{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:E7NN3VPQKPKXGZCM3FBQWQXB2J","short_pith_number":"pith:E7NN3VPQ","canonical_record":{"source":{"id":"1404.6632","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2014-04-26T10:53:57Z","cross_cats_sorted":[],"title_canon_sha256":"5c88f122821587744b84ef509694c286be29d62ed8c3ac37df08a8e16fb1228f","abstract_canon_sha256":"8da49a6ddaf1707c5a2c930029d2e3d6b4f910f21288fc337b3ee35b222b8f5a"},"schema_version":"1.0"},"canonical_sha256":"27daddd5f053d573644cd9430b42e1d27b6e2c5305e90b79280bbc803b67ad44","source":{"kind":"arxiv","id":"1404.6632","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1404.6632","created_at":"2026-05-18T01:59:38Z"},{"alias_kind":"arxiv_version","alias_value":"1404.6632v3","created_at":"2026-05-18T01:59:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.6632","created_at":"2026-05-18T01:59:38Z"},{"alias_kind":"pith_short_12","alias_value":"E7NN3VPQKPKX","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"E7NN3VPQKPKXGZCM","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"E7NN3VPQ","created_at":"2026-05-18T12:28:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:E7NN3VPQKPKXGZCM3FBQWQXB2J","target":"record","payload":{"canonical_record":{"source":{"id":"1404.6632","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2014-04-26T10:53:57Z","cross_cats_sorted":[],"title_canon_sha256":"5c88f122821587744b84ef509694c286be29d62ed8c3ac37df08a8e16fb1228f","abstract_canon_sha256":"8da49a6ddaf1707c5a2c930029d2e3d6b4f910f21288fc337b3ee35b222b8f5a"},"schema_version":"1.0"},"canonical_sha256":"27daddd5f053d573644cd9430b42e1d27b6e2c5305e90b79280bbc803b67ad44","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:59:38.805222Z","signature_b64":"HeV/lnIQXNY9HE+Nef3DXiLGbig/EZEPPrnc3vgl+9XfgusEqE5Y90mvxWQdDfny8rJ5ayajvdamm6oRZQPmCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27daddd5f053d573644cd9430b42e1d27b6e2c5305e90b79280bbc803b67ad44","last_reissued_at":"2026-05-18T01:59:38.804477Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:59:38.804477Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1404.6632","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:59:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T9Vn/FzihxI+M4wY4mjM3ZXo64z3KXojbIXTIEE/l/0oiVUh1EugLE959W6PDyRQpc3EnheG/mKKxQvwzhwVDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T06:41:10.478883Z"},"content_sha256":"e6fbe694362d7032dc5ae9567f5977505465f88e371740136510ce795ecc09d2","schema_version":"1.0","event_id":"sha256:e6fbe694362d7032dc5ae9567f5977505465f88e371740136510ce795ecc09d2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:E7NN3VPQKPKXGZCM3FBQWQXB2J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Complexity of Atoms, Combinatorially","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.FL","authors_text":"Szabolcs Ivan","submitted_at":"2014-04-26T10:53:57Z","abstract_excerpt":"Atoms of a (regular) language $L$ were introduced by Brzozowski and Tamm in 2011 as intersections of complemented and uncomplemented quotients of $L$. They derived tight upper bounds on the complexity of atoms in 2013. In 2014, Brzozowski and Davies characterized the regular languages meeting these bounds. To achieve these results, they used the so-called \"atomaton\" of a language, introduced by Brzozowski and Tamm in 2011. In this note we give an alternative proof of their characterization, via a purely combinatorial approach."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.6632","kind":"arxiv","version":3},"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":"2026-05-18T01:59:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FK64PZRb7wxjicHFvuCev0XiJcZwN9Lz17RZ/1UElOAorYgcooRpTth/eKTUVxbpMIfcW6OOazqEBCsG/Mr/Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T06:41:10.479228Z"},"content_sha256":"2a53bb7f14b46073c553594bde4c0eadbc1c866121cfea34bf8fe4a008aa62da","schema_version":"1.0","event_id":"sha256:2a53bb7f14b46073c553594bde4c0eadbc1c866121cfea34bf8fe4a008aa62da"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E7NN3VPQKPKXGZCM3FBQWQXB2J/bundle.json","state_url":"https://pith.science/pith/E7NN3VPQKPKXGZCM3FBQWQXB2J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E7NN3VPQKPKXGZCM3FBQWQXB2J/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":"2026-06-24T06:41:10Z","links":{"resolver":"https://pith.science/pith/E7NN3VPQKPKXGZCM3FBQWQXB2J","bundle":"https://pith.science/pith/E7NN3VPQKPKXGZCM3FBQWQXB2J/bundle.json","state":"https://pith.science/pith/E7NN3VPQKPKXGZCM3FBQWQXB2J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E7NN3VPQKPKXGZCM3FBQWQXB2J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:E7NN3VPQKPKXGZCM3FBQWQXB2J","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":"8da49a6ddaf1707c5a2c930029d2e3d6b4f910f21288fc337b3ee35b222b8f5a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2014-04-26T10:53:57Z","title_canon_sha256":"5c88f122821587744b84ef509694c286be29d62ed8c3ac37df08a8e16fb1228f"},"schema_version":"1.0","source":{"id":"1404.6632","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1404.6632","created_at":"2026-05-18T01:59:38Z"},{"alias_kind":"arxiv_version","alias_value":"1404.6632v3","created_at":"2026-05-18T01:59:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.6632","created_at":"2026-05-18T01:59:38Z"},{"alias_kind":"pith_short_12","alias_value":"E7NN3VPQKPKX","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"E7NN3VPQKPKXGZCM","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"E7NN3VPQ","created_at":"2026-05-18T12:28:25Z"}],"graph_snapshots":[{"event_id":"sha256:2a53bb7f14b46073c553594bde4c0eadbc1c866121cfea34bf8fe4a008aa62da","target":"graph","created_at":"2026-05-18T01:59:38Z","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":"Atoms of a (regular) language $L$ were introduced by Brzozowski and Tamm in 2011 as intersections of complemented and uncomplemented quotients of $L$. They derived tight upper bounds on the complexity of atoms in 2013. In 2014, Brzozowski and Davies characterized the regular languages meeting these bounds. To achieve these results, they used the so-called \"atomaton\" of a language, introduced by Brzozowski and Tamm in 2011. In this note we give an alternative proof of their characterization, via a purely combinatorial approach.","authors_text":"Szabolcs Ivan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2014-04-26T10:53:57Z","title":"Complexity of Atoms, Combinatorially"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.6632","kind":"arxiv","version":3},"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:e6fbe694362d7032dc5ae9567f5977505465f88e371740136510ce795ecc09d2","target":"record","created_at":"2026-05-18T01:59:38Z","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":"8da49a6ddaf1707c5a2c930029d2e3d6b4f910f21288fc337b3ee35b222b8f5a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2014-04-26T10:53:57Z","title_canon_sha256":"5c88f122821587744b84ef509694c286be29d62ed8c3ac37df08a8e16fb1228f"},"schema_version":"1.0","source":{"id":"1404.6632","kind":"arxiv","version":3}},"canonical_sha256":"27daddd5f053d573644cd9430b42e1d27b6e2c5305e90b79280bbc803b67ad44","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"27daddd5f053d573644cd9430b42e1d27b6e2c5305e90b79280bbc803b67ad44","first_computed_at":"2026-05-18T01:59:38.804477Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:59:38.804477Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HeV/lnIQXNY9HE+Nef3DXiLGbig/EZEPPrnc3vgl+9XfgusEqE5Y90mvxWQdDfny8rJ5ayajvdamm6oRZQPmCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:59:38.805222Z","signed_message":"canonical_sha256_bytes"},"source_id":"1404.6632","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e6fbe694362d7032dc5ae9567f5977505465f88e371740136510ce795ecc09d2","sha256:2a53bb7f14b46073c553594bde4c0eadbc1c866121cfea34bf8fe4a008aa62da"],"state_sha256":"9a84f44cbfaec379f4336f38c7ebcb23eba91ebb91c43d123c18400c12f2944b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xP4WpEuaWI/IGlb61k0y5HllM5RVRf4tybIjHHY28MmjUL2jCu/qQ4v294i5ZqVBpLMiFC3E4SgoXFjXprd4BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T06:41:10.481209Z","bundle_sha256":"9dc14d24d86f552934a5771db2eab100f78ecc466952f2fb9b7a7d5a1a370d7a"}}