{"paper":{"title":"The sum $2^{\\mathit{KA}(x)-\\mathit{KP}(x)}$ over all prefixes $x$ of some binary sequence can be infinite","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Akim Kumok, Mikhail Andreev","submitted_at":"2014-01-07T18:30:52Z","abstract_excerpt":"We consider two quantities that measure complexity of binary strings: $\\mathit{KA}(x)$ is defined as the minus logarithm of continuous a priori probability on the binary tree, and $\\mathit{KP}(x)$ denotes prefix complexity of a binary string $x$. In this paper we answer a question posed by Joseph Miller and prove that there exists an infinite binary sequence $\\omega$ such that the sum of $2^{\\mathit{KA}(x)-\\mathit{KP}(x)}$ over all prefixes $x$ of $\\omega$ is infinite. Such a sequence can be chosen among characteristic sequences of computably enumerable sets."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.1467","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"}