{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:NRCGX72ORBORCS4OHYX5HR7ZDW","short_pith_number":"pith:NRCGX72O","schema_version":"1.0","canonical_sha256":"6c446bff4e885d114b8e3e2fd3c7f91da558e722828e3f7dca27f2a16c19651d","source":{"kind":"arxiv","id":"2507.04110","version":4},"attestation_state":"computed","paper":{"title":"Low Sets and Closure Properties of Counting Function Classes","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Yaroslav Ivanashev","submitted_at":"2025-07-05T17:40:08Z","abstract_excerpt":"A language L is low for a relativizable complexity class C, if C$^L$ = C. For the classes #P, GapP, and SpanP the exact low classes of languages are known: Low(#P) = UP $\\cap$ coUP, Low(GapP) = SPP, and Low(SpanP) = NP $\\cap$ coNP. In this paper, we prove that Low(TotP) = P, and give characterizations of low function classes for #P, GapP, TotP, and SpanP. In particular, we prove that Low$_f$(#P) = UPSV$_t$ and Low$_f$(SpanP) = NPSV$_t$. We establish the inclusion relations between NPSV$_t$, UPSV$_t$, and the counting function classes by giving for each of these inclusions an equivalent inclusi"},"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":"2507.04110","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"cs.CC","submitted_at":"2025-07-05T17:40:08Z","cross_cats_sorted":[],"title_canon_sha256":"ae6ca7e3f7c5e7f317d3abc3ad00bdd5938636086348950f2116bafab540e13b","abstract_canon_sha256":"e2e2af3c5fd788ffe2074834fa0a53c66409a9ad5aa8207117c6191360ae9aae"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-28T01:22:20.992652Z","signature_b64":"zG658ql4kCXXtBslA7PE8/3fN6KCM+nFSqGCGQfRQCzN6o0uOo8Vg0S9H1rl66rySLgZUC6MExz+/sUR5onpBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6c446bff4e885d114b8e3e2fd3c7f91da558e722828e3f7dca27f2a16c19651d","last_reissued_at":"2026-07-28T01:22:20.991719Z","signature_status":"signed_v1","first_computed_at":"2026-07-28T01:22:20.991719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Low Sets and Closure Properties of Counting Function Classes","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Yaroslav Ivanashev","submitted_at":"2025-07-05T17:40:08Z","abstract_excerpt":"A language L is low for a relativizable complexity class C, if C$^L$ = C. For the classes #P, GapP, and SpanP the exact low classes of languages are known: Low(#P) = UP $\\cap$ coUP, Low(GapP) = SPP, and Low(SpanP) = NP $\\cap$ coNP. In this paper, we prove that Low(TotP) = P, and give characterizations of low function classes for #P, GapP, TotP, and SpanP. In particular, we prove that Low$_f$(#P) = UPSV$_t$ and Low$_f$(SpanP) = NPSV$_t$. We establish the inclusion relations between NPSV$_t$, UPSV$_t$, and the counting function classes by giving for each of these inclusions an equivalent inclusi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.04110","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.04110/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2507.04110","created_at":"2026-07-28T01:22:20.992150+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.04110v4","created_at":"2026-07-28T01:22:20.992150+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.04110","created_at":"2026-07-28T01:22:20.992150+00:00"},{"alias_kind":"pith_short_12","alias_value":"NRCGX72ORBOR","created_at":"2026-07-28T01:22:20.992150+00:00"},{"alias_kind":"pith_short_16","alias_value":"NRCGX72ORBORCS4O","created_at":"2026-07-28T01:22:20.992150+00:00"},{"alias_kind":"pith_short_8","alias_value":"NRCGX72O","created_at":"2026-07-28T01:22:20.992150+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/NRCGX72ORBORCS4OHYX5HR7ZDW","json":"https://pith.science/pith/NRCGX72ORBORCS4OHYX5HR7ZDW.json","graph_json":"https://pith.science/api/pith-number/NRCGX72ORBORCS4OHYX5HR7ZDW/graph.json","events_json":"https://pith.science/api/pith-number/NRCGX72ORBORCS4OHYX5HR7ZDW/events.json","paper":"https://pith.science/paper/NRCGX72O"},"agent_actions":{"view_html":"https://pith.science/pith/NRCGX72ORBORCS4OHYX5HR7ZDW","download_json":"https://pith.science/pith/NRCGX72ORBORCS4OHYX5HR7ZDW.json","view_paper":"https://pith.science/paper/NRCGX72O","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.04110&json=true","fetch_graph":"https://pith.science/api/pith-number/NRCGX72ORBORCS4OHYX5HR7ZDW/graph.json","fetch_events":"https://pith.science/api/pith-number/NRCGX72ORBORCS4OHYX5HR7ZDW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NRCGX72ORBORCS4OHYX5HR7ZDW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NRCGX72ORBORCS4OHYX5HR7ZDW/action/storage_attestation","attest_author":"https://pith.science/pith/NRCGX72ORBORCS4OHYX5HR7ZDW/action/author_attestation","sign_citation":"https://pith.science/pith/NRCGX72ORBORCS4OHYX5HR7ZDW/action/citation_signature","submit_replication":"https://pith.science/pith/NRCGX72ORBORCS4OHYX5HR7ZDW/action/replication_record"}},"created_at":"2026-07-28T01:22:20.992150+00:00","updated_at":"2026-07-28T01:22:20.992150+00:00"}