{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-regular-language",
  "semantic_family_id": "sf-regular-language",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "正規語言"
  },
  "summary": {
    "zh-Hant": "正規語言是存在一個完整DFA可辨識其全部字串的語言。"
  },
  "statement": {
    "zh-Hant": "DFA certificate須涵蓋宣告alphabet上的任意有限word。mod-count/parity語言以state invariant對任意字長證明；有限語言可用prefix-trie DFA構造。有限測試字不足以證regularity或language equivalence。"
  },
  "formula": {
    "tex": "R(L)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>R</mi><mo>(</mo><mi>L</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "R",
        "glyph": "R"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "L",
          "glyph": "L"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "de101212519023af5935fcf7a65618d910f55bf91f9958d482de38758e4ffa58"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "有限alphabet與完整DFA總轉移。"
    },
    {
      "id": "a2",
      "zh-Hant": "本輪不引入regex、NFA、epsilon transition、minimization或pumping lemma。"
    }
  ],
  "symbols": [
    {
      "token": "R",
      "role": "regular_language",
      "role_zh": "正規語言證書",
      "scope": "representative_formula"
    },
    {
      "token": "L",
      "role": "language",
      "role_zh": "字串集合",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-automaton",
      "title_zh": "確定有限自動機",
      "reason_zh": "DFA辨識證書。"
    },
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "language為word集合。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "word是有限符號序列。"
    },
    {
      "id": "mko-decision-problem",
      "title_zh": "決策問題",
      "reason_zh": "membership接受／拒絕。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "存在完整DFA是全語言證書。",
      "parity/mod-count用任意prefix invariant。",
      "有限語言可構造DFA；有限測試通過不足。"
    ],
    "common_misconception_zh": "用少數字串測試就宣稱regular或兩語言相等。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理DFA與明示invariant／有限語言構造，不外推其他模型。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入NFA、regular expression與closure construction。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-regular-language.md 整理；2026-10-14 脈衝 R07（使用者手動接續 2026-11-02 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}