{"paper":{"title":"Loop Termination and Generalized Collatz Sequences","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Termination of one-variable linear-constraint loops is decidable in polynomial time if a conjecture on generalized Collatz sequences holds.","cross_cats":[],"primary_cat":"cs.LO","authors_text":"Mishel Carelli","submitted_at":"2026-05-14T17:13:03Z","abstract_excerpt":"Linear-constraint loops are programs whose transition relation is specified by a system of linear inequalities. The termination problem asks, given a loop, whether it admits an infinite computation. Decidability of termination remains open for linear-constraint loops over integers, rationals, and reals. We focus on loops over integers and show that they are tightly connected to generalized Collatz sequences - integer sequences generated by maps that are linear on each residue class modulo a fixed natural number. We prove that termination of one-variable linear-constraint loops is decidable in "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that termination of one-variable linear-constraint loops is decidable in polynomial time, provided a long-standing conjecture about generalized Collatz sequences holds.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The long-standing conjecture about generalized Collatz sequences holds.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Termination of one-variable linear-constraint loops over integers is decidable in polynomial time if the generalized Collatz conjecture holds, with any such procedure also settling specific instances of the conjecture.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Termination of one-variable linear-constraint loops is decidable in polynomial time if a conjecture on generalized Collatz sequences holds.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"124e0d6ba7610f66155055f5986636a3b7d91ae2933746b03e3731edb717649c"},"source":{"id":"2605.15094","kind":"arxiv","version":1},"verdict":{"id":"b8172661-6b63-4160-9ffa-43552cb7406c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T03:09:47.102227Z","strongest_claim":"We prove that termination of one-variable linear-constraint loops is decidable in polynomial time, provided a long-standing conjecture about generalized Collatz sequences holds.","one_line_summary":"Termination of one-variable linear-constraint loops over integers is decidable in polynomial time if the generalized Collatz conjecture holds, with any such procedure also settling specific instances of the conjecture.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The long-standing conjecture about generalized Collatz sequences holds.","pith_extraction_headline":"Termination of one-variable linear-constraint loops is decidable in polynomial time if a conjecture on generalized Collatz sequences holds."},"references":{"count":12,"sample":[{"doi":"","year":2013,"title":"1 Amir M. Ben-Amram. Mortality of Iterated Piecewise Affine Functions over the Integers: Decid- ability and Complexity. In Natacha Portier and Thomas Wilke, editors,30th International Sym- posium on T","work_id":"e7202e57-6ac0-478a-8a52-81e9870cab56","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.4230/lipics.stacs.2013.514","year":2013,"title":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik. URL:https://drops.dagstuhl.de/entities/ document/10.4230/LIPIcs.STACS.2013.514,doi:10.4230/LIPIcs.STACS.2013.514. 2 Amir M. Ben-Amram, Samir Genaim, ","work_id":"96a5edcc-4487-4bf5-ae00-97d8a84888e7","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1145/2400676","year":null,"title":"doi:10.1145/2400676. 2400679. 3 Amir M. Ben-Amram, Samir Genaim, Joël Ouaknine, and James Worrell. Termination analysis of linear-constraint programs,","work_id":"d9dde632-b7f9-4546-9847-3fc8192c3e5f","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"4 Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars.Computational Geometry: Algorithms and Applications","work_id":"2bdb1388-6c89-494d-8f95-a66d1c2aa46d","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/s0304-3975(00)00399-6","year":2001,"title":"Springer Berlin Heidelberg. 6 Vincent D. Blondel, Olivier Bournez, Pascal Koiran, Christos H. Papadimitriou, and John N. Tsitsiklis. Deciding stability and mortality of piecewise affine dynamical syst","work_id":"a5ab2758-ef09-4898-824f-b63f51a059c8","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":12,"snapshot_sha256":"e32dae3c9916109469a9d69624fc464fff7c8c4e6fa088826d5d4214e4530bc8","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"324fd1351e4ad91c8c6cf1c778cf313efcefbaf485ddfdae474871bef7bcef21"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}