{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:NO5PQVEFV2YOCGEYW6EZ4U7JWL","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":"e98e2b58554e53cd678db22830444daab483ddd4caac96d065d279ac8152508e","cross_cats_sorted":["cs.LO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.FL","submitted_at":"2026-01-27T10:49:13Z","title_canon_sha256":"1ce3218e737d1d5dc46f11f50c1c8f90b16f9db2838cde1fc08f22a9f8b562b1"},"schema_version":"1.0","source":{"id":"2601.19466","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2601.19466","created_at":"2026-05-28T02:04:45Z"},{"alias_kind":"arxiv_version","alias_value":"2601.19466v2","created_at":"2026-05-28T02:04:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2601.19466","created_at":"2026-05-28T02:04:45Z"},{"alias_kind":"pith_short_12","alias_value":"NO5PQVEFV2YO","created_at":"2026-05-28T02:04:45Z"},{"alias_kind":"pith_short_16","alias_value":"NO5PQVEFV2YOCGEY","created_at":"2026-05-28T02:04:45Z"},{"alias_kind":"pith_short_8","alias_value":"NO5PQVEF","created_at":"2026-05-28T02:04:45Z"}],"graph_snapshots":[{"event_id":"sha256:33d4685c25b87ac18fe65090e0fb8419f3d9ca6868315bb70b43d35cae7ff865","target":"graph","created_at":"2026-05-28T02:04:45Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2601.19466/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Indexed languages are a classical notion in formal language theory, which has attracted attention in recent decades due to its role in higher-order model checking: They are precisely the languages accepted by order-2 pushdown automata.\n  The downward closure of an indexed language -- the set of all (scattered) subwords of its members -- is well-known to be a regular over-approximation. It is known since 2015 that the downward closure of a given indexed language is effectively computable. However, the algorithm comes with no complexity bounds, and it has remained open whether a primitive-recurs","authors_text":"Corto Mascle, Georg Zetzsche, Richard Mandel","cross_cats":["cs.LO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.FL","submitted_at":"2026-01-27T10:49:13Z","title":"The complexity of downward closures of indexed languages"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.19466","kind":"arxiv","version":2},"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":"sha256:69ac1779416dcd2391530454ec2c2daa882181bc190a9e72ff7d379a29714888","target":"record","created_at":"2026-05-28T02:04:45Z","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":"e98e2b58554e53cd678db22830444daab483ddd4caac96d065d279ac8152508e","cross_cats_sorted":["cs.LO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.FL","submitted_at":"2026-01-27T10:49:13Z","title_canon_sha256":"1ce3218e737d1d5dc46f11f50c1c8f90b16f9db2838cde1fc08f22a9f8b562b1"},"schema_version":"1.0","source":{"id":"2601.19466","kind":"arxiv","version":2}},"canonical_sha256":"6bbaf85485aeb0e11898b7899e53e9b2fa8d04c66c29f545c693bc49f976cbee","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6bbaf85485aeb0e11898b7899e53e9b2fa8d04c66c29f545c693bc49f976cbee","first_computed_at":"2026-05-28T02:04:45.552104Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-28T02:04:45.552104Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"C4KYjvv4YHYR2iobjXbai/eFefKZMbcWL4etWSCkzyU07BuyLGQIpKacP7Y2ks4I46GtKW2OKi8tZjtAivE3Dg==","signature_status":"signed_v1","signed_at":"2026-05-28T02:04:45.552653Z","signed_message":"canonical_sha256_bytes"},"source_id":"2601.19466","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:69ac1779416dcd2391530454ec2c2daa882181bc190a9e72ff7d379a29714888","sha256:33d4685c25b87ac18fe65090e0fb8419f3d9ca6868315bb70b43d35cae7ff865"],"state_sha256":"f927a294c048c937adc69fd93dc6831517ced298d2778b8655ad26e7bfbb14ed"}