{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:CJSBZHIBTFSMVAHCR6XHDZTW5B","short_pith_number":"pith:CJSBZHIB","schema_version":"1.0","canonical_sha256":"12641c9d019964ca80e28fae71e676e8439f5141b7347663490a6ebaa89734ca","source":{"kind":"arxiv","id":"1504.07753","version":1},"attestation_state":"computed","paper":{"title":"Hydras: Directed Hypergraphs and Horn Formulas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LO","math.CO"],"primary_cat":"cs.DM","authors_text":"Despina Stasi, Gyorgy Turan, Robert H. Sloan","submitted_at":"2015-04-29T08:08:39Z","abstract_excerpt":"We introduce a new graph parameter, the hydra number, arising from the minimization problem for Horn formulas in propositional logic. The hydra number of a graph $G=(V,E)$ is the minimal number of hyperarcs of the form $u,v\\rightarrow w$ required in a directed hypergraph $H=(V,F)$, such that for every pair $(u, v)$, the set of vertices reachable in $H$ from $\\{u, v\\}$ is the entire vertex set $V$ if $(u, v) \\in E$, and it is $\\{u, v\\}$ otherwise. Here reachability is defined by forward chaining, a standard marking algorithm.\n  Various bounds are given for the hydra number. We show that the hyd"},"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":"1504.07753","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-04-29T08:08:39Z","cross_cats_sorted":["cs.LO","math.CO"],"title_canon_sha256":"563b4694c3a98635b5f7e1cef5013732cc65480a5716a54a4cdd9b0f5a58bb8c","abstract_canon_sha256":"95dd8fbb542079baf5447e1ad442f5e46676af69ade9a57514d5bc703b957a80"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:32.344554Z","signature_b64":"77WMp/UZyPm25SDouB1uTnFklPbhLEeX8/4Jptu1noG/Bac994Qxgn46CPDrK/CKrQ6ywi8EL5uBuzlSsypTBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"12641c9d019964ca80e28fae71e676e8439f5141b7347663490a6ebaa89734ca","last_reissued_at":"2026-05-18T02:17:32.343807Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:32.343807Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hydras: Directed Hypergraphs and Horn Formulas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LO","math.CO"],"primary_cat":"cs.DM","authors_text":"Despina Stasi, Gyorgy Turan, Robert H. Sloan","submitted_at":"2015-04-29T08:08:39Z","abstract_excerpt":"We introduce a new graph parameter, the hydra number, arising from the minimization problem for Horn formulas in propositional logic. The hydra number of a graph $G=(V,E)$ is the minimal number of hyperarcs of the form $u,v\\rightarrow w$ required in a directed hypergraph $H=(V,F)$, such that for every pair $(u, v)$, the set of vertices reachable in $H$ from $\\{u, v\\}$ is the entire vertex set $V$ if $(u, v) \\in E$, and it is $\\{u, v\\}$ otherwise. Here reachability is defined by forward chaining, a standard marking algorithm.\n  Various bounds are given for the hydra number. We show that the hyd"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07753","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":"1504.07753","created_at":"2026-05-18T02:17:32.343915+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.07753v1","created_at":"2026-05-18T02:17:32.343915+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.07753","created_at":"2026-05-18T02:17:32.343915+00:00"},{"alias_kind":"pith_short_12","alias_value":"CJSBZHIBTFSM","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_16","alias_value":"CJSBZHIBTFSMVAHC","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_8","alias_value":"CJSBZHIB","created_at":"2026-05-18T12:29:14.074870+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/CJSBZHIBTFSMVAHCR6XHDZTW5B","json":"https://pith.science/pith/CJSBZHIBTFSMVAHCR6XHDZTW5B.json","graph_json":"https://pith.science/api/pith-number/CJSBZHIBTFSMVAHCR6XHDZTW5B/graph.json","events_json":"https://pith.science/api/pith-number/CJSBZHIBTFSMVAHCR6XHDZTW5B/events.json","paper":"https://pith.science/paper/CJSBZHIB"},"agent_actions":{"view_html":"https://pith.science/pith/CJSBZHIBTFSMVAHCR6XHDZTW5B","download_json":"https://pith.science/pith/CJSBZHIBTFSMVAHCR6XHDZTW5B.json","view_paper":"https://pith.science/paper/CJSBZHIB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.07753&json=true","fetch_graph":"https://pith.science/api/pith-number/CJSBZHIBTFSMVAHCR6XHDZTW5B/graph.json","fetch_events":"https://pith.science/api/pith-number/CJSBZHIBTFSMVAHCR6XHDZTW5B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CJSBZHIBTFSMVAHCR6XHDZTW5B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CJSBZHIBTFSMVAHCR6XHDZTW5B/action/storage_attestation","attest_author":"https://pith.science/pith/CJSBZHIBTFSMVAHCR6XHDZTW5B/action/author_attestation","sign_citation":"https://pith.science/pith/CJSBZHIBTFSMVAHCR6XHDZTW5B/action/citation_signature","submit_replication":"https://pith.science/pith/CJSBZHIBTFSMVAHCR6XHDZTW5B/action/replication_record"}},"created_at":"2026-05-18T02:17:32.343915+00:00","updated_at":"2026-05-18T02:17:32.343915+00:00"}