{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:3MSLHT35QUKL6HDLUHRLC27Y7Y","short_pith_number":"pith:3MSLHT35","schema_version":"1.0","canonical_sha256":"db24b3cf7d8514bf1c6ba1e2b16bf8fe1a786eda46bb4aa84fb28409354b60f9","source":{"kind":"arxiv","id":"1603.09333","version":2},"attestation_state":"computed","paper":{"title":"The subpower membership problem for semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.GR","authors_text":"Andrei Bulatov, Marcin Kozik, Markus Steindl, Peter Mayr","submitted_at":"2016-03-29T22:50:33Z","abstract_excerpt":"Fix a finite semigroup $S$ and let $a_1,\\ldots,a_k, b$ be tuples in a direct power $S^n$. The subpower membership problem (SMP) asks whether $b$ can be generated by $a_1,\\ldots,a_k$. If $S$ is a finite group, then there is a folklore algorithm that decides this problem in time polynomial in $nk$. For semigroups this problem always lies in PSPACE. We show that the SMP for a full transformation semigroup on 3 letters or more is actually PSPACE-complete, while on 2 letters it is in P. For commutative semigroups, we provide a dichotomy result: if a commutative semigroup $S$ embeds into a direct pr"},"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":"1603.09333","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-03-29T22:50:33Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"58d4217b01bf2bd804b1aa7819e90000a674209f135a20531dda86643291559f","abstract_canon_sha256":"d099a199c1c44a00e0a252f03d5cd96fe40ea6e96270d7ee813a516cefe293dc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:48.974622Z","signature_b64":"uiTYLNXlNmjxzokOjFEVAoQuN8bAJeRt6aW6+3c3+69JC8JfVz/gR847/I9y1gRnLXDxgCarOC8t6LjmNIprAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db24b3cf7d8514bf1c6ba1e2b16bf8fe1a786eda46bb4aa84fb28409354b60f9","last_reissued_at":"2026-05-18T01:07:48.974166Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:48.974166Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The subpower membership problem for semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.GR","authors_text":"Andrei Bulatov, Marcin Kozik, Markus Steindl, Peter Mayr","submitted_at":"2016-03-29T22:50:33Z","abstract_excerpt":"Fix a finite semigroup $S$ and let $a_1,\\ldots,a_k, b$ be tuples in a direct power $S^n$. The subpower membership problem (SMP) asks whether $b$ can be generated by $a_1,\\ldots,a_k$. If $S$ is a finite group, then there is a folklore algorithm that decides this problem in time polynomial in $nk$. For semigroups this problem always lies in PSPACE. We show that the SMP for a full transformation semigroup on 3 letters or more is actually PSPACE-complete, while on 2 letters it is in P. For commutative semigroups, we provide a dichotomy result: if a commutative semigroup $S$ embeds into a direct pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.09333","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1603.09333","created_at":"2026-05-18T01:07:48.974241+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.09333v2","created_at":"2026-05-18T01:07:48.974241+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.09333","created_at":"2026-05-18T01:07:48.974241+00:00"},{"alias_kind":"pith_short_12","alias_value":"3MSLHT35QUKL","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"3MSLHT35QUKL6HDL","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"3MSLHT35","created_at":"2026-05-18T12:29:55.572404+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/3MSLHT35QUKL6HDLUHRLC27Y7Y","json":"https://pith.science/pith/3MSLHT35QUKL6HDLUHRLC27Y7Y.json","graph_json":"https://pith.science/api/pith-number/3MSLHT35QUKL6HDLUHRLC27Y7Y/graph.json","events_json":"https://pith.science/api/pith-number/3MSLHT35QUKL6HDLUHRLC27Y7Y/events.json","paper":"https://pith.science/paper/3MSLHT35"},"agent_actions":{"view_html":"https://pith.science/pith/3MSLHT35QUKL6HDLUHRLC27Y7Y","download_json":"https://pith.science/pith/3MSLHT35QUKL6HDLUHRLC27Y7Y.json","view_paper":"https://pith.science/paper/3MSLHT35","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.09333&json=true","fetch_graph":"https://pith.science/api/pith-number/3MSLHT35QUKL6HDLUHRLC27Y7Y/graph.json","fetch_events":"https://pith.science/api/pith-number/3MSLHT35QUKL6HDLUHRLC27Y7Y/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3MSLHT35QUKL6HDLUHRLC27Y7Y/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3MSLHT35QUKL6HDLUHRLC27Y7Y/action/storage_attestation","attest_author":"https://pith.science/pith/3MSLHT35QUKL6HDLUHRLC27Y7Y/action/author_attestation","sign_citation":"https://pith.science/pith/3MSLHT35QUKL6HDLUHRLC27Y7Y/action/citation_signature","submit_replication":"https://pith.science/pith/3MSLHT35QUKL6HDLUHRLC27Y7Y/action/replication_record"}},"created_at":"2026-05-18T01:07:48.974241+00:00","updated_at":"2026-05-18T01:07:48.974241+00:00"}