{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:G2WASIQXYLKUJB2SHYVFOALNRR","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":"72a1dbb8087fee8a45387cc1bb496d3a89cee00094f51d1281e5de7e5c428c39","cross_cats_sorted":["cs.LO"],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"cs.FL","submitted_at":"2010-01-17T22:32:49Z","title_canon_sha256":"5b9b31b0bbe237a076a24fb8396105d6e7e216939fb387c7c2604a746d1f9dd2"},"schema_version":"1.0","source":{"id":"1001.2932","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.2932","created_at":"2026-05-18T03:08:59Z"},{"alias_kind":"arxiv_version","alias_value":"1001.2932v2","created_at":"2026-05-18T03:08:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.2932","created_at":"2026-05-18T03:08:59Z"},{"alias_kind":"pith_short_12","alias_value":"G2WASIQXYLKU","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"G2WASIQXYLKUJB2S","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"G2WASIQX","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:83ee21613d23f2b8c309b205b4908bdfa5defaea957b6c86f616b593e2174e25","target":"graph","created_at":"2026-05-18T03:08:59Z","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":"Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition $S+T=\\makeset{m+n}{m \\in S, \\: n \\in T}$ and with ultimately periodic constants is exactly the class of hyper-arithmetical sets. Equations using addition only can represent every hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can also be represented by equations over sets of natural numbers equipped with union, addition and subtraction $S \\dotminus T=\\makeset{m-n}{m \\in S,","authors_text":"Alexander Okhotin, Artur Je\\.z","cross_cats":["cs.LO"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"cs.FL","submitted_at":"2010-01-17T22:32:49Z","title":"On equations over sets of integers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.2932","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":"sha256:8a48ffb6de047330b73725eccb4adaf84cb66c072e53f7faf3c065384c079c39","target":"record","created_at":"2026-05-18T03:08:59Z","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":"72a1dbb8087fee8a45387cc1bb496d3a89cee00094f51d1281e5de7e5c428c39","cross_cats_sorted":["cs.LO"],"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"cs.FL","submitted_at":"2010-01-17T22:32:49Z","title_canon_sha256":"5b9b31b0bbe237a076a24fb8396105d6e7e216939fb387c7c2604a746d1f9dd2"},"schema_version":"1.0","source":{"id":"1001.2932","kind":"arxiv","version":2}},"canonical_sha256":"36ac092217c2d54487523e2a57016d8c7f9f8604ead10c0763ecd7e37c3d8823","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"36ac092217c2d54487523e2a57016d8c7f9f8604ead10c0763ecd7e37c3d8823","first_computed_at":"2026-05-18T03:08:59.948242Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:08:59.948242Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tbFY5zDiqBQByptFoBRooXP87roNl9nVTAZoBXb5JHxoWHi+t6skyEPpVQueY1Q71V66bqqf3KLdVpkhYlCKBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:08:59.948911Z","signed_message":"canonical_sha256_bytes"},"source_id":"1001.2932","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8a48ffb6de047330b73725eccb4adaf84cb66c072e53f7faf3c065384c079c39","sha256:83ee21613d23f2b8c309b205b4908bdfa5defaea957b6c86f616b593e2174e25"],"state_sha256":"a6fa3c704d1daa7ef75fafc092e555f8b6435582b32e60438c211e84f54c1d09"}