{"paper":{"title":"Termination of Innermost-Terminating Right-Linear Overlay Term Rewrite Systems","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"Right-linear overlay term rewrite systems terminate if and only if they innermost terminate.","cross_cats":[],"primary_cat":"cs.LO","authors_text":"Naoki Nishida","submitted_at":"2026-04-22T16:44:24Z","abstract_excerpt":"It has been shown that, regarding a terminating right-linear overlay term rewrite system (TRS), any rewrite sequence terminating in a normal form can be simulated by an innermost reduction. In this paper, using this simulation property, we show that for a right-linear overlay TRS, there is no infinite minimal dependency-pair chain if and only if there is no infinite innermost minimal dependency-pair chain. As a consequence, termination and innermost termination coincide for the class of right-linear overlay TRSs."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"for a right-linear overlay TRS, there is no infinite minimal dependency-pair chain if and only if there is no infinite innermost minimal dependency-pair chain. This implies that a right-linear overlay TRS is terminating if and only if it is innermost terminating.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The previously established simulation property that, for a terminating right-linear overlay TRS, any rewrite sequence ending in a normal form can be simulated by innermost reduction.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Right-linear overlay TRSs terminate if and only if they are innermost terminating, because absence of infinite minimal dependency-pair chains is equivalent to absence of infinite innermost minimal dependency-pair chains.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Right-linear overlay term rewrite systems terminate if and only if they innermost terminate.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"07fc7c7bed0517ad2806e8c28e1ad2ba7fe070a172333ed1175ebd07874cea03"},"source":{"id":"2604.20754","kind":"arxiv","version":2},"verdict":{"id":"40f0d808-82c1-4880-91c5-af1d2a1f6d8d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-09T23:00:10.560713Z","strongest_claim":"for a right-linear overlay TRS, there is no infinite minimal dependency-pair chain if and only if there is no infinite innermost minimal dependency-pair chain. This implies that a right-linear overlay TRS is terminating if and only if it is innermost terminating.","one_line_summary":"Right-linear overlay TRSs terminate if and only if they are innermost terminating, because absence of infinite minimal dependency-pair chains is equivalent to absence of infinite innermost minimal dependency-pair chains.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The previously established simulation property that, for a terminating right-linear overlay TRS, any rewrite sequence ending in a normal form can be simulated by innermost reduction.","pith_extraction_headline":"Right-linear overlay term rewrite systems terminate if and only if they innermost terminate."},"integrity":{"clean":false,"summary":{"advisory":0,"critical":1,"by_detector":{"doi_compliance":{"total":1,"advisory":0,"critical":1,"informational":0}},"informational":0},"endpoint":"/pith/2604.20754/integrity.json","findings":[{"note":"DOI '10.48550/arxiv.xxxx.xxxxx' as printed in the bibliography is syntactically invalid and cannot resolve.","detector":"doi_compliance","severity":"critical","ref_index":159,"audited_at":"2026-05-20T01:37:40.026367Z","detected_doi":"10.48550/arxiv.xxxx.xxxxx","finding_type":"broken_identifier","verdict_class":"incontrovertible","detected_arxiv_id":null}],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T13:41:25.741703Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-20T01:37:40.026367Z","status":"completed","version":"1.0.0","findings_count":1}],"snapshot_sha256":"06abd1f9e3edec8cb9745cdcea95b661bb3256e0f1ef25a73bcc7208c548be42"},"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"}