{"paper":{"title":"Enhanced CAD-Based Quantifier Elimination With Multiple Equational Constraints","license":"http://creativecommons.org/licenses/by/4.0/","headline":"CAD-based quantifier elimination gains a detailed parameter partition and a stronger projection reduction when multiple equational constraints are present.","cross_cats":["math.AG"],"primary_cat":"cs.SC","authors_text":"James H. Davenport, Matthew England, Scott McCallum","submitted_at":"2026-04-26T20:32:37Z","abstract_excerpt":"This paper presents two enhancements to cylindrical algebraic decomposition (CAD) based quantifier elimination (QE) for cases in which multiple equational constraints are present in the given input formula $\\phi^*$. The first enhancement provides more detail in the output when there is a conceptual partition of the set of variables of $\\phi^*$ into parameters and unknowns. In such cases, we describe how to partition the parameter space so that: (1) in each open set of the partition the number $\\nu$ of associated unknowns is a finite constant or is infinite; and (2) for each such open set for w"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The paper presents two enhancements to CAD-based QE for cases in which multiple equational constraints are present: a detailed partition of parameter space indicating finite or infinite unknowns with expressions where finite, and an efficiency gain reducing the second CAD equational projection step more significantly than prior theory when conditions are met.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the input formula phi* has multiple equational constraints and that the specific conditions for the efficiency gain in the second projection step are satisfied in the given situations.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Two enhancements to CAD-based quantifier elimination enable detailed partitioning of parameter spaces into regions with finite or infinite unknowns and allow greater reduction in the second equational projection step.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"CAD-based quantifier elimination gains a detailed parameter partition and a stronger projection reduction when multiple equational constraints are present.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7ef597955d0dbaf6370ef1fac0874576cc4dd5337d3eac550aaeca974c49606e"},"source":{"id":"2604.23873","kind":"arxiv","version":2},"verdict":{"id":"a71e2338-d5e1-4d9d-af49-feab9d27ef93","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T04:39:23.360287Z","strongest_claim":"The paper presents two enhancements to CAD-based QE for cases in which multiple equational constraints are present: a detailed partition of parameter space indicating finite or infinite unknowns with expressions where finite, and an efficiency gain reducing the second CAD equational projection step more significantly than prior theory when conditions are met.","one_line_summary":"Two enhancements to CAD-based quantifier elimination enable detailed partitioning of parameter spaces into regions with finite or infinite unknowns and allow greater reduction in the second equational projection step.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the input formula phi* has multiple equational constraints and that the specific conditions for the efficiency gain in the second projection step are satisfied in the given situations.","pith_extraction_headline":"CAD-based quantifier elimination gains a detailed parameter partition and a stronger projection reduction when multiple equational constraints are present."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.23873/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T07:43:04.202181Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:42:33.225829Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"90dbdd5056a54ea0628c1b0a7b6cbed8dbf350d8132f96e17a02712cfd2fd04c"},"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"}