{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R818",
  "content_hash": null,
  "slug": "twnfa-claim-one-way-liveness-quadratic-bound",
  "type": "claim",
  "title": "One-way liveness forces at least h(h+1)/4 deterministic states",
  "summary": "For the alphabet of all \\(h\\) by \\(h\\) Boolean matrices, an \\(h\\)-state 1NFA recognizes strings with nonzero Boolean product, and every equivalent 2DFA has at least \\(h(h+1)/4\\) states.",
  "relevance": "For Polynomial determinization of two-way finite automata, record twnfa-claim-one-way-liveness-quadratic-bound (“One-way liveness forces at least h(h+1)/4 deterministic states”) records a bound, answer, status fact, or structural consequence. The record states: For the alphabet of all \\(h\\) by \\(h\\) Boolean matrices, an \\(h\\)-state 1NFA recognizes strings with nonzero Boolean product, and every equivalent 2DFA has at least \\(h(h+1)/4\\) states.",
  "relevance_source": "recorded",
  "body": "Let \\(\\Sigma_h=\\{0,1\\}^{h\\times h}\\). A word \\(A_1\\cdots A_m\\) is live when the Boolean product \\(A_1\\cdots A_m\\) has a nonzero entry. Equivalently, the layered graph described by the matrices has a path joining its first and last columns. An \\(h\\)-state 1NFA follows a path by storing its current row index.\n\nAdeogun and Kapoutsis construct \\(1+\\binom{h+1}{2}\\) smooth connectivity properties. Each successive pair is separated by liveness and satisfies the suffix-of-choice condition in their main lemma. The two exit-size sequences of any solving 2DFA must decrease strictly a total of at least \\(\\binom{h+1}{2}\\) times. One sequence therefore has at least half that many strict decreases, which yields\n\\[\n|Q|\\ge \\frac12\\binom{h+1}{2}=\\frac{h(h+1)}4.\n\\]\nThe alphabet size is \\(|\\Sigma_h|=2^{h^2}\\), so this statement by itself uses an alphabet that grows with \\(h\\).",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "one-way liveness over the full alphabet of h by h Boolean matrices for every integer h at least 1",
    "family": "the one-way-liveness languages owl_h"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2602.24279",
      "locator": "Adeogun and Kapoutsis, arXiv:2602.24279v2, Sections 2.2 and 4.1-4.4, especially Theorem 1 on p. 15"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2602.24279",
    "locator": "Adeogun and Kapoutsis, arXiv:2602.24279v2, Sections 2.2 and 4.1-4.4, especially Theorem 1 on p. 15"
  },
  "relations": [
    {
      "slug": "R815",
      "title": "The fixed-alphabet determinization question remains open",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R817",
      "title": "Two singleton contexts isolate any Boolean-matrix entry",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R816",
      "title": "The one-way-liveness bound transfers to a fixed binary alphabet",
      "object_type": "claim",
      "relation": "depends_on",
      "direction": "incoming"
    },
    {
      "slug": "R813",
      "title": "Test the proposed maximum length of the smooth-property chain",
      "object_type": "attempt",
      "relation": "uses",
      "direction": "incoming"
    },
    {
      "slug": "two-way-nfa-polynomial-determinization",
      "title": "two way nfa polynomial determinization",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}