{
  "abstract": "We investigate decidability of properties in the lattice $\\mathop{\\mathsf{NExt}} \\mathsf{Grz}_t$ of extensions of the Grzegorczyk tense logic $\\mathsf{Grz}_t$ and the lattice $\\mathop{\\mathsf{NExt}} \\mathsf{S4}_t$ of reflexive and transitive tense logics, with applications to the lattice $\\mathop{\\mathsf{Ext}} \\mathsf{biIPC}$ of bi-superintuitionistic logics. We prove that a broad class of properties is undecidable in $\\mathop{\\mathsf{NExt}} \\mathsf{Grz}_t$, including tabularity, Kripke completeness, the finite model property, and decidability, which also yields their undecidability in $\\mathop{\\mathsf{NExt}} \\mathsf{S4}_t$. We also construct infinitely many tabular extensions of $\\mathsf{Grz}_t$ (and thus of $\\mathsf{S4}_t$) whose coincidence problems are undecidable, while presenting one tabular extension of $\\mathsf{Grz}_t$ and infinitely many ones of $\\mathsf{S4}_t$ with a decidable coincidence problem. As a consequence, we obtain that the finite model property and tabularity are undecidable in $\\mathop{\\mathsf{Ext}} \\mathsf{biIPC}$, and that there are infinitely many tabular extensions of $\\mathsf{biIPC}$ whose coincidence problems are undecidable. These results clarify some similarities and differences between $\\mathop{\\mathsf{NExt}} \\mathsf{Grz}_t$ and $\\mathop{\\mathsf{NExt}} \\mathsf{Grz}$, $\\mathop{\\mathsf{NExt}} \\mathsf{S4}_t$ and $\\mathop{\\mathsf{NExt}} \\mathsf{S4}$, as well as $\\mathop{\\mathsf{Ext}} \\mathsf{biIPC}$ and $\\mathop{\\mathsf{Ext}} \\mathsf{IPC}$. The proofs adapt Chagrov's method of reducing from an undecidable problem for Minsky machines. We isolate and explicitly formulate the method of good valuations, a recurring technique underlying several proofs in the literature that use large frames, making it available for further applications.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30816",
  "author_total": 2,
  "authors": [
    {
      "name": "Qian Chen"
    },
    {
      "name": "Tenyo Takahashi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3jzrpaqvt4cat.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30816",
  "fulltext_url": null,
  "id": "3jzrpaqvt4cat",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30816",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7204956309",
  "pdf_url": "https://arxiv.org/pdf/2608.30816",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3jzrpaqvt4cat.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30816",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30816"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Most properties are undecidable even in $\\mathop{\\mathsf{NExt}} \\mathsf{Grz}_t$",
  "title_text": "Most properties are undecidable even in mathopNExt Grz_t",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/3jzrpaqvt4cat",
  "venue": "arXiv",
  "versions": []
}