{"paper":{"title":"Optimal asymptotic bounds on the oracle use in computations from Chaitin's Omega","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.LO"],"primary_cat":"math.LO","authors_text":"Andrew Lewis-Pye, George Barmpalias, Nan Fang","submitted_at":"2016-02-05T08:59:56Z","abstract_excerpt":"Chaitin's number Omega is the halting probability of a universal prefix-free machine, and although it depends on the underlying enumeration of prefix-free machines, it is always Turing-complete. It can be observed, in fact, that for every computably enumerable (c.e.) real, there exists a Turing functional via which Omega computes it, and such that the number of bits of omega that are needed for the computation of the first n bits of the given number (i.e. the use on argument n) is bounded above by a computable function h(n) = n+o(n). We characterise the asymptotic upper bounds on the use of Ch"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.03208","kind":"arxiv","version":2},"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"}