{"paper":{"title":"The Average Sensitivity of Bounded-Depth Formulas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Benjamin Rossman","submitted_at":"2015-08-31T04:09:03Z","abstract_excerpt":"We show that unbounded fan-in boolean formulas of depth $d+1$ and size $s$ have average sensitivity $O(\\frac{1}{d}\\log s)^d$. In particular, this gives a tight $2^{\\Omega(d(n^{1/d}-1))}$ lower bound on the size of depth $d+1$ formulas computing the \\textsc{parity} function. These results strengthen the corresponding $2^{\\Omega(n^{1/d})}$ and $O(\\log s)^d$ bounds for circuits due to H{\\aa}stad (1986) and Boppana (1997). Our proof technique studies a random process where the Switching Lemma is applied to formulas in an efficient manner."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07677","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"}