{
  "id": "1305.0651",
  "version": "v1",
  "published": "2013-05-03T09:34:12.000Z",
  "updated": "2013-05-03T09:34:12.000Z",
  "title": "Associative and commutative tree representations for Boolean functions",
  "authors": [
    "Antoine Genitrini",
    "Bernhard Gittenberger",
    "Veronika Kraus",
    "Cécile Mailler"
  ],
  "comment": "36 pages, 9 figures",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "Since the 90's, several authors have studied a probability distribution on the set of Boolean functions on $n$ variables induced by some probability distributions on formulas built upon the connectors $And$ and $Or$ and the literals $\\{x_{1}, \\bar{x}_{1}, \\dots, x_{n}, \\bar{x}_{n}\\}$. These formulas rely on plane binary labelled trees, known as Catalan trees. We extend all the results, in particular the relation between the probability and the complexity of a Boolean function, to other models of formulas: non-binary or non-plane labelled trees (i.e. Polya trees). This includes the natural tree class where associativity and commutativity of the connectors $And$ and $Or$ are realised.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-03T09:34:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A16",
      "05C05",
      "06E30",
      "60C05"
    ],
    "keywords": [
      "boolean function",
      "commutative tree representations",
      "probability distribution",
      "plane binary labelled trees",
      "natural tree class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.0651G"
    }
  }
}