{
  "abstract": "We study definability in fixed-point logic with counting (FPC) for a class of constraint satisfaction problems (CSPs) that simultaneously generalises finite-domain CSPs and temporal CSPs. The templates for the CSPs are first-order expansions of the structure $([k] \\times {\\mathbb Q};<_1,\\dots,<_k,\\sim)$ where $<_i \\, := \\{((i,x),(i,x')) \\mid x<x'\\}$ and $\\sim \\, := \\{((i,x),(i,x')) \\mid i \\in [k], x,x' \\in {\\mathbb Q}\\}$, i.e., $k$ copies of $({\\mathbb Q};<)$. Such a CSP is definable in FPC precisely when the template pp-constructs neither the ternary temporal relation $X$, which holds on three elements if the minimum is attained exactly twice, nor a nontrivial finite affine structure. Equivalently, the template has global polymorphisms that witness the tractability of the temporal CSPs on the copies, and that provide 3-4 weak near-unanimity operations on the set of copies of ${\\mathbb Q}$. In this case, we prove that the CSP is even expressible in the less expressive logic FP$^*$, which is the restriction of FPC without counting terms (i.e., the extension of first-order logic by a fixed point operator and an ordered auxiliary number sort). Our algorithm samples finitely many values in each copy and then performs nested constraint-singleton consistency tests (which is a uniform polynomial-time algorithm for finite-domain CSPs). The correctness proof uses finite-domain CSP theory to combine consistency certificates by a single polymorphism term, fixing each variable to one copy without destroying the certificates needed for the temporal algorithm. We then use a known normal form for algebraic products to control the remaining interactions between different copies.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18287",
  "author_total": 2,
  "authors": [
    {
      "name": "Manuel Bodirsky"
    },
    {
      "name": "Philipp A. Grzywaczyk"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/607cqkvgeyg9e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18287",
  "fulltext_url": null,
  "id": "607cqkvgeyg9e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18287",
  "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",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7221753611",
  "pdf_url": "https://arxiv.org/pdf/2610.18287",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/607cqkvgeyg9e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18287",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18287"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Finite-on-Top-of-Temporal CSPs in Fixed-Point Logic",
  "title_text": "Finite-on-Top-of-Temporal CSPs in Fixed-Point Logic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/607cqkvgeyg9e",
  "venue": "arXiv",
  "versions": []
}