{"paper":{"title":"A Fixed-Depth Size-Hierarchy Theorem for AC$^0[\\oplus]$ via the Coin Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Karteek Sreenivasaiah, Nutan Limaye, Srikanth Srinivasan, S. Venkitesh, Utkarsh Tripathi","submitted_at":"2018-09-11T18:04:36Z","abstract_excerpt":"We prove the first Fixed-depth Size-hierarchy Theorem for uniform AC$^0[\\oplus]$ circuits; in particular, for fixed $d$, the class $\\mathcal{C}_{d,k}$ of uniform AC$^0[\\oplus]$ formulas of depth $d$ and size $n^k$ form an infinite hierarchy. For this, we find the first class of explicit functions giving (up to polynomial factor) matching upper and lower bounds for AC$^0[\\oplus]$ formulas, derived from the $\\delta$-Coin Problem, the computational problem of distinguishing between coins that are heads with probability $(1+\\delta)/2$ or $(1-\\delta)/2,$ where $\\delta$ is a parameter going to $0$. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.04092","kind":"arxiv","version":2},"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"}