{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:E2RO66CSPX4U7VMS7NP5ROL5RZ","short_pith_number":"pith:E2RO66CS","schema_version":"1.0","canonical_sha256":"26a2ef78527df94fd592fb5fd8b97d8e55b1783af18100d3be9f595eccdf6649","source":{"kind":"arxiv","id":"1710.01934","version":2},"attestation_state":"computed","paper":{"title":"Model-Theoretic Characterizations of Boolean and Arithmetic Circuit Classes of Small Depth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Anselm Haak, Arnaud Durand, Heribert Vollmer","submitted_at":"2017-10-05T09:23:29Z","abstract_excerpt":"In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes $\\textrm{NC}^1$, $\\textrm{SAC}^1$ and $\\textrm{AC}^1$ as well as their arithmetic counterparts $\\#\\textrm{NC}^1$, $\\#\\textrm{SAC}^1$ and $\\#\\textrm{AC}^1$. We build on Immerman's characterization of constant-depth polynomial-size circuits by formulas of first-order logic, i.e., $\\textrm{AC}^0 = \\textrm{FO}$, and augment the logical language with an operator for defining relations in an inductive way. Considering sli"},"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":"1710.01934","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2017-10-05T09:23:29Z","cross_cats_sorted":[],"title_canon_sha256":"da7fd4a7eabdd8a35d374b71efef9a0d88c92b6eb094fd464ebac71945bd8e49","abstract_canon_sha256":"02c9c7d8162673c01173209724e51e223ae668f20cf23a90fba8fc19495ccd35"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:34.371351Z","signature_b64":"d/RyvVMzaO6dSoZlR+wiqZcsqK4x1ptOUFJec4BYc//UdHiV4TysU9T3wx2MWhWPS4AQGPGDifLCFpBWu19LBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"26a2ef78527df94fd592fb5fd8b97d8e55b1783af18100d3be9f595eccdf6649","last_reissued_at":"2026-05-18T00:33:34.370725Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:34.370725Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Model-Theoretic Characterizations of Boolean and Arithmetic Circuit Classes of Small Depth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Anselm Haak, Arnaud Durand, Heribert Vollmer","submitted_at":"2017-10-05T09:23:29Z","abstract_excerpt":"In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes $\\textrm{NC}^1$, $\\textrm{SAC}^1$ and $\\textrm{AC}^1$ as well as their arithmetic counterparts $\\#\\textrm{NC}^1$, $\\#\\textrm{SAC}^1$ and $\\#\\textrm{AC}^1$. We build on Immerman's characterization of constant-depth polynomial-size circuits by formulas of first-order logic, i.e., $\\textrm{AC}^0 = \\textrm{FO}$, and augment the logical language with an operator for defining relations in an inductive way. Considering sli"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.01934","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1710.01934","created_at":"2026-05-18T00:33:34.370804+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.01934v2","created_at":"2026-05-18T00:33:34.370804+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.01934","created_at":"2026-05-18T00:33:34.370804+00:00"},{"alias_kind":"pith_short_12","alias_value":"E2RO66CSPX4U","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_16","alias_value":"E2RO66CSPX4U7VMS","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_8","alias_value":"E2RO66CS","created_at":"2026-05-18T12:31:12.930513+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/E2RO66CSPX4U7VMS7NP5ROL5RZ","json":"https://pith.science/pith/E2RO66CSPX4U7VMS7NP5ROL5RZ.json","graph_json":"https://pith.science/api/pith-number/E2RO66CSPX4U7VMS7NP5ROL5RZ/graph.json","events_json":"https://pith.science/api/pith-number/E2RO66CSPX4U7VMS7NP5ROL5RZ/events.json","paper":"https://pith.science/paper/E2RO66CS"},"agent_actions":{"view_html":"https://pith.science/pith/E2RO66CSPX4U7VMS7NP5ROL5RZ","download_json":"https://pith.science/pith/E2RO66CSPX4U7VMS7NP5ROL5RZ.json","view_paper":"https://pith.science/paper/E2RO66CS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.01934&json=true","fetch_graph":"https://pith.science/api/pith-number/E2RO66CSPX4U7VMS7NP5ROL5RZ/graph.json","fetch_events":"https://pith.science/api/pith-number/E2RO66CSPX4U7VMS7NP5ROL5RZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/E2RO66CSPX4U7VMS7NP5ROL5RZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/E2RO66CSPX4U7VMS7NP5ROL5RZ/action/storage_attestation","attest_author":"https://pith.science/pith/E2RO66CSPX4U7VMS7NP5ROL5RZ/action/author_attestation","sign_citation":"https://pith.science/pith/E2RO66CSPX4U7VMS7NP5ROL5RZ/action/citation_signature","submit_replication":"https://pith.science/pith/E2RO66CSPX4U7VMS7NP5ROL5RZ/action/replication_record"}},"created_at":"2026-05-18T00:33:34.370804+00:00","updated_at":"2026-05-18T00:33:34.370804+00:00"}