{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-automaton",
  "semantic_family_id": "sf-finite-automaton",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "確定有限自動機"
  },
  "summary": {
    "zh-Hant": "DFA由有限狀態集、有限字母表、總轉移函數、唯一初始狀態與接受狀態子集組成。"
  },
  "statement": {
    "zh-Hant": "完整核Q×Σ每一輸入恰有一個合法輸出；依序追蹤完整有限word。空字接受恰當初始狀態為接受狀態。一組有限trace不證語言等價或regularity。"
  },
  "formula": {
    "tex": "D(Q)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>D</mi><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "D",
        "glyph": "D"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "Q",
          "glyph": "Q"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "75d4cf9f7b6bbfb14b5f0a9c715ee0398f908dcb921221bb12bc1f617c57f755"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "有限非空state set與有限非空alphabet，轉移表完整且決定。"
    },
    {
      "id": "a2",
      "zh-Hant": "無epsilon transition，不引入NFA語義。"
    }
  ],
  "symbols": [
    {
      "token": "D",
      "role": "deterministic_finite_automaton",
      "role_zh": "DFA紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "Q",
      "role": "finite_state_set",
      "role_zh": "有限狀態集",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-computational-model",
      "title_zh": "計算模型",
      "reason_zh": "DFA為有限計算模型。"
    },
    {
      "id": "mko-discrete-dynamical-system",
      "title_zh": "離散動力系統",
      "reason_zh": "轉移迭代。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "轉移須為總函數。"
    },
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "狀態、字母與接受集。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "word為有限有序符號列。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "完整轉移表保證deterministic與total。",
      "從唯一初始狀態逐符號更新。",
      "空字不走轉移；有限trace不證兩DFA語言等價。"
    ],
    "common_misconception_zh": "漏轉移、把NFA多分支或epsilon transition混入DFA，或由少數測試字宣稱語言等價。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理完整DFA與有限word trace；不宣稱一般regular language結果。",
    "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": "加入正規語言、NFA與等價構造。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-automaton.md 整理；2026-10-14 脈衝 R07（使用者手動接續 2026-11-01 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"
  }
}