{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261013-advance-20261018-r06-decision-problem-0050",
  "pilot_number": "R0016-08725",
  "number": 8725,
  "family_id": "qf-ext18-decision-decider-agreement",
  "title": {
    "zh-Hant": "第 8725 題：有限決定問題",
    "en": "Question 8725: Whole Finite Decider Table against Its Specification"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261013-advance-20261018-r06-decision-problem-0050",
    "batch_id": "batch-20261013-advance-20261018-r06-decision-problem-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-decision-problem"
    ],
    "title_zh": "有限決定問題",
    "stem_zh": "明示有限輸入域。操作=decision-decider-agreement；資料={\"n\":6,\"yes\":[3,4],\"decider\":[false,false,false,true,true,false]}。請依 [n,yes_set,decider,mismatches,agrees] 的欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(6, [3, 4], [否, 否, 否, 是, 是, 否], [], 是)",
      "explanation_zh": "擴展基礎核對=(6, [3, 4], [否, 否, 否, 是, 是, 否], [], 是)；依完整明示資料逐欄精確重算；有限／合成資料核對不是一般形式證明；規格與decider分離；有限完整表不外推無限可判定性。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限決定問題"
      ]
    },
    "variant": {
      "family_id": "qf-ext18-decision-decider-agreement",
      "seed": 202610186050,
      "parameters": {
        "n": 6,
        "yes": [
          3,
          4
        ],
        "decider": [
          false,
          false,
          false,
          true,
          true,
          false
        ],
        "domain": "Fin_decision_problem",
        "operation": "decision-decider-agreement",
        "semantic_template": "qf-ext18-decision-decider-agreement/{\"decider\":[false,false,false,true,true,false],\"n\":6,\"yes\":[3,4]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_arithmetic",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做驗收，不構成人工審定、普遍證明、Lean執行或形式Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-13",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261013-advance-20261018-r06-decision-problem-0050",
    "batch_id": "batch-20261013-advance-20261018-r06-decision-problem-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "92e4355d6ed2bbd7ed0b2df14d9b0d454450091ad39b7a27e988ff1b8ce76ea3",
      "family_id": "qf-ext18-decision-decider-agreement",
      "seed": 202610186050,
      "semantic_template": "qf-ext18-decision-decider-agreement/{\"decider\":[false,false,false,true,true,false],\"n\":6,\"yes\":[3,4]}"
    },
    "fields": {
      "title": "Whole Finite Decider Table against Its Specification",
      "stem": "The explicit finite input set is Fin(6)={0,…,5} and its yes-instance set is [3, 4], normalized as a set. The specification is separate from the supplied Boolean decider table; finite table agreement implies no general infinite-domain decidability result. Complete decider=[false, false, false, true, true, false]. Give (n, yes-instance set, decider values, all mismatch indices, complete agreement?).",
      "answer": {
        "kind": "expression",
        "value": "(6, [3, 4], [No, No, No, Yes, Yes, No], [], Yes)",
        "explanation": "Check every table cell, not a sample; the explicit finite table does not establish termination of a general program. Exact finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (6, [3, 4], [No, No, No, Yes, Yes, No], [], Yes)."
      },
      "skills": [
        "Whole Finite Decider Table against Its Specification"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "明示有限輸入域。操作=decision-decider-agreement；資料={\"n\":6,\"yes\":[3,4],\"decider\":[false,false,false,true,true,false]}。請依 [n,yes_set,decider,mismatches,agrees] 的欄序回答。",
        "en": "The explicit finite input set is Fin(6)={0,…,5} and its yes-instance set is [3, 4], normalized as a set. The specification is separate from the supplied Boolean decider table; finite table agreement implies no general infinite-domain decidability result. Complete decider=[false, false, false, true, true, false]. Give (n, yes-instance set, decider values, all mismatch indices, complete agreement?)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "domain Fin(6);raw yes instances=[3, 4]",
      "after": "normalized yes set=[3, 4]",
      "relation": "implies",
      "rule_id": "finite_decision_spec_not_program_execution",
      "conditions": [],
      "explanation": {
        "zh-Hant": "規格的yes集合與候選decider分開；重複標籤按集合去重，域外標籤不允許。完整有限表不代表任意程式在無限輸入上可判定或會終止。",
        "en": "Keep the yes-instance specification separate from the candidate decider.Deduplicate labels as a set and reject outside-domain labels.A complete finite table does not prove an arbitrary program decidable or terminating on infinite inputs."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "input=0;spec=否;declared decider=否",
      "after": "mismatch=否",
      "relation": "implies",
      "rule_id": "finite_decision_each_decider_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "規格與decider在同一原輸入比對，所有格都核，不用decider自己定義正確答案。",
        "en": "Compare specification and decider at the same original input,checking every cell.The decider does not define its own correctness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "input=1;spec=否;declared decider=否",
      "after": "mismatch=否",
      "relation": "implies",
      "rule_id": "finite_decision_each_decider_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "規格與decider在同一原輸入比對，所有格都核，不用decider自己定義正確答案。",
        "en": "Compare specification and decider at the same original input,checking every cell.The decider does not define its own correctness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "input=2;spec=否;declared decider=否",
      "after": "mismatch=否",
      "relation": "implies",
      "rule_id": "finite_decision_each_decider_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "規格與decider在同一原輸入比對，所有格都核，不用decider自己定義正確答案。",
        "en": "Compare specification and decider at the same original input,checking every cell.The decider does not define its own correctness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "input=3;spec=是;declared decider=是",
      "after": "mismatch=否",
      "relation": "implies",
      "rule_id": "finite_decision_each_decider_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "規格與decider在同一原輸入比對，所有格都核，不用decider自己定義正確答案。",
        "en": "Compare specification and decider at the same original input,checking every cell.The decider does not define its own correctness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "input=4;spec=是;declared decider=是",
      "after": "mismatch=否",
      "relation": "implies",
      "rule_id": "finite_decision_each_decider_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "規格與decider在同一原輸入比對，所有格都核，不用decider自己定義正確答案。",
        "en": "Compare specification and decider at the same original input,checking every cell.The decider does not define its own correctness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "input=5;spec=否;declared decider=否",
      "after": "mismatch=否",
      "relation": "implies",
      "rule_id": "finite_decision_each_decider_cell",
      "conditions": [],
      "explanation": {
        "zh-Hant": "規格與decider在同一原輸入比對，所有格都核，不用decider自己定義正確答案。",
        "en": "Compare specification and decider at the same original input,checking every cell.The decider does not define its own correctness."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "complete mismatch labels=[]",
      "after": "complete agreement=是",
      "relation": "implies",
      "rule_id": "complete_decision_agreement_gate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整一致當且僅當整個有限域沒有不符格，空域條件空泛成立。",
        "en": "Complete agreement means no mismatch anywhere in the finite domain;the condition holds vacuously on an empty domain."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      6,
      [
        3,
        4
      ],
      [
        false,
        false,
        false,
        true,
        true,
        false
      ],
      [],
      true
    ]
  },
  "conclusion": {
    "canonical_value": "(6, [3, 4], [否, 否, 否, 是, 是, 否], [], 是)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
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      "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261018-r06-decision-problem-foundations.en.jsonl"
    },
    "mko_bindings": {
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    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
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    "original_candidate_sha256": "d426f9fca23c09aa81b3a9b9f86dea9292b21ce3d0a68467cb763546054fa125",
    "original_candidate_line_sha256": "d426f9fca23c09aa81b3a9b9f86dea9292b21ce3d0a68467cb763546054fa125",
    "mko_binding_revision": {
      "id": "edition-symbol-bindings-20261101",
      "path": "content/edition-symbol-bindings-20261101/revision.json",
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  "historical_source_state": {
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      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-005-p17",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
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      "question_first_witness_sha256": "198e04380b2d32aef2375b07ca0df1f8afa454d1184993173fad8a3c9187a3d1"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 8,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "d5bbad60984489865dc3cf2a8f4bdba6985044134644c7badfb3c63221ffb200",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}