{"paper":{"title":"Satisfiability in {\\L}ukasiewicz logic and its unbounded relative","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The existential theory of the additive ℓ-group on the reals with -1 is NP-complete.","cross_cats":["cs.LO"],"primary_cat":"math.LO","authors_text":"Filip Jankovec, Zuzana Hanikov\\'a","submitted_at":"2025-12-22T19:57:02Z","abstract_excerpt":"Unbounded {\\L}ukasiewicz logic is a substructural logic that combines features of infinite-valued {\\L}ukasiewicz logic with those of abelian logic. The logic is finitely strongly complete w.r.t.~the additive $\\ell$-group on the reals expanded with a distinguished element $-1$. We show that the existential theory of this structure is NP-complete. This provides a complexity upper bound for the set of theorems and the finite consequence relation of unbounded {\\L}ukasiewicz logic. The result is obtained by reducing the problem to the existential theory of the MV-algebra on the reals, the standard "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that the existential theory of this structure is NP-complete. This provides a complexity upper bound for the set of theorems and the finite consequence relation of unbounded Łukasiewicz logic.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The reduction from the existential theory of the unbounded Łukasiewicz structure (additive ℓ-group on reals with -1) to the existential theory of the standard MV-algebra on the reals is polynomial-time and preserves satisfiability.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The existential theory of the real additive ℓ-group with -1 is NP-complete, providing a complexity bound for satisfiability in unbounded Łukasiewicz logic via reduction to the MV-algebra.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The existential theory of the additive ℓ-group on the reals with -1 is NP-complete.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f2e7a4bcd592fa0a0388fe43d3179db89d2e26b31b0a45d3f8f0a908b4dd08e5"},"source":{"id":"2601.00817","kind":"arxiv","version":2},"verdict":{"id":"abb77d57-504f-4b53-b6fa-2169ce194a7e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T20:04:51.044479Z","strongest_claim":"We show that the existential theory of this structure is NP-complete. This provides a complexity upper bound for the set of theorems and the finite consequence relation of unbounded Łukasiewicz logic.","one_line_summary":"The existential theory of the real additive ℓ-group with -1 is NP-complete, providing a complexity bound for satisfiability in unbounded Łukasiewicz logic via reduction to the MV-algebra.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The reduction from the existential theory of the unbounded Łukasiewicz structure (additive ℓ-group on reals with -1) to the existential theory of the standard MV-algebra on the reals is polynomial-time and preserves satisfiability.","pith_extraction_headline":"The existential theory of the additive ℓ-group on the reals with -1 is NP-complete."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2601.00817/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}