-14T01:44:39Z","abstract_excerpt":"We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of $\\mathsf{AND}$, $\\mathsf{OR}$, and $\\mathsf{NOT}$ gates. Our hierarchy theorem says that for every $d \\geq 2$, there is an explicit $n$-variable Boolean function $f$, computed by a linear-size depth-$d$ formula, which is such that any depth-$(d-1)$ circuit that agrees with $f$ on $(1/2 + o_n(1))$ fraction of all inputs must have size $\\exp({n^{\\Omega(1/d)}}).$ This answers an open question posed by H{\\aa}stad in his Ph.D. thesis.\n  Our average-case depth hierarchy theorem implies that the polynomia"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03398","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"}