{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261102-r07-regular-language-0061", "pilot_number": "R0033-17211", "number": 17211, "family_id": "qf-mod33-regular-finite-language-construction", "title": { "zh-Hant": "第 17211 題：正規語言", "en": "Question 17211: Construct a Complete Prefix-Trie DFA for an Exact Finite Language" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261102-r07-regular-language-0061", "batch_id": "batch-20261014-advance-20261102-r07-regular-language-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-regular-language" ], "title_zh": "正規語言", "stem_zh": "完整DFA總轉移與任意word invariant或明示有限語言構造。操作=regular-finite-language-construction；資料={\"alphabet_count\":2,\"words\":[[0],[1,0,1,0,1,0]],\"probe\":[1,0,1,0,1,0]}。請依 [prefix_states,dead_state,accepting,transition,probe,trace,accepted,certificate] 欄序回答。", "answer": { "kind": "expression", "value": "([[], [0], [1], [1, 0], [1, 0, 1], [1, 0, 1, 0], [1, 0, 1, 0, 1], [1, 0, 1, 0, 1, 0]], 8, [1, 7], [[1, 2], [8, 8], [3, 8], [8, 4], [5, 8], [8, 6], [7, 8], [8, 8], [8, 8]], [1, 0, 1, 0, 1, 0], [0, 2, 3, 4, 5, 6, 7], 是, explicit_DFA_recognizes_exact_finite_language)", "explanation_zh": "Wave5第二層核對=([[], [0], [1], [1, 0], [1, 0, 1], [1, 0, 1, 0], [1, 0, 1, 0, 1], [1, 0, 1, 0, 1, 0]], 8, [1, 7], [[1, 2], [8, 8], [3, 8], [8, 4], [5, 8], [8, 6], [7, 8], [8, 8], [8, 8]], [1, 0, 1, 0, 1, 0], [0, 2, 3, 4, 5, 6, 7], 是, explicit_DFA_recognizes_exact_finite_language)；依完整明示資料重算；mod-count證書須涵蓋任意word length而不只是測試集；不引入regex、NFA、epsilon、minimization或pumping lemma；有限測試不足。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "正規語言" ] }, "variant": { "family_id": "qf-mod33-regular-finite-language-construction", "seed": 202611027061, "parameters": { "alphabet_count": 2, "words": [ [ 0 ], [ 1, 0, 1, 0, 1, 0 ] ], "probe": [ 1, 0, 1, 0, 1, 0 ], "domain": "Exact_DFA_regular_language_certificate", "operation": "regular-finite-language-construction", "semantic_template": "qf-mod33-regular-finite-language-construction/{\"alphabet_count\":2,\"probe\":[1,0,1,0,1,0],\"words\":[[0],[1,0,1,0,1,0]]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須由主端獨立重算；不構成人工審定、任意維basis theorem、bounded-to-global kernel/image、一般compactness、因果推論、language equivalence、Lean或形式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-20261102-r07-regular-language-0061", "batch_id": "batch-20261014-advance-20261102-r07-regular-language-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "ab1b523b4a36ca9984d221a26f3fea1943b53cfca76d08861c952924658dd502", "family_id": "qf-mod33-regular-finite-language-construction", "seed": 202611027061, "semantic_template": "qf-mod33-regular-finite-language-construction/{\"alphabet_count\":2,\"probe\":[1,0,1,0,1,0],\"words\":[[0],[1,0,1,0,1,0]]}" }, "fields": { "title": "Construct a Complete Prefix-Trie DFA for an Exact Finite Language", "stem": "Use alphabet=Fin(2)={0,…,1} and the complete nonempty duplicate-free word family=[[0], [1, 0, 1, 0, 1, 0]]. The family may include the empty word; this constructor does not handle an empty family. Probe=[1, 0, 1, 0, 1, 0]. Give (prefix_states,dead_state,accepting,transition,probe,trace,accepted?,explicit_DFA_recognizes_exact_finite_language).", "answer": { "kind": "expression", "value": "([[], [0], [1], [1, 0], [1, 0, 1], [1, 0, 1, 0], [1, 0, 1, 0, 1], [1, 0, 1, 0, 1, 0]], 8, [1, 7], [[1, 2], [8, 8], [3, 8], [8, 4], [5, 8], [8, 6], [7, 8], [8, 8], [8, 8]], [1, 0, 1, 0, 1, 0], [0, 2, 3, 4, 5, 6, 7], Yes, explicit_DFA_recognizes_exact_finite_language)", "explanation": "Collect all prefixes including the empty prefix, sort first by length then by their JSON lexicographic order, and use those positions as states. Append a dead state, send absent extensions to it and loop there. Accept exactly the states equal to full supplied words. Retain all table rows and the complete probe trace. This construction covers arbitrary words of the declared alphabet without claiming minimization. The exact result, retaining the requested order, is ([[], [0], [1], [1, 0], [1, 0, 1], [1, 0, 1, 0], [1, 0, 1, 0, 1], [1, 0, 1, 0, 1, 0]], 8, [1, 7], [[1, 2], [8, 8], [3, 8], [8, 4], [5, 8], [8, 6], [7, 8], [8, 8], [8, 8]], [1, 0, 1, 0, 1, 0], [0, 2, 3, 4, 5, 6, 7], Yes, explicit_DFA_recognizes_exact_finite_language)." }, "skills": [ "Construct a Complete Prefix-Trie DFA for an Exact Finite Language" ], "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DFA總轉移與任意word invariant或明示有限語言構造。操作=regular-finite-language-construction；資料={\"alphabet_count\":2,\"words\":[[0],[1,0,1,0,1,0]],\"probe\":[1,0,1,0,1,0]}。請依 [prefix_states,dead_state,accepting,transition,probe,trace,accepted,certificate] 欄序回答。", "en": "Use alphabet=Fin(2)={0,…,1} and the complete nonempty duplicate-free word family=[[0], [1, 0, 1, 0, 1, 0]]. The family may include the empty word; this constructor does not handle an empty family. Probe=[1, 0, 1, 0, 1, 0]. Give (prefix_states,dead_state,accepting,transition,probe,trace,accepted?,explicit_DFA_recognizes_exact_finite_language)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "DECLARED complete nonempty duplicate-free word FAMILY=[[0], [1, 0, 1, 0, 1, 0]];alphabet Fin(2)", "after": "all prefixes including empty sorted by length then source JSON order=[[], [0], [1], [1, 0], [1, 0, 1], [1, 0, 1, 0], [1, 0, 1, 0, 1], [1, 0, 1, 0, 1, 0]];dead state appended index=8", "relation": "implies", "rule_id": "regular_complete_finite_language_prefix_states_not_sampled_tests", "conditions": [], "explanation": { "zh-Hant": "原族是整個有限語言，不是測試集；可含空字但族本身不可空。前綴依長度與來源JSON序排序，符號界0..9使同長排序相容；不套另一題族的30狀態上限。", "en": "This family is the entire finite language,not test samples.It may contain the empty word but the family is nonempty.Sort prefixes by length and the source JSON order;symbols0..9 make same-length ordering compatible.Do not inherit another family's30-state cap." }, "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": "prefix state=0;represented prefix=[];next symbol=0;extended word=[0]", "after": "extension is declared prefix=是;next state=1", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "prefix state=0;represented prefix=[];next symbol=1;extended word=[1]", "after": "extension is declared prefix=是;next state=2", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "prefix state=1;represented prefix=[0];next symbol=0;extended word=[0, 0]", "after": "extension is declared prefix=否;next state=8", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "prefix state=1;represented prefix=[0];next symbol=1;extended word=[0, 1]", "after": "extension is declared prefix=否;next state=8", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "prefix state=2;represented prefix=[1];next symbol=0;extended word=[1, 0]", "after": "extension is declared prefix=是;next state=3", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "prefix state=2;represented prefix=[1];next symbol=1;extended word=[1, 1]", "after": "extension is declared prefix=否;next state=8", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "prefix state=3;represented prefix=[1, 0];next symbol=0;extended word=[1, 0, 0]", "after": "extension is declared prefix=否;next state=8", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "prefix state=3;represented prefix=[1, 0];next symbol=1;extended word=[1, 0, 1]", "after": "extension is declared prefix=是;next state=4", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "prefix state=4;represented prefix=[1, 0, 1];next symbol=0;extended word=[1, 0, 1, 0]", "after": "extension is declared prefix=是;next state=5", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "prefix state=4;represented prefix=[1, 0, 1];next symbol=1;extended word=[1, 0, 1, 1]", "after": "extension is declared prefix=否;next state=8", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "prefix state=5;represented prefix=[1, 0, 1, 0];next symbol=0;extended word=[1, 0, 1, 0, 0]", "after": "extension is declared prefix=否;next state=8", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "prefix state=5;represented prefix=[1, 0, 1, 0];next symbol=1;extended word=[1, 0, 1, 0, 1]", "after": "extension is declared prefix=是;next state=6", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "prefix state=6;represented prefix=[1, 0, 1, 0, 1];next symbol=0;extended word=[1, 0, 1, 0, 1, 0]", "after": "extension is declared prefix=是;next state=7", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "prefix state=6;represented prefix=[1, 0, 1, 0, 1];next symbol=1;extended word=[1, 0, 1, 0, 1, 1]", "after": "extension is declared prefix=否;next state=8", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "prefix state=7;represented prefix=[1, 0, 1, 0, 1, 0];next symbol=0;extended word=[1, 0, 1, 0, 1, 0, 0]", "after": "extension is declared prefix=否;next state=8", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "S017", "uses": [ "G001", "S016" ], "before": "prefix state=7;represented prefix=[1, 0, 1, 0, 1, 0];next symbol=1;extended word=[1, 0, 1, 0, 1, 0, 1]", "after": "extension is declared prefix=否;next state=8", "relation": "implies", "rule_id": "regular_trie_every_prefix_symbol_total_transition", "conditions": [], "explanation": { "zh-Hant": "若延伸仍是某個供給字的前綴就進該前綴狀態，否則進唯一dead；每個prefix與每個字母都填一格，非NFA分支。", "en": "Move to the extended prefix if it is a prefix of a supplied word;otherwise enter the unique dead state.Fill every prefix-symbol cell,with no NFA branching." }, "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": "S018", "uses": [ "G001", "S017" ], "before": "dead state=8;all alphabet symbols Fin(2);full supplied words=[[0], [1, 0, 1, 0, 1, 0]]", "after": "dead loops row=[8, 8];accepting prefix-state indices=[1, 7];complete table=[[1, 2], [8, 8], [3, 8], [8, 4], [5, 8], [8, 6], [7, 8], [8, 8], [8, 8]]", "relation": "implies", "rule_id": "regular_trie_dead_absorbing_accepts_full_words_only", "conditions": [], "explanation": { "zh-Hant": "dead對每個符號都留dead；只把等於完整供給字的狀態列為接受，普通真前綴不自動接受。含空字時start0可在接受集。", "en": "Every symbol loops at dead.Accept exactly states equal to full supplied words,not every proper prefix.If the empty word is supplied,start0 is accepting." }, "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": "S019", "uses": [ "G001", "S018" ], "before": "same complete prefix-state table and full word family=[[0], [1, 0, 1, 0, 1, 0]];start state0 represents empty prefix", "after": "for EVERY finite input prefix,its state is that exact prefix index if present,otherwise dead;prefix extension closure and dead absorption preserve this by induction;accepted iff input is one FULL supplied word", "relation": "implies", "rule_id": "regular_trie_all_word_induction_exact_language_not_probe_only", "conditions": [], "explanation": { "zh-Hant": "基底是空前綴；若仍在前綴表，下一格恰按延伸定義。若已不是任何供給字前綴，後續延伸亦不可能變回前綴，所以dead吸收正確。對任意字長歸納，最後接受恰等於完整字清單，不由probe成功外推全語言。", "en": "The empty prefix is the base case.Each transition follows exact extension.If a consumed word is not a supplied-word prefix,no later extension can become one,so dead absorption is valid.Induction over arbitrary word length gives acceptance exactly for full listed words,not an inference from a successful probe." }, "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": "S020", "uses": [ "G001", "S019" ], "before": "finite-language probe:original ordered word=[1, 0, 1, 0, 1, 0];start=0", "after": "trace before consuming symbols=[0]", "relation": "implies", "rule_id": "regular_word_initial_state_empty_word_retained", "conditions": [], "explanation": { "zh-Hant": "先放初始狀態，空字不做轉移；符號順序不可為了結果相同而重排。", "en": "Include the initial state.An empty word performs no transition;do not reorder symbols merely because a final answer might coincide." }, "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": "S021", "uses": [ "G001", "S020" ], "before": "finite-language probe:word position=0;symbol=1;current state=0;complete original row=[1, 2]", "after": "next state=2;ordered prefix trace=[0, 2]", "relation": "implies", "rule_id": "regular_every_symbol_source_bound_transition", "conditions": [], "explanation": { "zh-Hant": "每次只讀原字一個符號，按state列與symbol欄查表，保留每個中間狀態；不跳過拒絕路徑或dead狀態。", "en": "Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state." }, "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": "S022", "uses": [ "G001", "S021" ], "before": "finite-language probe:word position=1;symbol=0;current state=2;complete original row=[3, 8]", "after": "next state=3;ordered prefix trace=[0, 2, 3]", "relation": "implies", "rule_id": "regular_every_symbol_source_bound_transition", "conditions": [], "explanation": { "zh-Hant": "每次只讀原字一個符號，按state列與symbol欄查表，保留每個中間狀態；不跳過拒絕路徑或dead狀態。", "en": "Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state." }, "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": "S023", "uses": [ "G001", "S022" ], "before": "finite-language probe:word position=2;symbol=1;current state=3;complete original row=[8, 4]", "after": "next state=4;ordered prefix trace=[0, 2, 3, 4]", "relation": "implies", "rule_id": "regular_every_symbol_source_bound_transition", "conditions": [], "explanation": { "zh-Hant": "每次只讀原字一個符號，按state列與symbol欄查表，保留每個中間狀態；不跳過拒絕路徑或dead狀態。", "en": "Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state." }, "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": "S024", "uses": [ "G001", "S023" ], "before": "finite-language probe:word position=3;symbol=0;current state=4;complete original row=[5, 8]", "after": "next state=5;ordered prefix trace=[0, 2, 3, 4, 5]", "relation": "implies", "rule_id": "regular_every_symbol_source_bound_transition", "conditions": [], "explanation": { "zh-Hant": "每次只讀原字一個符號，按state列與symbol欄查表，保留每個中間狀態；不跳過拒絕路徑或dead狀態。", "en": "Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state." }, "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": "S025", "uses": [ "G001", "S024" ], "before": "finite-language probe:word position=4;symbol=1;current state=5;complete original row=[8, 6]", "after": "next state=6;ordered prefix trace=[0, 2, 3, 4, 5, 6]", "relation": "implies", "rule_id": "regular_every_symbol_source_bound_transition", "conditions": [], "explanation": { "zh-Hant": "每次只讀原字一個符號，按state列與symbol欄查表，保留每個中間狀態；不跳過拒絕路徑或dead狀態。", "en": "Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state." }, "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": "S026", "uses": [ "G001", "S025" ], "before": "finite-language probe:word position=5;symbol=0;current state=6;complete original row=[7, 8]", "after": "next state=7;ordered prefix trace=[0, 2, 3, 4, 5, 6, 7]", "relation": "implies", "rule_id": "regular_every_symbol_source_bound_transition", "conditions": [], "explanation": { "zh-Hant": "每次只讀原字一個符號，按state列與symbol欄查表，保留每個中間狀態；不跳過拒絕路徑或dead狀態。", "en": "Read one original symbol per lookup,using the state row and symbol column.Retain every intermediate state,including rejecting paths and the dead state." }, "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": "S027", "uses": [ "G001", "S026" ], "before": "original probe=[1, 0, 1, 0, 1, 0];final state=7;accepting indices=[1, 7]", "after": "probe accepted=是;separate all-word construction certificate retained", "relation": "implies", "rule_id": "regular_probe_result_separate_universal_construction", "conditions": [], "explanation": { "zh-Hant": "probe只是一個實例的運行；上面的完整構造與歸納才是整個有限語言正規性的證書。", "en": "The probe is one run.The complete construction and induction above supply the full finite-language certificate." }, "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": [ [ [], [ 0 ], [ 1 ], [ 1, 0 ], [ 1, 0, 1 ], [ 1, 0, 1, 0 ], [ 1, 0, 1, 0, 1 ], [ 1, 0, 1, 0, 1, 0 ] ], 8, [ 1, 7 ], [ [ 1, 2 ], [ 8, 8 ], [ 3, 8 ], [ 8, 4 ], [ 5, 8 ], [ 8, 6 ], [ 7, 8 ], [ 8, 8 ], [ 8, 8 ] ], [ 1, 0, 1, 0, 1, 0 ], [ 0, 2, 3, 4, 5, 6, 7 ], true, "explicit_DFA_recognizes_exact_finite_language" ] }, "conclusion": { "canonical_value": "([[], [0], [1], [1, 0], [1, 0, 1], [1, 0, 1, 0], [1, 0, 1, 0, 1], [1, 0, 1, 0, 1, 0]], 8, [1, 7], [[1, 2], [8, 8], [3, 8], [8, 4], [5, 8], [8, 6], [7, 8], [8, 8], [8, 8]], [1, 0, 1, 0, 1, 0], [0, 2, 3, 4, 5, 6, 7], 是, explicit_DFA_recognizes_exact_finite_language)", "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-20261102-r07-regular-language-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261102-r07-regular-language-foundations/questions.jsonl", "batch_sha256": "6b354df25a379577a6b550ace23ccd63aa6511564e13109eb1a705a45232f831", "original_line": 61, "original_line_sha256": "b8a91ff74885023436930d75bfd30733554b2e72af31d8260f6337dfa46eec93", "question_compact_sha256": "ab1b523b4a36ca9984d221a26f3fea1943b53cfca76d08861c952924658dd502", "english_row_compact_sha256": "76cee782b917bff54bfc847ffbd07988ac98ba81dcc3a399162f9ba46c5bf2f9", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261102-r07-regular-language-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261102-r07-regular-language-foundations.en.jsonl" }, "mko_bindings": { "mko-regular-language": { "sha256": "5b76e4da673cf35d7a7014ce76f103fa504f5f9986498711ed877c15dc35e57a", "english_sha256": "b6bfb3d62bc7ec660a984b86c35cabf78b5826bddd2394009ea358db21f4038a", "original_path": "public/data/mko/regular-language.json", "english_path": "public/data/i18n/editions/mko-regular-language.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "c6f506c0e91ce68861b828c110bfee6002681c5a6e691f1ba5ae296376b30ad5", "original_candidate_sha256": "91c48ad34f9833db2ef3ce0fd0d610231af7793439fd8818648753ad8c40a5d8", "original_candidate_line_sha256": "91c48ad34f9833db2ef3ce0fd0d610231af7793439fd8818648753ad8c40a5d8" }, "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-20261026-008-p34", "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": "4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3", "question_first_witness_sha256": "22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 27, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "f935935a5b47d02dfc74c410e551035dee41817d0478c4f26e0f4f8385ae9028", "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." } }