{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:HLL2CC4TNI2YOIGSIAV3Z64VDP","short_pith_number":"pith:HLL2CC4T","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"},"canonical_sha256":"3ad7a10b936a358720d2402bbcfb951bdc4a57d7c073accd69cfc2693d6eb080","source":{"kind":"arxiv","id":"0912.1034","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.1034","created_at":"2026-05-18T02:10:26Z"},{"alias_kind":"arxiv_version","alias_value":"0912.1034v1","created_at":"2026-05-18T02:10:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.1034","created_at":"2026-05-18T02:10:26Z"},{"alias_kind":"pith_short_12","alias_value":"HLL2CC4TNI2Y","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"HLL2CC4TNI2YOIGS","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"HLL2CC4T","created_at":"2026-05-18T12:25:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:HLL2CC4TNI2YOIGSIAV3Z64VDP","target":"record","payload":{"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"},"canonical_sha256":"3ad7a10b936a358720d2402bbcfb951bdc4a57d7c073accd69cfc2693d6eb080","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"},"source_kind":"arxiv","source_id":"0912.1034","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:10:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SPxK0OFP2gPtCsdXoQMp6nDx8VxuITTMfmZ9VKJMul2bBgSmKSnaM+5S7V7rneAtTi0PPubWx6zWE9nFvlHNDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T06:17:57.877161Z"},"content_sha256":"4a76a9061b81572343cdab418776bd3ebfb9b3f050a030800cddf251c064b318","schema_version":"1.0","event_id":"sha256:4a76a9061b81572343cdab418776bd3ebfb9b3f050a030800cddf251c064b318"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:HLL2CC4TNI2YOIGSIAV3Z64VDP","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:10:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3kWzsBpHa7DJ/VH/Kmj5SG0hwWAgzLvCIWc5pYO3Kx2bst44orZzK5/sZpn3AwtnHRWvEEi9mXtTddOq9EVsDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T06:17:57.877537Z"},"content_sha256":"a5f0e0a34a87a0f354942b56c469ba9bb2c42dfc47e7d4da6edc5ccb3ae261d9","schema_version":"1.0","event_id":"sha256:a5f0e0a34a87a0f354942b56c469ba9bb2c42dfc47e7d4da6edc5ccb3ae261d9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP/bundle.json","state_url":"https://pith.science/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-04T06:17:57Z","links":{"resolver":"https://pith.science/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP","bundle":"https://pith.science/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP/bundle.json","state":"https://pith.science/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HLL2CC4TNI2YOIGSIAV3Z64VDP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:HLL2CC4TNI2YOIGSIAV3Z64VDP","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f83e83923060f17685dfead57dfcf2ecfc8a07871e719b56fa06343cc941d994","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2009-12-05T16:08:39Z","title_canon_sha256":"a0b922207f9015821f16871ab91fc23aa2bc9299c0b132f54313b5b87a61c8db"},"schema_version":"1.0","source":{"id":"0912.1034","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.1034","created_at":"2026-05-18T02:10:26Z"},{"alias_kind":"arxiv_version","alias_value":"0912.1034v1","created_at":"2026-05-18T02:10:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.1034","created_at":"2026-05-18T02:10:26Z"},{"alias_kind":"pith_short_12","alias_value":"HLL2CC4TNI2Y","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"HLL2CC4TNI2YOIGS","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"HLL2CC4T","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:a5f0e0a34a87a0f354942b56c469ba9bb2c42dfc47e7d4da6edc5ccb3ae261d9","target":"graph","created_at":"2026-05-18T02:10:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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.","authors_text":"C. Zou, G. Jir\\'askov\\'a, J. Brzozowski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2009-12-05T16:08:39Z","title":"Quotient Complexity of Closed Languages"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.1034","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4a76a9061b81572343cdab418776bd3ebfb9b3f050a030800cddf251c064b318","target":"record","created_at":"2026-05-18T02:10:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f83e83923060f17685dfead57dfcf2ecfc8a07871e719b56fa06343cc941d994","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2009-12-05T16:08:39Z","title_canon_sha256":"a0b922207f9015821f16871ab91fc23aa2bc9299c0b132f54313b5b87a61c8db"},"schema_version":"1.0","source":{"id":"0912.1034","kind":"arxiv","version":1}},"canonical_sha256":"3ad7a10b936a358720d2402bbcfb951bdc4a57d7c073accd69cfc2693d6eb080","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3ad7a10b936a358720d2402bbcfb951bdc4a57d7c073accd69cfc2693d6eb080","first_computed_at":"2026-05-18T02:10:26.247713Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:10:26.247713Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hQXLqCmmuZ4a0beui4oPwmJwPmUc9Leok0A36hlvY7DwWvJEFumJwWV9J9Bvdi6ChNBifW4DFuTuRDgCS2IJAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:10:26.248778Z","signed_message":"canonical_sha256_bytes"},"source_id":"0912.1034","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a76a9061b81572343cdab418776bd3ebfb9b3f050a030800cddf251c064b318","sha256:a5f0e0a34a87a0f354942b56c469ba9bb2c42dfc47e7d4da6edc5ccb3ae261d9"],"state_sha256":"3dbdfc963c3255862acdfbf86034beb773900cdd602e480d3cb85f706345628e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZcimD4kVMC4rHblqiMQtSjrXAB9x0t48/EJbUePCvHNMOlyoRXxwAWrdS/qGRwWF3iqpMftna8gyZP7D4jgPBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T06:17:57.880246Z","bundle_sha256":"dddf1c5dcf2761899be7969864f85646ab187d344044a0f4c9a227ccea7b708a"}}