{"paper":{"title":"Deep $\\Pi^0_1$ Classes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Christopher P. Porter, Laurent Bienvenu","submitted_at":"2014-03-03T14:47:15Z","abstract_excerpt":"A set of infinite binary sequences $\\mathcal{C}\\subseteq2^\\omega$ is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of $\\Pi^0_1$ classes. In this paper, we introduce the notion of depth for $\\Pi^0_1$ classes, which is a stronger form of negligibility. Whereas a negligible $\\Pi^0_1$ class $\\mathcal{C}$ has the property that one cannot probabilistically compute a member of $\\mathcal{C}$ with positive probability, a deep $\\Pi^0_1$ class $\\mathcal{C}$ has t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.0450","kind":"arxiv","version":3},"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"}