{
  "id": "1210.4937",
  "version": "v2",
  "published": "2012-10-17T20:02:04.000Z",
  "updated": "2013-03-20T13:06:03.000Z",
  "title": "From Bi-immunity to Absolute Undecidability",
  "authors": [
    "Laurent Bienvenu",
    "Rupert Hölzl",
    "Adam R. Day"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "An infinite binary sequence A is absolutely undecidable if it is impossible to compute A on a set of positions of positive upper density. Absolute undecidability is a weakening of bi-immunity. Downey, Jockusch and Schupp asked whether, unlike the case for bi-immunity, there is an absolutely undecidable set in every non-zero Turing degree. We provide a positive answer to this question by applying techniques from coding theory. We show how to use Walsh-Hadamard codes to build a truth-table functional which maps any sequence A to a sequence B, such that given any restriction of B to a set of positive upper density, one can recover A. This implies that if A is non-computable, then B is absolutely undecidable. Using a forcing construction, we show that this result cannot be strengthened in any significant fashion.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-03-20T13:06:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03D30"
    ],
    "keywords": [
      "absolute undecidability",
      "bi-immunity",
      "positive upper density",
      "absolutely undecidable",
      "infinite binary sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.4937B"
    }
  }
}