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Suppose that L and L' are binary regular languages with quotient complexities m and n, and that the transition semigroups of the minimal deterministic automata accepting L and L' are the symmetric groups S_m and S_n of degrees m and n, respectively. Denote by o any binary boolean operation that is not a constant and not a function of one argument only. For m,n >= 2 with (m,n) not in {(2,2),(3,4),(4,3),(4,4)} we prove that the quotient complexity of LoL' is mn if"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1310.1841","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2013-10-07T16:06:22Z","cross_cats_sorted":[],"title_canon_sha256":"1041c67cf69d401351e9bf6718db5538e4a90decded5aca966bd9baa6848722d","abstract_canon_sha256":"51d916929576f613e881018abf8ee46ffa20ce6a2128ddc81833c65078727f51"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:10.422217Z","signature_b64":"1dyg5ro0hZeYmhXfLGYMGIBXo6dFPoerAkuWZwZLTpDCQj4Gr76A+/W52ekSea0JsOqJe9oN689evsFl+SRLCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3703848ac3b229afa9ec625f229f03d26d41279d584004438b96386e808494c7","last_reissued_at":"2026-05-18T03:11:10.421286Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:10.421286Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Symmetric Groups and Quotient Complexity of Boolean Operations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.FL","authors_text":"Janusz Brzozowski, Jason Bell, Nelma Moreira, Rog\\'erio Reis","submitted_at":"2013-10-07T16:06:22Z","abstract_excerpt":"The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. 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For m,n >= 2 with (m,n) not in {(2,2),(3,4),(4,3),(4,4)} we prove that the quotient complexity of LoL' is mn if"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.1841","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1310.1841","created_at":"2026-05-18T03:11:10.421422+00:00"},{"alias_kind":"arxiv_version","alias_value":"1310.1841v1","created_at":"2026-05-18T03:11:10.421422+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.1841","created_at":"2026-05-18T03:11:10.421422+00:00"},{"alias_kind":"pith_short_12","alias_value":"G4BYJCWDWIU2","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_16","alias_value":"G4BYJCWDWIU27KPM","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_8","alias_value":"G4BYJCWD","created_at":"2026-05-18T12:27:45.050594+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/G4BYJCWDWIU27KPMMJPSFHYD2J","json":"https://pith.science/pith/G4BYJCWDWIU27KPMMJPSFHYD2J.json","graph_json":"https://pith.science/api/pith-number/G4BYJCWDWIU27KPMMJPSFHYD2J/graph.json","events_json":"https://pith.science/api/pith-number/G4BYJCWDWIU27KPMMJPSFHYD2J/events.json","paper":"https://pith.science/paper/G4BYJCWD"},"agent_actions":{"view_html":"https://pith.science/pith/G4BYJCWDWIU27KPMMJPSFHYD2J","download_json":"https://pith.science/pith/G4BYJCWDWIU27KPMMJPSFHYD2J.json","view_paper":"https://pith.science/paper/G4BYJCWD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1310.1841&json=true","fetch_graph":"https://pith.science/api/pith-number/G4BYJCWDWIU27KPMMJPSFHYD2J/graph.json","fetch_events":"https://pith.science/api/pith-number/G4BYJCWDWIU27KPMMJPSFHYD2J/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/G4BYJCWDWIU27KPMMJPSFHYD2J/action/timestamp_anchor","attest_storage":"https://pith.science/pith/G4BYJCWDWIU27KPMMJPSFHYD2J/action/storage_attestation","attest_author":"https://pith.science/pith/G4BYJCWDWIU27KPMMJPSFHYD2J/action/author_attestation","sign_citation":"https://pith.science/pith/G4BYJCWDWIU27KPMMJPSFHYD2J/action/citation_signature","submit_replication":"https://pith.science/pith/G4BYJCWDWIU27KPMMJPSFHYD2J/action/replication_record"}},"created_at":"2026-05-18T03:11:10.421422+00:00","updated_at":"2026-05-18T03:11:10.421422+00:00"}