{"paper":{"title":"Separation of AC$^0[\\oplus]$ Formulas and Circuits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Benjamin Rossman, Srikanth Srinivasan","submitted_at":"2017-02-13T04:23:37Z","abstract_excerpt":"This paper gives the first separation between the power of {\\em formulas} and {\\em circuits} of equal depth in the $\\mathrm{AC}^0[\\oplus]$ basis (unbounded fan-in AND, OR, NOT and MOD$_2$ gates). We show, for all $d(n) \\le O(\\frac{\\log n}{\\log\\log n})$, that there exist {\\em polynomial-size depth-$d$ circuits} that are not equivalent to {\\em depth-$d$ formulas of size $n^{o(d)}$} (moreover, this is optimal in that $n^{o(d)}$ cannot be improved to $n^{O(d)}$). This result is obtained by a combination of new lower and upper bounds for {\\em Approximate Majorities}, the class of Boolean functions "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.03625","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"}