{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:EZQIPUCAKFS2FGUY4CXVIPLVE4","short_pith_number":"pith:EZQIPUCA","canonical_record":{"source":{"id":"1412.2281","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2014-12-06T20:58:05Z","cross_cats_sorted":[],"title_canon_sha256":"e09dde8553e0342711a2c1414d34d609753b8480289f3431e46844b00478baa0","abstract_canon_sha256":"8f31d7c6a324f62d2d98d58ff626f37bf1d29c7488bfe74b78a90b323cb54485"},"schema_version":"1.0"},"canonical_sha256":"266087d0405165a29a98e0af543d752724916a6b70cc2acac41c88c6afa9ab1d","source":{"kind":"arxiv","id":"1412.2281","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.2281","created_at":"2026-05-18T01:30:08Z"},{"alias_kind":"arxiv_version","alias_value":"1412.2281v2","created_at":"2026-05-18T01:30:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.2281","created_at":"2026-05-18T01:30:08Z"},{"alias_kind":"pith_short_12","alias_value":"EZQIPUCAKFS2","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"EZQIPUCAKFS2FGUY","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"EZQIPUCA","created_at":"2026-05-18T12:28:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:EZQIPUCAKFS2FGUY4CXVIPLVE4","target":"record","payload":{"canonical_record":{"source":{"id":"1412.2281","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2014-12-06T20:58:05Z","cross_cats_sorted":[],"title_canon_sha256":"e09dde8553e0342711a2c1414d34d609753b8480289f3431e46844b00478baa0","abstract_canon_sha256":"8f31d7c6a324f62d2d98d58ff626f37bf1d29c7488bfe74b78a90b323cb54485"},"schema_version":"1.0"},"canonical_sha256":"266087d0405165a29a98e0af543d752724916a6b70cc2acac41c88c6afa9ab1d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:08.232345Z","signature_b64":"0dy2fpv+AijR2FZ3c+i0IXNMdCVf8zY/QXlmAUWTwqXRJUUdWrL+76B+9iMWeXjlGEC0C9TwUTo2+mMLkzT8BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"266087d0405165a29a98e0af543d752724916a6b70cc2acac41c88c6afa9ab1d","last_reissued_at":"2026-05-18T01:30:08.231810Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:08.231810Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1412.2281","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:30:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uU0HgrfcgYwwXxY/pWPITtRMxiK07fsI85EJMNTX9MjYga2hZiUIzwV7LZ02NWMr42Xz1KN15PSv6ZhTeNImDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T09:25:27.857873Z"},"content_sha256":"1f2848f972d12ee0e371f1e674136c85514e8b813f49fe0aff9dc75482f63c98","schema_version":"1.0","event_id":"sha256:1f2848f972d12ee0e371f1e674136c85514e8b813f49fe0aff9dc75482f63c98"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:EZQIPUCAKFS2FGUY4CXVIPLVE4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Syntactic Complexity of Suffix-Free Languages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.FL","authors_text":"Janusz Brzozowski, Marek Szyku{\\l}a","submitted_at":"2014-12-06T20:58:05Z","abstract_excerpt":"We solve an open problem concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a suffix-free language with $n$ left quotients (that is, with state complexity $n$) is at most $(n-1)^{n-2}+n-2$ for $n\\ge 6$. Since this bound is known to be reachable, this settles the problem. We also reduce the alphabet of the witness languages reaching this bound to five letters instead of $n+2$, and show that it cannot be any smaller. Finally, we prove that the transition semigroup of a minimal deterministic automaton accepting a witness language is unique for each $n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.2281","kind":"arxiv","version":2},"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":"2026-05-18T01:30:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iVLjQkYcXSpniz1pBSdA1uI6V9AsLXrzh5iX94ZOHyRKm9xgkUyEQ62vCxuYZFLR6qTfyHvTRfi/Jfm2EEGqDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T09:25:27.858241Z"},"content_sha256":"012464ef92a34afe8727f609e68490bfb72823052651cf72cb70b522be9d1211","schema_version":"1.0","event_id":"sha256:012464ef92a34afe8727f609e68490bfb72823052651cf72cb70b522be9d1211"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EZQIPUCAKFS2FGUY4CXVIPLVE4/bundle.json","state_url":"https://pith.science/pith/EZQIPUCAKFS2FGUY4CXVIPLVE4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EZQIPUCAKFS2FGUY4CXVIPLVE4/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":"2026-05-30T09:25:27Z","links":{"resolver":"https://pith.science/pith/EZQIPUCAKFS2FGUY4CXVIPLVE4","bundle":"https://pith.science/pith/EZQIPUCAKFS2FGUY4CXVIPLVE4/bundle.json","state":"https://pith.science/pith/EZQIPUCAKFS2FGUY4CXVIPLVE4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EZQIPUCAKFS2FGUY4CXVIPLVE4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:EZQIPUCAKFS2FGUY4CXVIPLVE4","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":"8f31d7c6a324f62d2d98d58ff626f37bf1d29c7488bfe74b78a90b323cb54485","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2014-12-06T20:58:05Z","title_canon_sha256":"e09dde8553e0342711a2c1414d34d609753b8480289f3431e46844b00478baa0"},"schema_version":"1.0","source":{"id":"1412.2281","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.2281","created_at":"2026-05-18T01:30:08Z"},{"alias_kind":"arxiv_version","alias_value":"1412.2281v2","created_at":"2026-05-18T01:30:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.2281","created_at":"2026-05-18T01:30:08Z"},{"alias_kind":"pith_short_12","alias_value":"EZQIPUCAKFS2","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_16","alias_value":"EZQIPUCAKFS2FGUY","created_at":"2026-05-18T12:28:28Z"},{"alias_kind":"pith_short_8","alias_value":"EZQIPUCA","created_at":"2026-05-18T12:28:28Z"}],"graph_snapshots":[{"event_id":"sha256:012464ef92a34afe8727f609e68490bfb72823052651cf72cb70b522be9d1211","target":"graph","created_at":"2026-05-18T01:30:08Z","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":"We solve an open problem concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a suffix-free language with $n$ left quotients (that is, with state complexity $n$) is at most $(n-1)^{n-2}+n-2$ for $n\\ge 6$. Since this bound is known to be reachable, this settles the problem. We also reduce the alphabet of the witness languages reaching this bound to five letters instead of $n+2$, and show that it cannot be any smaller. Finally, we prove that the transition semigroup of a minimal deterministic automaton accepting a witness language is unique for each $n$.","authors_text":"Janusz Brzozowski, Marek Szyku{\\l}a","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2014-12-06T20:58:05Z","title":"Syntactic Complexity of Suffix-Free Languages"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.2281","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":"sha256:1f2848f972d12ee0e371f1e674136c85514e8b813f49fe0aff9dc75482f63c98","target":"record","created_at":"2026-05-18T01:30:08Z","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":"8f31d7c6a324f62d2d98d58ff626f37bf1d29c7488bfe74b78a90b323cb54485","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2014-12-06T20:58:05Z","title_canon_sha256":"e09dde8553e0342711a2c1414d34d609753b8480289f3431e46844b00478baa0"},"schema_version":"1.0","source":{"id":"1412.2281","kind":"arxiv","version":2}},"canonical_sha256":"266087d0405165a29a98e0af543d752724916a6b70cc2acac41c88c6afa9ab1d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"266087d0405165a29a98e0af543d752724916a6b70cc2acac41c88c6afa9ab1d","first_computed_at":"2026-05-18T01:30:08.231810Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:30:08.231810Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0dy2fpv+AijR2FZ3c+i0IXNMdCVf8zY/QXlmAUWTwqXRJUUdWrL+76B+9iMWeXjlGEC0C9TwUTo2+mMLkzT8BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:30:08.232345Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.2281","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1f2848f972d12ee0e371f1e674136c85514e8b813f49fe0aff9dc75482f63c98","sha256:012464ef92a34afe8727f609e68490bfb72823052651cf72cb70b522be9d1211"],"state_sha256":"bfeb5266f6ad604850889f1b5081b8e8f0e1cdb7f2f1eec9fefacb5a66c9ca4d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZnbncM1cINmtZVjVy2cnl+yQZp9FsPkvh3hccX4EPaeAR24lI7EochtwIZepIXJ5I+sMPGI7xukOrv/nADUKBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T09:25:27.860412Z","bundle_sha256":"7ed6450eb59dc855f1f03421d04b5669c648e532288db8cfe1ca4292b5a4e618"}}