{"paper":{"title":"Delta-Decidability over the Reals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.LO","authors_text":"Edmund Clarke, Jeremy Avigad, Sicun Gao","submitted_at":"2012-04-30T15:35:10Z","abstract_excerpt":"Given any collection F of computable functions over the reals, we show that there exists an algorithm that, given any L_F-sentence \\varphi containing only bounded quantifiers, and any positive rational number \\delta, decides either \"\\varphi is true\", or \"a \\delta-strengthening of \\varphi is false\". Under mild assumptions, for a C-computable signature F, the \\delta-decision problem for bounded \\Sigma_k-sentences in L_F resides in (\\Sigma_k^P)^C. The results stand in sharp contrast to the well-known undecidability results, and serve as a theoretical basis for the use of numerical methods in deci"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.6671","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}