{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:Q47ROL4TMK6S4653ZTKJW75RLX","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":"8e8781485107efdd1b300f7060bceef81aaaaec8313a1cb839e1399d46e52eb2","cross_cats_sorted":["cs.FL","cs.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2009-11-17T13:40:29Z","title_canon_sha256":"9032976700b2ccfa63fd7eb2826d3e4145f937bb672a11e778339f10e663dce4"},"schema_version":"1.0","source":{"id":"0911.3307","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0911.3307","created_at":"2026-05-18T04:40:20Z"},{"alias_kind":"arxiv_version","alias_value":"0911.3307v1","created_at":"2026-05-18T04:40:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.3307","created_at":"2026-05-18T04:40:20Z"},{"alias_kind":"pith_short_12","alias_value":"Q47ROL4TMK6S","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"Q47ROL4TMK6S4653","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"Q47ROL4T","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:a65419c9be4e49bac505d1ea2351f5c60eb822f14156f7b2a79b784739291f3e","target":"graph","created_at":"2026-05-18T04:40:20Z","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 dictionary is a set of finite words over some finite alphabet X. The omega-power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [Omega-powers and descriptive set theory, JSL 2005] the complexity of the set of dictionaries whose associated omega-powers have a given complexity. In particular, he considered the sets $W({\\bf\\Si}^0_{k})$ (respectively, $W({\\bf\\Pi}^0_{k})$, $W({\\bf\\Delta}_1^1)$) of dictionaries $V \\subseteq 2^\\star$ whose omega-powers are ${\\bf\\Si}^0_{k}$-sets (respectively, ${\\bf\\Pi}^0_{k}$-sets, Borel sets). In","authors_text":"Olivier Finkel (ELM)","cross_cats":["cs.FL","cs.LO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2009-11-17T13:40:29Z","title":"On Some Sets of Dictionaries Whose omega-Powers Have a Given Complexity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.3307","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":"sha256:52a90e13f1bdc7e98cea78c9aa680a1d88c826c6eee81f3a6b447137474f23d3","target":"record","created_at":"2026-05-18T04:40:20Z","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":"8e8781485107efdd1b300f7060bceef81aaaaec8313a1cb839e1399d46e52eb2","cross_cats_sorted":["cs.FL","cs.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2009-11-17T13:40:29Z","title_canon_sha256":"9032976700b2ccfa63fd7eb2826d3e4145f937bb672a11e778339f10e663dce4"},"schema_version":"1.0","source":{"id":"0911.3307","kind":"arxiv","version":1}},"canonical_sha256":"873f172f9362bd2e7bbbccd49b7fb15dfdf05f178c507169b59ea6bb2caa3cb4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"873f172f9362bd2e7bbbccd49b7fb15dfdf05f178c507169b59ea6bb2caa3cb4","first_computed_at":"2026-05-18T04:40:20.489930Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:40:20.489930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"L4EKuwvRlJRjOkelPtpvhHIFrroV2dvFlbj/jxLdbVu6J8C+fnbJoyWOPa5NdjQpOc35WHd5zen4n5/aHLS2Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:40:20.490661Z","signed_message":"canonical_sha256_bytes"},"source_id":"0911.3307","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:52a90e13f1bdc7e98cea78c9aa680a1d88c826c6eee81f3a6b447137474f23d3","sha256:a65419c9be4e49bac505d1ea2351f5c60eb822f14156f7b2a79b784739291f3e"],"state_sha256":"e599d6424e0335db2d530541f678a299ede49416909b645b3ad12bf65822dd83"}