{
  "abstract": "Tree decompositions describe large graphs as gluings of smaller component parts. Here we study their categorical duals, tree co-decompositions, which construct large graphs via fibered products of small constituent parts. Co-decompositions are compressed representations of graphs or relational structures: the decomposed object can be exponentially larger than the space required to specify the co-decomposition. Our main contribution is an algorithmic meta-theorem for first order logic: given only a co-decomposition of a $σ$-structure $G$ with $b$ bags, each of size at most $w$ and an FO sentence $ϕ$, there is an algorithm which, without ever constructing or inspecting $G$, operates on the co-decomposition and determines whether $G \\models ϕ$ in time $f_σ(w,|φ|)\\cdot b$. We pair this algorithm with structural results on tree co-width, including its relationship to tree-width and a separation from twin-width. Furthermore, we characterize bounded path co-width using a graph grammar. In contrast to our logical meta-theorem, we prove that $(s,t)$-Path becomes NP-hard on co-decomposition inputs. These results demonstrate how categorical constructions can enable efficient computation on exponentially compressed structures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04658",
  "author_total": 4,
  "authors": [
    {
      "name": "Benjamin Merlin Bumpus"
    },
    {
      "name": "Guilherme C. M. Gomes"
    },
    {
      "name": "Laura Larios-Jones"
    },
    {
      "name": "Will J. Turner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/73j68rnz732gt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04658",
  "fulltext_url": null,
  "id": "73j68rnz732gt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04658",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.CT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04658",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/73j68rnz732gt.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04658",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04658"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Co-Decompositions and FO Model Checking",
  "title_text": "Co-Decompositions and FO Model Checking",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/73j68rnz732gt",
  "venue": "arXiv",
  "versions": []
}