{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-many-one-reduction",
  "semantic_family_id": "sf-many-one-reduction",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "多對一歸約"
  },
  "summary": {
    "zh-Hant": "A≤mB表示存在total computable f，使每個x皆滿足x∈A恰當f(x)∈B。"
  },
  "statement": {
    "zh-Hant": "有限題族明示source carrier、target carrier、decision predicates與完整total mapping table，逐x雙向驗x∈A iff f(x)∈B；只驗yes instances不足。composition須保留方向：A≤mB且B≤mC推出A≤mC。若A≤mB且B decidable，先算f再跑B decider可得A decider；反向transfer不成立。partial map、倒轉方向與Turing reduction混同均須拒絕。有限mapping table只證指定finite domain；無限domain仍需完整可計算算法及termination certificate。"
  },
  "formula": {
    "tex": "R(A,B,f)=1",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>R</mi><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>f</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "R",
          "glyph": "R"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "A",
            "glyph": "A"
          },
          {
            "type": "symbol",
            "name": "B",
            "glyph": "B"
          },
          {
            "type": "symbol",
            "name": "f",
            "glyph": "f"
          }
        ]
      },
      "rhs": {
        "type": "number",
        "value": 1
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "5c64c055c722e6b0689d2bbb44106533352fc8b4bf64cbdbdb71c6cf6954dc65"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "本輪exact families限明示bounded finite carriers與complete mapping arrays。"
    },
    {
      "id": "a2",
      "zh-Hant": "decider transfer使用完整finite truth tables或明示total termination。"
    }
  ],
  "symbols": [
    {
      "token": "R",
      "role": "many_one_reduction_predicate",
      "role_zh": "多對一歸約述詞",
      "scope": "representative_formula"
    },
    {
      "token": "A",
      "role": "source_decision_problem",
      "role_zh": "來源判定問題",
      "scope": "representative_formula"
    },
    {
      "token": "B",
      "role": "target_decision_problem",
      "role_zh": "目標判定問題",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "total_computable_map",
      "role_zh": "全可計算映射",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-decision-problem",
      "title_zh": "判定問題",
      "reason_zh": "yes/no predicates。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "complete total map。"
    },
    {
      "id": "mko-composition",
      "title_zh": "函數合成",
      "reason_zh": "reduction transitivity。"
    },
    {
      "id": "mko-turing-machine-decision-model",
      "title_zh": "Turing機判定模型",
      "reason_zh": "decider與termination界線。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "total map資格先於reduction。",
      "membership equivalence需正反雙向。",
      "composition與decider transfer都保持正確方向。"
    ],
    "common_misconception_zh": "只驗yes instances、使用partial map，或由A decidable倒推出B decidable。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "finite table不證無限domain computability；不混同Turing reduction。",
    "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undecidability reductions與recognizability boundaries。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-many-one-reduction.md 整理；2026-10-15 脈衝 R07（使用者手動接續 2026-11-15 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}