{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:HLL2CC4TNI2YOIGSIAV3Z64VDP","short_pith_number":"pith:HLL2CC4T","schema_version":"1.0","canonical_sha256":"3ad7a10b936a358720d2402bbcfb951bdc4a57d7c073accd69cfc2693d6eb080","source":{"kind":"arxiv","id":"0912.1034","version":1},"attestation_state":"computed","paper":{"title":"Quotient Complexity of Closed Languages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.FL","authors_text":"C. Zou, G. Jir\\'askov\\'a, J. Brzozowski","submitted_at":"2009-12-05T16:08:39Z","abstract_excerpt":"A language L is prefix-closed if, whenever a word w is in L, then every prefix of w is also in L. We define suffix-, factor-, and subword-closed languages in the same way, where by subword we mean subsequence. We study the quotient complexity (usually called state complexity) of operations on prefix-, suffix-, factor-, and subword-closed languages. We find tight upper bounds on the complexity of the prefix-, suffix-, factor-, and subword-closure of arbitrary languages, and on the complexity of boolean operations, concatenation, star and reversal in each of the four classes of closed languages."},"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":"0912.1034","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2009-12-05T16:08:39Z","cross_cats_sorted":[],"title_canon_sha256":"a0b922207f9015821f16871ab91fc23aa2bc9299c0b132f54313b5b87a61c8db","abstract_canon_sha256":"f83e83923060f17685dfead57dfcf2ecfc8a07871e719b56fa06343cc941d994"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:10:26.248778Z","signature_b64":"hQXLqCmmuZ4a0beui4oPwmJwPmUc9Leok0A36hlvY7DwWvJEFumJwWV9J9Bvdi6ChNBifW4DFuTuRDgCS2IJAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3ad7a10b936a358720d2402bbcfb951bdc4a57d7c073accd69cfc2693d6eb080","last_reissued_at":"2026-05-18T02:10:26.247713Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:10:26.247713Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quotient Complexity of Closed Languages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.FL","authors_text":"C. Zou, G. Jir\\'askov\\'a, J. Brzozowski","submitted_at":"2009-12-05T16:08:39Z","abstract_excerpt":"A language L is prefix-closed if, whenever a word w is in L, then every prefix of w is also in L. We define suffix-, factor-, and subword-closed languages in the same way, where by subword we mean subsequence. We study the quotient complexity (usually called state complexity) of operations on prefix-, suffix-, factor-, and subword-closed languages. We find tight upper bounds on the complexity of the prefix-, suffix-, factor-, and subword-closure of arbitrary languages, and on the complexity of boolean operations, concatenation, star and reversal in each of the four classes of closed languages."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.1034","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":"0912.1034","created_at":"2026-05-18T02:10:26.247850+00:00"},{"alias_kind":"arxiv_version","alias_value":"0912.1034v1","created_at":"2026-05-18T02:10:26.247850+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.1034","created_at":"2026-05-18T02:10:26.247850+00:00"},{"alias_kind":"pith_short_12","alias_value":"HLL2CC4TNI2Y","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_16","alias_value":"HLL2CC4TNI2YOIGS","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_8","alias_value":"HLL2CC4T","created_at":"2026-05-18T12:25:59.703012+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/HLL2CC4TNI2YOIGSIAV3Z64VDP","json":"https://pith.science/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP.json","graph_json":"https://pith.science/api/pith-number/HLL2CC4TNI2YOIGSIAV3Z64VDP/graph.json","events_json":"https://pith.science/api/pith-number/HLL2CC4TNI2YOIGSIAV3Z64VDP/events.json","paper":"https://pith.science/paper/HLL2CC4T"},"agent_actions":{"view_html":"https://pith.science/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP","download_json":"https://pith.science/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP.json","view_paper":"https://pith.science/paper/HLL2CC4T","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0912.1034&json=true","fetch_graph":"https://pith.science/api/pith-number/HLL2CC4TNI2YOIGSIAV3Z64VDP/graph.json","fetch_events":"https://pith.science/api/pith-number/HLL2CC4TNI2YOIGSIAV3Z64VDP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP/action/storage_attestation","attest_author":"https://pith.science/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP/action/author_attestation","sign_citation":"https://pith.science/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP/action/citation_signature","submit_replication":"https://pith.science/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP/action/replication_record"}},"created_at":"2026-05-18T02:10:26.247850+00:00","updated_at":"2026-05-18T02:10:26.247850+00:00"}