{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261013-advance-20261019-r05-algorithm-correctness-0038", "pilot_number": "R0018-09538", "number": 9538, "family_id": "qf-ext19-correctness-partial-audit", "title": { "zh-Hant": "第 9538 題：算法正確性責任", "en": "Question 9538: Partial Correctness and Its Vacuous Case" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261013-advance-20261019-r05-algorithm-correctness-0038", "batch_id": "batch-20261013-advance-20261019-r05-algorithm-correctness-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-algorithm-correctness" ], "title_zh": "算法正確性責任", "stem_zh": "完整有限正確性模型。操作=correctness-partial-audit；資料={\"n\":6,\"init\":[0],\"halt\":[5],\"steps\":[[0,1],[1,2],[2,5],[4,5],[2,1]],\"post\":[]}。請依 [reachable_halts,post,bad_halts,partial_correct,vacuous] 欄序回答。", "answer": { "kind": "expression", "value": "([5], [], [5], 否, 否)", "explanation_zh": "第二批擴展核對=([5], [], [5], 否, 否)；依完整明示資料重算；有限／合成資料核對不是一般形式證明；部分正確性與終止性分離；一條成功路徑不證總正確。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "算法正確性責任" ] }, "variant": { "family_id": "qf-ext19-correctness-partial-audit", "seed": 202610195038, "parameters": { "n": 6, "init": [ 0 ], "halt": [ 5 ], "steps": [ [ 0, 1 ], [ 1, 2 ], [ 2, 5 ], [ 4, 5 ], [ 2, 1 ] ], "post": [], "domain": "Fin_correctness_model", "operation": "correctness-partial-audit", "semantic_template": "qf-ext19-correctness-partial-audit/{\"halt\":[5],\"init\":[0],\"n\":6,\"post\":[],\"steps\":[[0,1],[1,2],[2,1],[2,5],[4,5]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "僅為產生端候選及自檢；主端須另驗，不構成人工審定、一般程式證明、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-20261019-r05-algorithm-correctness-0038", "batch_id": "batch-20261013-advance-20261019-r05-algorithm-correctness-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "25c99fb5eb17fb413754c2ac9233cdf566ce68b79c374bb726f92b8ea9c6a60f", "family_id": "qf-ext19-correctness-partial-audit", "seed": 202610195038, "semantic_template": "qf-ext19-correctness-partial-audit/{\"halt\":[5],\"init\":[0],\"n\":6,\"post\":[],\"steps\":[[0,1],[1,2],[2,1],[2,5],[4,5]]}" }, "fields": { "title": "Partial Correctness and Its Vacuous Case", "stem": "Use the complete finite, possibly nondeterministic transition model S=Fin(6)={0,…,5}, Init=[0], Halt=[5], Step=[[0, 1], [1, 2], [2, 5], [4, 5], [2, 1]]. Normalize sets and ordered transition pairs. Execution stops at a declared Halt state, ignoring all of its outgoing edges. A stuck nonhalting state is not a successful halt; finite prefixes prove no general termination. Postcondition set=[]. Give (reachable halts, postcondition, bad reachable halts, partial correctness?, no reachable halt?).", "answer": { "kind": "expression", "value": "([5], [], [5], No, No)", "explanation": "Partial correctness checks only reached halts; it may hold vacuously while no run terminates successfully. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5], [], [5], No, No)." }, "skills": [ "Partial Correctness and Its Vacuous Case" ], "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": "完整有限正確性模型。操作=correctness-partial-audit；資料={\"n\":6,\"init\":[0],\"halt\":[5],\"steps\":[[0,1],[1,2],[2,5],[4,5],[2,1]],\"post\":[]}。請依 [reachable_halts,post,bad_halts,partial_correct,vacuous] 欄序回答。", "en": "Use the complete finite, possibly nondeterministic transition model S=Fin(6)={0,…,5}, Init=[0], Halt=[5], Step=[[0, 1], [1, 2], [2, 5], [4, 5], [2, 1]]. Normalize sets and ordered transition pairs. Execution stops at a declared Halt state, ignoring all of its outgoing edges. A stuck nonhalting state is not a successful halt; finite prefixes prove no general termination. Postcondition set=[]. Give (reachable halts, postcondition, bad reachable halts, partial correctness?, no reachable halt?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "n=6;raw Init=[0];raw Halt=[5];raw Step=[[0, 1], [1, 2], [2, 5], [4, 5], [2, 1]]", "after": "Init=[0];Halt=[5];canonical raw Step=[[0, 1], [1, 2], [2, 1], [2, 5], [4, 5]]", "relation": "implies", "rule_id": "finite_transition_normalization", "conditions": [], "explanation": { "zh-Hant": "整張有限資料先合型，再去重原有向邊。保留halt的原始出邊在資料中，但執行語意在halt停止；不把stuck當成功halt，也不由一條prefix推一般程式終止。", "en": "Type the complete finite data and deduplicate directed edges.Retain raw outgoing halt edges as data,but execution stops at halt.Stuck is not successful halt,and one prefix proves no general program termination." }, "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": "Init=[0]", "after": "zero-step reachable=[0]", "relation": "implies", "rule_id": "zero_step_reachability", "conditions": [], "explanation": { "zh-Hant": "每個初始狀態以0步可達，包括初始即halt者；空Init沒有可達狀態。", "en": "Every initial state is reachable in zero steps,including an initial halt.An empty Init has no reachable states." }, "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": "depth=0;state=0;halt=否;raw successors=[1]", "after": "expanded successors=[1];newly reached=[1]", "relation": "implies", "rule_id": "reachability_halt_truncates_expansion", "conditions": [], "explanation": { "zh-Hant": "從非halt狀態展開全部後繼，不只挑一條路；已見狀態不重複加入。halt即使列有出邊也不展開。", "en": "Expand every successor of a nonhalt state,not just one chosen path.Add unseen states only.A halt is not expanded even when raw outgoing edges exist." }, "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": "completed frontier=[0];depth=0", "after": "next frontier=[1];all reached=[0, 1]", "relation": "implies", "rule_id": "complete_finite_reachability_layer", "conditions": [], "explanation": { "zh-Hant": "前緣清空時才到達完整閉包；有限狀態及只加未見者保證此列舉終止，不依賴路徑長度抽樣。", "en": "The closure is complete only when the frontier empties.Finitely many states and adding only unseen states ensure enumeration terminates,without sampling a path-length cutoff." }, "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": "depth=1;state=1;halt=否;raw successors=[2]", "after": "expanded successors=[2];newly reached=[2]", "relation": "implies", "rule_id": "reachability_halt_truncates_expansion", "conditions": [], "explanation": { "zh-Hant": "從非halt狀態展開全部後繼，不只挑一條路；已見狀態不重複加入。halt即使列有出邊也不展開。", "en": "Expand every successor of a nonhalt state,not just one chosen path.Add unseen states only.A halt is not expanded even when raw outgoing edges exist." }, "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": "completed frontier=[1];depth=1", "after": "next frontier=[2];all reached=[0, 1, 2]", "relation": "implies", "rule_id": "complete_finite_reachability_layer", "conditions": [], "explanation": { "zh-Hant": "前緣清空時才到達完整閉包；有限狀態及只加未見者保證此列舉終止，不依賴路徑長度抽樣。", "en": "The closure is complete only when the frontier empties.Finitely many states and adding only unseen states ensure enumeration terminates,without sampling a path-length cutoff." }, "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": "depth=2;state=2;halt=否;raw successors=[1, 5]", "after": "expanded successors=[1, 5];newly reached=[5]", "relation": "implies", "rule_id": "reachability_halt_truncates_expansion", "conditions": [], "explanation": { "zh-Hant": "從非halt狀態展開全部後繼，不只挑一條路；已見狀態不重複加入。halt即使列有出邊也不展開。", "en": "Expand every successor of a nonhalt state,not just one chosen path.Add unseen states only.A halt is not expanded even when raw outgoing edges exist." }, "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": "completed frontier=[2];depth=2", "after": "next frontier=[5];all reached=[0, 1, 2, 5]", "relation": "implies", "rule_id": "complete_finite_reachability_layer", "conditions": [], "explanation": { "zh-Hant": "前緣清空時才到達完整閉包；有限狀態及只加未見者保證此列舉終止，不依賴路徑長度抽樣。", "en": "The closure is complete only when the frontier empties.Finitely many states and adding only unseen states ensure enumeration terminates,without sampling a path-length cutoff." }, "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": "S009", "uses": [ "G001", "S008" ], "before": "depth=3;state=5;halt=是;raw successors=[]", "after": "expanded successors=[];newly reached=[]", "relation": "implies", "rule_id": "reachability_halt_truncates_expansion", "conditions": [], "explanation": { "zh-Hant": "從非halt狀態展開全部後繼，不只挑一條路；已見狀態不重複加入。halt即使列有出邊也不展開。", "en": "Expand every successor of a nonhalt state,not just one chosen path.Add unseen states only.A halt is not expanded even when raw outgoing edges exist." }, "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": "S010", "uses": [ "G001", "S009" ], "before": "completed frontier=[5];depth=3", "after": "next frontier=[];all reached=[0, 1, 2, 5]", "relation": "implies", "rule_id": "complete_finite_reachability_layer", "conditions": [], "explanation": { "zh-Hant": "前緣清空時才到達完整閉包；有限狀態及只加未見者保證此列舉終止，不依賴路徑長度抽樣。", "en": "The closure is complete only when the frontier empties.Finitely many states and adding only unseen states ensure enumeration terminates,without sampling a path-length cutoff." }, "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": "S011", "uses": [ "G001", "S010" ], "before": "reached state=0;declared halt=否;raw successors=[1]", "after": "status=active-nonhalt", "relation": "implies", "rule_id": "halt_stuck_active_are_distinct", "conditions": [], "explanation": { "zh-Hant": "halt標記優先；非halt且無後繼才是stuck，有後繼者為active。沒有出邊不會自動變成成功停止。", "en": "The halt designation takes precedence.Only a nonhalt with no successor is stuck;one with successors is active.No outgoing edge does not automatically mean successful termination." }, "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": "S012", "uses": [ "G001", "S011" ], "before": "reached state=1;declared halt=否;raw successors=[2]", "after": "status=active-nonhalt", "relation": "implies", "rule_id": "halt_stuck_active_are_distinct", "conditions": [], "explanation": { "zh-Hant": "halt標記優先；非halt且無後繼才是stuck，有後繼者為active。沒有出邊不會自動變成成功停止。", "en": "The halt designation takes precedence.Only a nonhalt with no successor is stuck;one with successors is active.No outgoing edge does not automatically mean successful termination." }, "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": "S013", "uses": [ "G001", "S012" ], "before": "reached state=2;declared halt=否;raw successors=[1, 5]", "after": "status=active-nonhalt", "relation": "implies", "rule_id": "halt_stuck_active_are_distinct", "conditions": [], "explanation": { "zh-Hant": "halt標記優先；非halt且無後繼才是stuck，有後繼者為active。沒有出邊不會自動變成成功停止。", "en": "The halt designation takes precedence.Only a nonhalt with no successor is stuck;one with successors is active.No outgoing edge does not automatically mean successful termination." }, "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": "S014", "uses": [ "G001", "S013" ], "before": "reached state=5;declared halt=是;raw successors=[]", "after": "status=halt", "relation": "implies", "rule_id": "halt_stuck_active_are_distinct", "conditions": [], "explanation": { "zh-Hant": "halt標記優先；非halt且無後繼才是stuck，有後繼者為active。沒有出邊不會自動變成成功停止。", "en": "The halt designation takes precedence.Only a nonhalt with no successor is stuck;one with successors is active.No outgoing edge does not automatically mean successful termination." }, "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": "S015", "uses": [ "G001", "S014" ], "before": "reachable Halt state=5;postcondition=[]", "after": "postcondition satisfied=否", "relation": "implies", "rule_id": "partial_correctness_each_reachable_halt", "conditions": [], "explanation": { "zh-Hant": "部分正確性只在可達halt狀態核後置條件；不可達halt不算反例，尚未停止的狀態也不冒充成功結果。", "en": "Partial correctness checks the postcondition only at reachable halts.Unreachable halts are not counterexamples,and nonterminated states are not successful results." }, "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": "S016", "uses": [ "G001", "S015" ], "before": "reachable halts=[5];bad halts=[5]", "after": "partial correctness=否;no reachable halt=否", "relation": "implies", "rule_id": "partial_correctness_vacuity_not_termination", "conditions": [], "explanation": { "zh-Hant": "沒有可達halt時，部分正確性可能空泛成立，但尚未證明會停。", "en": "With no reachable halt,partial correctness may hold vacuously without establishing termination." }, "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": [ [ 5 ], [], [ 5 ], false, false ] }, "conclusion": { "canonical_value": "([5], [], [5], 否, 否)", "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": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261013-advance-20261019-r05-algorithm-correctness-foundations", "batch_path": "public/data/questions/batches/batch-20261013-advance-20261019-r05-algorithm-correctness-foundations/questions.jsonl", "batch_sha256": "703aa1b3f31f48ec8b83f7f6c0f3fd482538ae384d981ee9c41e22de7727dc2f", "original_line": 38, "original_line_sha256": "944d4940e81a011ebcbfab9192423de74b75c05b625f338caffdcf2e78848fae", "question_compact_sha256": "25c99fb5eb17fb413754c2ac9233cdf566ce68b79c374bb726f92b8ea9c6a60f", "english_row_compact_sha256": "d68c74be77e504b98e9114874670ef21d247c7814431c24b571d269e5a62867b", "english_manifest_path": "public/data/i18n/editions/batch-20261013-advance-20261019-r05-algorithm-correctness-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261013-advance-20261019-r05-algorithm-correctness-foundations.en.jsonl" }, "mko_bindings": { "mko-algorithm-correctness": { "sha256": "d9789607b7391a897a2a291245a0699d81ef6dbbdfac86b9b86c04cf7dc0ad27", "english_sha256": "33d91b7ce0b46158c3d90a3ec3dfc6506626564456beeabe3e4337d620f07e87", "original_path": "public/data/mko/algorithm-correctness.json", "english_path": "public/data/i18n/editions/mko-algorithm-correctness.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "2cf5df7bf9c3d48dfd41d968723b60d910c4ecfec9c2089d22ac1cda059a6086", "original_candidate_sha256": "bcf60664f44ec213f6e218616bfb8c89143f2f51b818c93f06992868c04877c9", "original_candidate_line_sha256": "bcf60664f44ec213f6e218616bfb8c89143f2f51b818c93f06992868c04877c9" }, "historical_source_state": { "review_state": { "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-20261026-006-p19", "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": { "semantic_review_sha256": "bd35da62087a15d6b89b45585dc46e89b9efbe097cba7b3c707b5030496f013c", "question_first_witness_sha256": "fab45a10d8ac95c7d61d32e47d1bdd3ca862c17c3289a6bf4b01d4dc9492fbb6" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 16, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "1c303c0a1939100a1a2167b9715276d634eeb5ceafe0830eb34a5a082164b925", "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." } }