{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0035", "pilot_number": "R0045-23260", "number": 23260, "family_id": "qf-mod42-nfa-epsilon-closure", "title": { "zh-Hant": "第 23260 題：非確定有限自動機", "en": "Question 23260: Epsilon Closure Reaches States without Consuming Input" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0035", "batch_id": "batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-nondeterministic-finite-automaton" ], "title_zh": "非確定有限自動機", "stem_zh": "transition=[from,input-or-null,to]且允許empty alphabet/final set的finite epsilon-NFA。操作=nfa-epsilon-closure；資料={\"states\":[\"s9\",\"m9\",\"f9\",\"d9\"],\"alphabet\":[\"a9\",\"b9\"],\"start_state\":\"s9\",\"final_states\":[\"f9\"],\"transitions\":[[\"s9\",null,\"m9\"],[\"s9\",null,\"d9\"],[\"m9\",\"a9\",\"m9\"],[\"m9\",\"b9\",\"f9\"]],\"seed_states\":[\"s9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([s9], [d9, m9, s9], 是, finite_epsilon_reachability_without_input_consumption)", "explanation_zh": "Wave6第五層核對=([s9], [d9, m9, s9], 是, finite_epsilon_reachability_without_input_consumption)；依完整明示資料獨立重算；null epsilon不消耗input、transition relation可empty或multi-valued且一條finite full-input final path即可證membership；empty final set合法且辨識空語言；bounded search failure不證rejection，epsilon cycle本身不接受。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "非確定有限自動機" ] }, "variant": { "family_id": "qf-mod42-nfa-epsilon-closure", "seed": 202611117035, "parameters": { "states": [ "s9", "m9", "f9", "d9" ], "alphabet": [ "a9", "b9" ], "start_state": "s9", "final_states": [ "f9" ], "transitions": [ [ "s9", null, "m9" ], [ "s9", null, "d9" ], [ "m9", "a9", "m9" ], [ "m9", "b9", "f9" ] ], "seed_states": [ "s9" ], "domain": "Finite_epsilon_NFA_existential_acceptance", "operation": "nfa-epsilon-closure", "semantic_template": "qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a9\",\"b9\"],\"final_states\":[\"f9\"],\"seed_states\":[\"s9\"],\"start_state\":\"s9\",\"states\":[\"d9\",\"f9\",\"m9\",\"s9\"],\"transitions\":[[\"m9\",\"a9\",\"m9\"],[\"m9\",\"b9\",\"f9\"],[\"s9\",null,\"d9\"],[\"s9\",null,\"m9\"]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、Lagrange converse existence、higher-order analyticity、sampled Hausdorff、CDF density、unchecked Catalan division、Euler criterion、bounded NFA rejection、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0035", "batch_id": "batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "be311da8c14e769fb7da282f3f07b448dab7d96bdc0b1ba7fd44d3776c756e58", "family_id": "qf-mod42-nfa-epsilon-closure", "seed": 202611117035, "semantic_template": "qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a9\",\"b9\"],\"final_states\":[\"f9\"],\"seed_states\":[\"s9\"],\"start_state\":\"s9\",\"states\":[\"d9\",\"f9\",\"m9\",\"s9\"],\"transitions\":[[\"m9\",\"a9\",\"m9\"],[\"m9\",\"b9\",\"f9\"],[\"s9\",null,\"d9\"],[\"s9\",null,\"m9\"]]}" }, "fields": { "title": "Epsilon Closure Reaches States without Consuming Input", "stem": "Use the complete finite epsilon-NFA: states=[s9, m9, f9, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s9]. Give (seed_states, epsilon_closure, closure_certificate, certificate).", "answer": { "kind": "expression", "value": "([s9], [d9, m9, s9], Yes, finite_epsilon_reachability_without_input_consumption)", "explanation": "Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s9], [d9, m9, s9], Yes, finite_epsilon_reachability_without_input_consumption)." }, "skills": [ "Epsilon Closure Reaches States without Consuming Input" ], "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": "transition=[from,input-or-null,to]且允許empty alphabet/final set的finite epsilon-NFA。操作=nfa-epsilon-closure；資料={\"states\":[\"s9\",\"m9\",\"f9\",\"d9\"],\"alphabet\":[\"a9\",\"b9\"],\"start_state\":\"s9\",\"final_states\":[\"f9\"],\"transitions\":[[\"s9\",null,\"m9\"],[\"s9\",null,\"d9\"],[\"m9\",\"a9\",\"m9\"],[\"m9\",\"b9\",\"f9\"]],\"seed_states\":[\"s9\"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the complete finite epsilon-NFA: states=[s9, m9, f9, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s9]. Give (seed_states, epsilon_closure, closure_certificate, certificate)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=nfa-epsilon-closure;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[seed_states, epsilon_closure, closure_certificate, certificate]", "relation": "implies", "rule_id": "source_ordered_interface_explicit_without_javascript", "conditions": [], "explanation": { "zh-Hant": "原題快照保留歷史介面；此處直接列來源欄序，不依賴JavaScript。tag與No保聲稱範圍，固定來源排序不冒充數學證明。", "en": "Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof." }, "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": "ORIGINAL transition index=0;[from,input_or_null,to]=[s9, null, m9];state carrier=[s9, m9, f9, d9];alphabet=[a9, b9]", "after": "state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows", "relation": "implies", "rule_id": "nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal", "conditions": [], "explanation": { "zh-Hant": "這是relation不是單值total函數，duplicate／多值／缺邊皆保原資料；null消耗0但不是一個alphabet字串。", "en": "This is a relation,not a total single-valued function.Retain duplicate,multivalued and missing edges;null consumes0 and is not an alphabet string." }, "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": "ORIGINAL transition index=1;[from,input_or_null,to]=[s9, null, d9];state carrier=[s9, m9, f9, d9];alphabet=[a9, b9]", "after": "state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows", "relation": "implies", "rule_id": "nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal", "conditions": [], "explanation": { "zh-Hant": "這是relation不是單值total函數，duplicate／多值／缺邊皆保原資料；null消耗0但不是一個alphabet字串。", "en": "This is a relation,not a total single-valued function.Retain duplicate,multivalued and missing edges;null consumes0 and is not an alphabet string." }, "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": "ORIGINAL transition index=2;[from,input_or_null,to]=[m9, a9, m9];state carrier=[s9, m9, f9, d9];alphabet=[a9, b9]", "after": "state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows", "relation": "implies", "rule_id": "nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal", "conditions": [], "explanation": { "zh-Hant": "這是relation不是單值total函數，duplicate／多值／缺邊皆保原資料；null消耗0但不是一個alphabet字串。", "en": "This is a relation,not a total single-valued function.Retain duplicate,multivalued and missing edges;null consumes0 and is not an alphabet string." }, "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": "ORIGINAL transition index=3;[from,input_or_null,to]=[m9, b9, f9];state carrier=[s9, m9, f9, d9];alphabet=[a9, b9]", "after": "state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows", "relation": "implies", "rule_id": "nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal", "conditions": [], "explanation": { "zh-Hant": "這是relation不是單值total函數，duplicate／多值／缺邊皆保原資料；null消耗0但不是一個alphabet字串。", "en": "This is a relation,not a total single-valued function.Retain duplicate,multivalued and missing edges;null consumes0 and is not an alphabet string." }, "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": "ORIGINAL states=[s9, m9, f9, d9],alphabet=[a9, b9],start=s9,final_states=[f9];complete relation=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]]", "after": "states nonempty with actual start;alphabet/final set/relation MAY be EMPTY;definition qualification does NOT prove determinism,totality,DFA equivalence or a successful run", "relation": "implies", "rule_id": "nfa_qualified_definition_EMPTY_alphabet_finals_edges_allowed", "conditions": [], "explanation": { "zh-Hant": "空final set合法且辨識空語言；空alphabet也合法。定義合格不保存在accepting path，亦不推deterministic或DFA等價。", "en": "An empty final set is legal and recognizes the empty language;an empty alphabet is legal too.A qualified definition supplies no accepting path,determinism or DFA-equivalence proof." }, "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": "NFA ORIGINAL EPS closure:ORIGINAL seed list=[s9];complete state carrier=[s9, m9, f9, d9]", "after": "initialize unique closure=[s9];seeds themselves are reachable by zero EPS steps;RAW seed list not changed", "relation": "implies", "rule_id": "nfa_EPS_closure_contains_ORIGINAL_seeds_duplicates_preserved", "conditions": [], "explanation": { "zh-Hant": "seed可空或重複，輸出原seed列表保留；closure集合去重且包含零步可達的seed，不憑EPS cycle宣稱接受。", "en": "Seeds may be empty or repeated.Retain the raw seed list while the closure set deduplicates and includes zero-step reachability.An EPS cycle alone gives no acceptance." }, "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": "NFA ORIGINAL EPS closure:stage=0;ORIGINAL-relation EPS pass from previous set=[s9]", "after": "ALL [ORIGINAL index,from,to,from_in_previous] EPS rows=[[0, s9, m9, 是], [1, s9, d9, 是]];next unique set=[d9, m9, s9];stable=否", "relation": "implies", "rule_id": "nfa_EVERY_EPS_saturation_stage_least_reachable_set", "conditions": [], "explanation": { "zh-Hant": "每個新增state有前一集合的EPS前驅，故有有限seed路徑；任何含seed且EPS封閉的集合都須含每一層。有限states只會新增有限次，穩定時即完整且最小closure，不消耗input。", "en": "Each added state has an EPS predecessor in the preceding set and hence a finite seed path.Any EPS-closed superset of the seeds contains every stage.Finiteness gives stabilization at the complete least closure,without consuming input." }, "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": "NFA ORIGINAL EPS closure:stage=1;ORIGINAL-relation EPS pass from previous set=[d9, m9, s9]", "after": "ALL [ORIGINAL index,from,to,from_in_previous] EPS rows=[[0, s9, m9, 是], [1, s9, d9, 是]];next unique set=[d9, m9, s9];stable=是", "relation": "implies", "rule_id": "nfa_EVERY_EPS_saturation_stage_least_reachable_set", "conditions": [], "explanation": { "zh-Hant": "每個新增state有前一集合的EPS前驅，故有有限seed路徑；任何含seed且EPS封閉的集合都須含每一層。有限states只會新增有限次，穩定時即完整且最小closure，不消耗input。", "en": "Each added state has an EPS predecessor in the preceding set and hence a finite seed path.Any EPS-closed superset of the seeds contains every stage.Finiteness gives stabilization at the complete least closure,without consuming input." }, "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": [ [ "s9" ], [ "d9", "m9", "s9" ], true, "finite_epsilon_reachability_without_input_consumption" ] }, "conclusion": { "canonical_value": "([s9], [d9, m9, s9], 是, finite_epsilon_reachability_without_input_consumption)", "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-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations/questions.jsonl", "batch_sha256": "5ab150dafd2899ef1c33ab5fca8efa13a6e4b72ab22fe0a04ebbbd61b659ec1c", "original_line": 35, "original_line_sha256": "9c8dd877aae3636fc78fd96b5cf5787a8d045cee5c8f08485ec945c48c99766a", "question_compact_sha256": "be311da8c14e769fb7da282f3f07b448dab7d96bdc0b1ba7fd44d3776c756e58", "english_row_compact_sha256": "8dbc5e05aa5bf77321fa8d21a35467450e6607f7b7cef37bc33c11d2386a67bb", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations.en.jsonl" }, "mko_bindings": { "mko-nondeterministic-finite-automaton": { "sha256": "4ac739d18d7d5b29bc3d146c9707a5cbf0e98a1bd3b66ce6a5b064ec83f1380b", "english_sha256": "d3037e441fbf749593b9c7847ce52c1e4d383a9b8dacdbe3f2068aec86b4f406", "original_path": "public/data/mko/nondeterministic-finite-automaton.json", "english_path": "public/data/i18n/editions/mko-nondeterministic-finite-automaton.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "9732ef02cb296d2e89f9b6cade5a76a83e9d37bdcf6a3de7f8613ebce02311e8", "original_candidate_sha256": "d4c24f39042cc573722acb21028d154facaed26e217a124f37bd573999ee1cb8", "original_candidate_line_sha256": "d4c24f39042cc573722acb21028d154facaed26e217a124f37bd573999ee1cb8" }, "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-20261026-009-p46", "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": "357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438", "question_first_witness_sha256": "99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 9, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "2cad50936c12af92c8150fb2ca8f535e4e975829bcfc82d92861b3050b7d0731", "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." } }