{
  "id": "2101.11849",
  "version": "v1",
  "published": "2021-01-28T07:39:59.000Z",
  "updated": "2021-01-28T07:39:59.000Z",
  "title": "On computable aspects of algebraic and definable closure",
  "authors": [
    "Nathanael Ackerman",
    "Cameron Freer",
    "Rehana Patel"
  ],
  "comment": "20 pages",
  "doi": "10.1093/logcom/exaa070",
  "categories": [
    "math.LO",
    "cs.LO"
  ],
  "abstract": "We investigate the computability of algebraic closure and definable closure with respect to a collection of formulas. We show that for a computable collection of formulas of quantifier rank at most $n$, in any given computable structure, both algebraic and definable closure with respect to that collection are $\\Sigma^0_{n+2}$ sets. We further show that these bounds are tight.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-28T07:39:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "definable closure",
      "computable aspects",
      "quantifier rank",
      "algebraic closure",
      "computable collection"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}