{
  "abstract": "We study the amalgamation decision problem (ADP): given a universal first-order sentence $Φ$, decide whether the class $\\mathrm{fm}(Φ)$ of its finite models has the amalgamation property. We call $Φ$ semantic Horn if $\\mathrm{fm}(Φ)$ is closed under binary direct products. By McKinsey's theorem, this is equivalent to $Φ$ being logically equivalent to a universal Horn sentence. The distinction concerns the input representation: $Φ$ itself need not be given in Horn form, and converting it into an explicit Horn normal form can incur an exponential blow-up. We prove that the ADP is decidable under this semantic promise. To this end, we associate with $Φ$ a finite-domain CSP template, called its completion template, and a distinguished infinite family of CSP instances, called its atlas instances. The class $\\mathrm{fm}(Φ)$ has the amalgamation property precisely when all atlas instances have a solution. For semantic Horn inputs, we prove that the completion template admits a semilattice polymorphism. Consequently, its CSP has bounded width and is decided by a fixed level of local consistency. We introduce uniform contextual strategies, finite certificates expressing this local-consistency condition simultaneously for all atlas instances, and give an effective fixed-point procedure for deciding whether such a certificate exists. The resulting algorithm runs in 2ExpTime in general and in ExpTime under any fixed bound on the arities of the input relations. More generally, the construction gives a decision procedure whenever the associated completion template has bounded width. These bounds are optimal: the semantic Horn ADP is 2ExpTime-complete in general and ExpTime-complete for every fixed arity bound of at least three. Both hardness results already hold for syntactic universal Horn sentences.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.12206",
  "author_total": 1,
  "authors": [
    {
      "name": "Jakub Rydval"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/81g2x9gzexatv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.12206",
  "fulltext_url": null,
  "id": "81g2x9gzexatv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.12206",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7202376834",
  "pdf_url": "https://arxiv.org/pdf/2608.12206",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/81g2x9gzexatv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2608.12206",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.12206"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Deciding Amalgamation Beyond Arity Two: The Semantic Horn Case",
  "title_text": "Deciding Amalgamation Beyond Arity Two: The Semantic Horn Case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/81g2x9gzexatv",
  "venue": "arXiv",
  "versions": []
}