{"paper":{"title":"Solovay functions and their applications in algorithmic randomness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Andr\\'e Nies, Laurent Bienvenu, Rod Downey, Wolfgang Merkle","submitted_at":"2016-03-28T09:48:26Z","abstract_excerpt":"Classical versions of Kolmogorov complexity are incomputable. Nevertheless, in 1975 Solovay showed that there are computable functions $f > K+O(1)$ such that for infinitely many strings $\\sigma$, $f(\\sigma)=K(\\sigma)+O(1)$, where $K$ denotes prefix-free Kolmogorov complexity (while $C$ denotes plain Kolmogorov complexity). Such an $f$ is now called a Solovay function. We prove that many classical results about $K$ can be obtained by replacing $K$ by a Solovay function. For example, the three following properties of a function $g$ all hold for the function $K$.\n  (i) The sum of the terms $\\sum_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08351","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"}