{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:NNLXBFNIVSVOCR4K4I5YUS3JSK","short_pith_number":"pith:NNLXBFNI","schema_version":"1.0","canonical_sha256":"6b577095a8acaae1478ae23b8a4b6992bb7f658944bde1afd94e98115b019327","source":{"kind":"arxiv","id":"1012.2394","version":1},"attestation_state":"computed","paper":{"title":"NE is not NP Turing Reducible to Nonexpoentially Dense NP Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Bin Fu","submitted_at":"2010-12-10T21:19:11Z","abstract_excerpt":"A long standing open problem in the computational complexity theory is to separate NE from BPP, which is a subclass of $NP_T(NP\\cap P/poly)$. In this paper, we show that $NE\\not\\subseteq NP_(NP \\cap$ Nonexponentially-Dense-Class), where Nonexponentially-Dense-Class is the class of languages A without exponential density (for each constant c>0,$|A^{\\le n}|\\le 2^{n^c}$ for infinitely many integers n).\n  Our result implies $NE\\not\\subseteq NP_T({pad(NP, g(n))})$ for every time constructible super-polynomial function g(n) such as $g(n)=n^{\\ceiling{\\log\\ceiling{\\log n}}}$, where Pad(NP, g(n)) is cl"},"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":"1012.2394","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2010-12-10T21:19:11Z","cross_cats_sorted":[],"title_canon_sha256":"e41268d34344f1497fa4a5e6a203e41e7d241197a5604c6b32cfe38de3d98c03","abstract_canon_sha256":"8dce1b73a1e1333e800f6edbb463e0afe68a4c3986155f25fd1b0f8f72b0e684"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:33:28.671783Z","signature_b64":"Lqef35uG3cWZwdv7eVpn2PkT1xYgj68FxRBBHwW1rLSivA8W1zZX+IAExvI8T4hpv1ocDYD9iOIeGKjm9Hc1Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b577095a8acaae1478ae23b8a4b6992bb7f658944bde1afd94e98115b019327","last_reissued_at":"2026-05-18T04:33:28.671338Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:33:28.671338Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"NE is not NP Turing Reducible to Nonexpoentially Dense NP Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Bin Fu","submitted_at":"2010-12-10T21:19:11Z","abstract_excerpt":"A long standing open problem in the computational complexity theory is to separate NE from BPP, which is a subclass of $NP_T(NP\\cap P/poly)$. In this paper, we show that $NE\\not\\subseteq NP_(NP \\cap$ Nonexponentially-Dense-Class), where Nonexponentially-Dense-Class is the class of languages A without exponential density (for each constant c>0,$|A^{\\le n}|\\le 2^{n^c}$ for infinitely many integers n).\n  Our result implies $NE\\not\\subseteq NP_T({pad(NP, g(n))})$ for every time constructible super-polynomial function g(n) such as $g(n)=n^{\\ceiling{\\log\\ceiling{\\log n}}}$, where Pad(NP, g(n)) is cl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2394","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":"1012.2394","created_at":"2026-05-18T04:33:28.671402+00:00"},{"alias_kind":"arxiv_version","alias_value":"1012.2394v1","created_at":"2026-05-18T04:33:28.671402+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.2394","created_at":"2026-05-18T04:33:28.671402+00:00"},{"alias_kind":"pith_short_12","alias_value":"NNLXBFNIVSVO","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_16","alias_value":"NNLXBFNIVSVOCR4K","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_8","alias_value":"NNLXBFNI","created_at":"2026-05-18T12:26:10.704358+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/NNLXBFNIVSVOCR4K4I5YUS3JSK","json":"https://pith.science/pith/NNLXBFNIVSVOCR4K4I5YUS3JSK.json","graph_json":"https://pith.science/api/pith-number/NNLXBFNIVSVOCR4K4I5YUS3JSK/graph.json","events_json":"https://pith.science/api/pith-number/NNLXBFNIVSVOCR4K4I5YUS3JSK/events.json","paper":"https://pith.science/paper/NNLXBFNI"},"agent_actions":{"view_html":"https://pith.science/pith/NNLXBFNIVSVOCR4K4I5YUS3JSK","download_json":"https://pith.science/pith/NNLXBFNIVSVOCR4K4I5YUS3JSK.json","view_paper":"https://pith.science/paper/NNLXBFNI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1012.2394&json=true","fetch_graph":"https://pith.science/api/pith-number/NNLXBFNIVSVOCR4K4I5YUS3JSK/graph.json","fetch_events":"https://pith.science/api/pith-number/NNLXBFNIVSVOCR4K4I5YUS3JSK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NNLXBFNIVSVOCR4K4I5YUS3JSK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NNLXBFNIVSVOCR4K4I5YUS3JSK/action/storage_attestation","attest_author":"https://pith.science/pith/NNLXBFNIVSVOCR4K4I5YUS3JSK/action/author_attestation","sign_citation":"https://pith.science/pith/NNLXBFNIVSVOCR4K4I5YUS3JSK/action/citation_signature","submit_replication":"https://pith.science/pith/NNLXBFNIVSVOCR4K4I5YUS3JSK/action/replication_record"}},"created_at":"2026-05-18T04:33:28.671402+00:00","updated_at":"2026-05-18T04:33:28.671402+00:00"}