{
  "abstract": "Probabilistic models are often constructed separately for different test suites and then expected to share a single latent dynamical system. We study bounded-state shared realization for probabilistic automata from finite partial trace specifications. A shared realization uses one normalized automaton with named initial distributions, a common family of row-stochastic letter-transition matrices, and one final output vector. We compare it with separate local realizations and a labelled query-specific static relaxation. Local and labelled query-specific static optima need not compose. For every $k,L\\ge1$, we construct finite partial specifications for which each individual specification and the labelled query-specific static relaxation have minimum state count $k$, whereas every exact shared realization requires exactly $k(L+1)$ states. The factor $L+1$ reflects the prefix progress that must be tracked by a single shared transition family. In the linear-rank-tight calibrated regime, the existence of an exact $k$-state shared realization is equivalent to finding a $(k-1)$-simplex containing the root responses, contained in the response polytope, and forward-invariant under every single-letter affine update. For finite rational specifications, exact shared realization with at most $K$ states is $\\exists\\mathbb{R}$-complete, while distinguishing zero minimax defect from inverse-polynomial defect is $\\mathsf{PromiseNP}$-complete. Both classifications hold over a fixed five-letter alphabet when the local and labelled query-specific static optima already equal $K$. They remain valid with Boolean final output vectors, and the robust gap persists over the full cube $[0,1]^d$. Thus the hardness arises from shared word composition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19707",
  "author_total": 1,
  "authors": [
    {
      "name": "Yixin Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/84x95tdg8xs8x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19707",
  "fulltext_url": null,
  "id": "84x95tdg8xs8x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19707",
  "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",
      "quant-ph"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7213593707",
  "pdf_url": "https://arxiv.org/pdf/2609.19707",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/84x95tdg8xs8x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:04Z",
      "native_id": "2609.19707",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19707"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Local Optimality Does Not Compose: Invariant Simplices and the Complexity of Shared Probabilistic Automata",
  "title_text": "Local Optimality Does Not Compose: Invariant Simplices and the Complexity of Shared Probabilistic Automata",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:04Z",
  "url": "https://research.draconitic.com/paper/84x95tdg8xs8x",
  "venue": "arXiv",
  "versions": []
}