{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-decision-problem",
  "semantic_family_id": "sf-decision-problem",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "決定問題"
  },
  "summary": {
    "zh-Hant": "決定問題由輸入集合 X 與 yes-instance 子集合 Y⊆X 指定；decider 必須對每個合法輸入終止並回傳正確布林答案。"
  },
  "statement": {
    "zh-Hant": "決定問題可表示為輸入集合 X 與 yes-instance 集合 Y⊆X，等價於命題規格 P:X→Prop。相對一個明示計算模型，decider D 必須對每個 x∈X 終止，且 D(x)=true 當且僅當 x∈Y。"
  },
  "formula": {
    "tex": "D:X\\to B",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>D</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>B</mi></mrow></math>",
    "semantic_ast": {
      "type": "mapping",
      "function": {
        "type": "symbol",
        "name": "D",
        "glyph": "D"
      },
      "domain": {
        "type": "symbol",
        "name": "X",
        "glyph": "X"
      },
      "codomain": {
        "type": "symbol",
        "name": "B",
        "glyph": "B"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "a20c25652ca31f59a78bb11aee7149532f0da0bb463fe32dec7818a95d20ece8"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "輸入集合 X 與 yes-instance 集合 Y 已固定，Y⊆X。"
    },
    {
      "id": "a2",
      "zh-Hant": "談可判定性時必須另指定有效輸入表示與計算模型；有限查表只證該有限 X。"
    }
  ],
  "symbols": [
    {
      "token": "D",
      "role": "candidate_decider",
      "role_zh": "候選決定程序",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "input_domain",
      "role_zh": "合法輸入集合",
      "scope": "representative_formula"
    },
    {
      "token": "B",
      "role": "boolean_codomain",
      "role_zh": "布林答案集合",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "輸入域與 yes-instance 以集合表示。"
    },
    {
      "id": "mko-proposition",
      "title_zh": "命題",
      "reason_zh": "規格 P(x) 是依輸入而定的命題。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "總 decider 對每個合法輸入產生唯一布林輸出。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "問題規格與實作分開：Y 或 P 描述正確答案，D 是候選程序。只在所有輸入上與規格一致且終止，才是 decider。",
      "對明示有限 X，完整表格可以窮舉驗證候選 D；抽樣一致不足以證明整個論域。域外輸入不參與目前問題。",
      "一般無限輸入域還需要有效編碼與計算模型。這一輪不由有限表格外推可計算性或不可判定性。"
    ],
    "common_misconception_zh": "把只接受 yes-instance 的半決定程序當成總 decider，或以有限測試通過宣稱無限問題已可判定。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限輸入表可完整核對規格與輸出；未指定計算模型時不宣稱一般可判定性。",
    "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": "建立 Prop 規格、Bool decider、有效表示與抽象計算模型之間的明示語義橋。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-decision-problem.md 整理；2026-10-13 脈衝 R06（使用者手動接續 2026-10-18 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}