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We give a zero-error randomized algorithm which takes an $\\AC^0$ circuit as input and constructs a set of restrictions which partition $\\{0,1\\}^n$ so that under each restriction the value of the circuit is constant. Let $d$ denote the depth of the circuit and $cn$ denote the number of gates. This algorithm runs in time $|C| 2^{n(1-\\mu_{c.d})}$ where $|C|$ is the size of the circuit for $\\mu_{c,d} \\ge 1/\\bigO[\\lg c + d \\lg d]^{d-1}$ with probability at least $1-2^{-n}$.\n  As a result, we get impro"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1107.3127","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2011-07-15T18:37:49Z","cross_cats_sorted":["cs.LO"],"title_canon_sha256":"4ff9818b4537a2d7b425f619da9806397e834f81a20d0d771a3c1519fa551585","abstract_canon_sha256":"7554d800dbdcb1f5385a495a733d9209324d5d36342b1d4ca6b821b36e856d29"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:21:42.842184Z","signature_b64":"zvLJSM9B96Pot8QYBNzq4FL6wTECIrDkf55tl61jRg1BYk5ao9goai7G1L+ymuMLgwWsKJGXGQvdKGo0zFdEDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6fbddccb4636a2384ff1cc68378a43b3d110ec13602b6c51045f7209c7a227f7","last_reissued_at":"2026-05-18T02:21:42.841664Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:21:42.841664Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Satisfiability Algorithm for AC$^0$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LO"],"primary_cat":"cs.CC","authors_text":"Ramamohan Paturi, Russell Impagliazzo, William Matthews","submitted_at":"2011-07-15T18:37:49Z","abstract_excerpt":"We consider the problem of efficiently enumerating the satisfying assignments to $\\AC^0$ circuits. 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This algorithm runs in time $|C| 2^{n(1-\\mu_{c.d})}$ where $|C|$ is the size of the circuit for $\\mu_{c,d} \\ge 1/\\bigO[\\lg c + d \\lg d]^{d-1}$ with probability at least $1-2^{-n}$.\n  As a result, we get impro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3127","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1107.3127","created_at":"2026-05-18T02:21:42.841726+00:00"},{"alias_kind":"arxiv_version","alias_value":"1107.3127v1","created_at":"2026-05-18T02:21:42.841726+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3127","created_at":"2026-05-18T02:21:42.841726+00:00"},{"alias_kind":"pith_short_12","alias_value":"N665ZS2GG2RD","created_at":"2026-05-18T12:26:37.096874+00:00"},{"alias_kind":"pith_short_16","alias_value":"N665ZS2GG2RDQT7R","created_at":"2026-05-18T12:26:37.096874+00:00"},{"alias_kind":"pith_short_8","alias_value":"N665ZS2G","created_at":"2026-05-18T12:26:37.096874+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/N665ZS2GG2RDQT7RZRUDPCSDWP","json":"https://pith.science/pith/N665ZS2GG2RDQT7RZRUDPCSDWP.json","graph_json":"https://pith.science/api/pith-number/N665ZS2GG2RDQT7RZRUDPCSDWP/graph.json","events_json":"https://pith.science/api/pith-number/N665ZS2GG2RDQT7RZRUDPCSDWP/events.json","paper":"https://pith.science/paper/N665ZS2G"},"agent_actions":{"view_html":"https://pith.science/pith/N665ZS2GG2RDQT7RZRUDPCSDWP","download_json":"https://pith.science/pith/N665ZS2GG2RDQT7RZRUDPCSDWP.json","view_paper":"https://pith.science/paper/N665ZS2G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1107.3127&json=true","fetch_graph":"https://pith.science/api/pith-number/N665ZS2GG2RDQT7RZRUDPCSDWP/graph.json","fetch_events":"https://pith.science/api/pith-number/N665ZS2GG2RDQT7RZRUDPCSDWP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N665ZS2GG2RDQT7RZRUDPCSDWP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N665ZS2GG2RDQT7RZRUDPCSDWP/action/storage_attestation","attest_author":"https://pith.science/pith/N665ZS2GG2RDQT7RZRUDPCSDWP/action/author_attestation","sign_citation":"https://pith.science/pith/N665ZS2GG2RDQT7RZRUDPCSDWP/action/citation_signature","submit_replication":"https://pith.science/pith/N665ZS2GG2RDQT7RZRUDPCSDWP/action/replication_record"}},"created_at":"2026-05-18T02:21:42.841726+00:00","updated_at":"2026-05-18T02:21:42.841726+00:00"}