{
  "id": "1802.00974",
  "version": "v1",
  "published": "2018-02-03T13:28:13.000Z",
  "updated": "2018-02-03T13:28:13.000Z",
  "title": "Parametric Presburger Arithmetic: Complexity of Counting and Quantifier Elimination",
  "authors": [
    "Tristram Bogart",
    "John Goodrick",
    "Danny Nguyen",
    "Kevin Woods"
  ],
  "comment": "14 pages, 1 figure",
  "categories": [
    "math.LO",
    "cs.CC",
    "math.CO"
  ],
  "abstract": "We consider an expansion of Presburger arithmetic which allows multiplication by $k$ parameters $t_1,\\ldots,t_k$. A formula in this language defines a parametric set $S_\\mathbf{t} \\subseteq \\mathbb{Z}^{d}$ as $\\mathbf{t}$ varies in $\\mathbb{Z}^k$, and we examine the counting function $|S_\\mathbf{t}|$ as a function of $\\mathbf{t}$. For a single parameter, it is known that $|S_t|$ can be expressed as an eventual quasi-polynomial (there is a period $m$ such that, for sufficiently large $t$, the function is polynomial on each of the residue classes mod $m$). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming \\textbf{P} $\\neq$ \\textbf{NP}) we construct a parametric set $S_{t_1,t_2}$ such that $|S_{t_1, t_2}|$ is not even polynomial-time computable on input $(t_1,t_2)$. In contrast, for parametric sets $S_\\mathbf{t} \\subseteq \\mathbb{Z}^d$ with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that $|S_\\mathbf{t}|$ is always polynomial-time computable in the size of $\\mathbf{t}$, and in fact can be represented using the gcd and similar functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-03T13:28:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C10",
      "68Q17",
      "52B20"
    ],
    "keywords": [
      "parametric presburger arithmetic",
      "quantifier elimination",
      "parametric set",
      "complexity",
      "residue classes mod"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}