{"paper":{"title":"Order-invariant cluster first-order logic on graph classes of bounded degree","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Order-invariant cluster first-order logic has the same expressive power as plain first-order logic on bounded-degree graph classes.","cross_cats":[],"primary_cat":"cs.LO","authors_text":"Fatemeh Ghasemi, Julien Grange","submitted_at":"2026-04-30T10:34:30Z","abstract_excerpt":"We introduce a new logic, called \\emph{cluster first-order logic}, a restricted fragment of first-order logic specifically designed to study order invariance. An order-invariant formula is one on a vocabulary that contains an order; however, whether a structure satisfies it or not is independent of the interpretation of the order.\n  We show that while order-invariant cluster first-order logic can define properties outside the scope of plain first-order logic in general, its expressive power is included in that of first-order logic when it comes to classes of bounded degree.\n  We establish this"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that while order-invariant cluster first-order logic can define properties outside the scope of plain first-order logic in general, its expressive power is included in that of first-order logic when it comes to classes of bounded degree.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That there exist linear orders on the universes of any two sufficiently similar bounded-degree structures such that the expanded structures remain similar (i.e., the local-to-global similarity-preserving construction succeeds).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Order-invariant cluster first-order logic has the same expressive power as first-order logic on every class of graphs with bounded degree.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Order-invariant cluster first-order logic has the same expressive power as plain first-order logic on bounded-degree graph classes.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"2c64ecc18e206e6adde31802461691185bc7d8b6f5f0ea51c362f982b08837d8"},"source":{"id":"2604.27693","kind":"arxiv","version":2},"verdict":{"id":"ade94221-8a4b-4afb-b8c6-0d88521815d1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T05:31:33.081076Z","strongest_claim":"We show that while order-invariant cluster first-order logic can define properties outside the scope of plain first-order logic in general, its expressive power is included in that of first-order logic when it comes to classes of bounded degree.","one_line_summary":"Order-invariant cluster first-order logic has the same expressive power as first-order logic on every class of graphs with bounded degree.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That there exist linear orders on the universes of any two sufficiently similar bounded-degree structures such that the expanded structures remain similar (i.e., the local-to-global similarity-preserving construction succeeds).","pith_extraction_headline":"Order-invariant cluster first-order logic has the same expressive power as plain first-order logic on bounded-degree graph classes."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.27693/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T21:41:08.088212Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T19:01:48.143778Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"99eb766c8c6314ae8ebf72b6b5209d59f62b9758fd6e29b8b8540f061fd25f0d"},"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"}