{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0089", "pilot_number": "R0045-23239", "number": 23239, "family_id": "qf-mod42-nfa-bounded-search-boundary", "title": { "zh-Hant": "第 23239 題：非確定有限自動機", "en": "Question 23239: A Depth-Limited No-Hit Is Not an NFA Rejection Proof" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0089", "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-bounded-search-boundary；資料={\"states\":[\"s13\",\"m13\",\"f13\",\"d13\"],\"alphabet\":[\"a13\",\"b13\"],\"start_state\":\"s13\",\"final_states\":[\"f13\"],\"transitions\":[[\"s13\",null,\"m13\"],[\"s13\",null,\"d13\"],[\"m13\",\"a13\",\"m13\"],[\"m13\",\"b13\",\"f13\"]],\"input\":[\"a13\",\"b13\"],\"max_steps\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "([a13, b13], 1, 否, 否, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)", "explanation_zh": "Wave6第五層核對=([a13, b13], 1, 否, 否, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)；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-bounded-search-boundary", "seed": 202611117089, "parameters": { "states": [ "s13", "m13", "f13", "d13" ], "alphabet": [ "a13", "b13" ], "start_state": "s13", "final_states": [ "f13" ], "transitions": [ [ "s13", null, "m13" ], [ "s13", null, "d13" ], [ "m13", "a13", "m13" ], [ "m13", "b13", "f13" ] ], "input": [ "a13", "b13" ], "max_steps": 1, "domain": "Finite_epsilon_NFA_existential_acceptance", "operation": "nfa-bounded-search-boundary", "semantic_template": "qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a13\",\"b13\"],\"final_states\":[\"f13\"],\"input\":[\"a13\",\"b13\"],\"max_steps\":1,\"start_state\":\"s13\",\"states\":[\"d13\",\"f13\",\"m13\",\"s13\"],\"transitions\":[[\"m13\",\"a13\",\"m13\"],[\"m13\",\"b13\",\"f13\"],[\"s13\",null,\"d13\"],[\"s13\",null,\"m13\"]]}" } }, "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-0089", "batch_id": "batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "b33408cf83a554d7187be7123ad12a9de6e101700eb7213bedf26dfd05449ff6", "family_id": "qf-mod42-nfa-bounded-search-boundary", "seed": 202611117089, "semantic_template": "qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a13\",\"b13\"],\"final_states\":[\"f13\"],\"input\":[\"a13\",\"b13\"],\"max_steps\":1,\"start_state\":\"s13\",\"states\":[\"d13\",\"f13\",\"m13\",\"s13\"],\"transitions\":[[\"m13\",\"a13\",\"m13\"],[\"m13\",\"b13\",\"f13\"],[\"s13\",null,\"d13\"],[\"s13\",null,\"m13\"]]}" }, "fields": { "title": "A Depth-Limited No-Hit Is Not an NFA Rejection Proof", "stem": "Use the complete finite epsilon-NFA: states=[s13, m13, f13, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]. 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. Input=[a13, b13], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).", "answer": { "kind": "expression", "value": "([a13, b13], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)", "explanation": "Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a13, b13], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)." }, "skills": [ "A Depth-Limited No-Hit Is Not an NFA Rejection Proof" ], "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-bounded-search-boundary；資料={\"states\":[\"s13\",\"m13\",\"f13\",\"d13\"],\"alphabet\":[\"a13\",\"b13\"],\"start_state\":\"s13\",\"final_states\":[\"f13\"],\"transitions\":[[\"s13\",null,\"m13\"],[\"s13\",null,\"d13\"],[\"m13\",\"a13\",\"m13\"],[\"m13\",\"b13\",\"f13\"]],\"input\":[\"a13\",\"b13\"],\"max_steps\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use the complete finite epsilon-NFA: states=[s13, m13, f13, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]. 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. Input=[a13, b13], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source operation=nfa-bounded-search-boundary;original snapshot and mathematics retained", "after": "explicit ordered source output fields=[input, max_steps, found_accepting_path, rejection_claim, boundary]", "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]=[s13, null, m13];state carrier=[s13, m13, f13, d13];alphabet=[a13, b13]", "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]=[s13, null, d13];state carrier=[s13, m13, f13, d13];alphabet=[a13, b13]", "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]=[m13, a13, m13];state carrier=[s13, m13, f13, d13];alphabet=[a13, b13]", "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]=[m13, b13, f13];state carrier=[s13, m13, f13, d13];alphabet=[a13, b13]", "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=[s13, m13, f13, d13],alphabet=[a13, b13],start=s13,final_states=[f13];complete relation=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]", "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": "ORIGINAL input=[a13, b13];actual start configuration=[s13, 0];max_steps=1", "after": "depth0 accepted=否;one finite FULL-input final path suffices;dead alternatives do not veto success", "relation": "implies", "rule_id": "nfa_bounded_initial_and_finite_path_acceptance_scope", "conditions": [], "explanation": { "zh-Hant": "搜尋步數計transition，EPS雖不耗input仍占一步。起點final可接受空字；有一條完整有限成功路徑即足夠，不需所有分支成功。", "en": "The bound counts transitions:an EPS move still costs a step despite consuming no input.An initially final start accepts empty input.One finite full-input final path suffices;not all branches must succeed." }, "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": "ORIGINAL depth=0;frontier in ORIGINAL-state/position display order=[[s13, 0]]", "after": "ALL [from,ORIGINAL transition index,next_or_null] rows=[[[s13, 0], 0, [m13, 0]], [[s13, 0], 1, [d13, 0]], [[s13, 0], 2, null], [[s13, 0], 3, null]];next distinct configs=[[m13, 0], [d13, 0]];hit FULL-input final=否", "relation": "implies", "rule_id": "nfa_ALL_nondeterministic_bounded_frontier_not_infinite_cycle", "conditions": [], "explanation": { "zh-Hant": "逐全部config與原relation列驗一步，按深度去重；input位置依symbol或null只增1或0。有限EPS循環只增加可達路徑，不是一條無窮接受run。", "en": "Check every configuration and original relation row,with per-depth deduplication.Input position increases1 or0 according to symbol or null.An EPS cycle permits paths but is not an infinite accepting run." }, "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": "ORIGINAL bound=1;input=[a13, b13];found_accepting_path=否", "after": "rejection_claim fixed No;bounded flag alone supplies no GENERAL rejection certificate;any actual completeness argument must be established separately", "relation": "implies", "rule_id": "nfa_bounded_No_rejection_claim_not_found_negation", "conditions": [], "explanation": { "zh-Hant": "最後No保留不由有限測試作一般拒絕推論，不否定found=True。某有限模型若另證此界已完整涵蓋configuration，可另判實際拒絕；不能把不同的完整判定偷換成本題found旗標。", "en": "The final No withholds a general rejection inference from finite tests,without negating found=True.If a finite model separately proves this bound complete,actual rejection may be decided;do not substitute that different completeness claim for this found flag." }, "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": [ [ "a13", "b13" ], 1, false, false, "bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs" ] }, "conclusion": { "canonical_value": "([a13, b13], 1, 否, 否, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)", "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": 89, "original_line_sha256": "39d458e43f6f5ff19e85f7a2d446c0ab52e9b826e5152e3ae88b8121696327b5", "question_compact_sha256": "b33408cf83a554d7187be7123ad12a9de6e101700eb7213bedf26dfd05449ff6", "english_row_compact_sha256": "03894462aa3560e2dde03b5d32d13a9b8e211ce52a66cce5042553aecf9f4cb5", "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": "e4a81fabf36007043ec9730dfea572f9cf4b8a34658cc2a3fef65ccff0b8c209", "original_candidate_line_sha256": "e4a81fabf36007043ec9730dfea572f9cf4b8a34658cc2a3fef65ccff0b8c209" }, "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": "201c0c8a38abbaacdaf7c947dcea26c25f08358ceb49e8be7fd30d91211d1f3b", "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." } }