# 第 21573 題：上下文無關文法 [返回所在分頁 108](https://ocme.evemisslab.com/zh-Hant/questions/page/108/) [← 逐題詳解目錄](https://ocme.evemisslab.com/zh-Hant/questions/) · [原題批次](https://ocme.evemisslab.com/zh-Hant/practice/batch-20261014-advance-20261109-r07-context-free-grammar-foundations/#q-20261014-advance-20261109-r07-context-free-grammar-0073) 本頁是已完成第二對話 AI 數理與中英語義覆核的教學詳解，只支持本題明示的模型與前提。沒有人工審查或新 Lean 證明，也不提升原題或 MKO 的 Canonical 真值狀態。 R0042-21573 · q-20261014-advance-20261109-r07-context-free-grammar-0073 ## 原題與已知條件 finite disjoint nonterminals/terminals、start與complete production set。操作=cfg-membership-witness；資料={"nonterminals":["S22","A22"],"terminals":["a22","b22"],"start":"S22","productions":[["S22",["a22","A22"]],["A22",["b22","A22"]],["A22",[]]],"allow_epsilon":true,"word":["a22","b22","b22"],"forms":[["S22"],["a22","A22"],["a22","b22","A22"],["a22","b22","b22","A22"],["a22","b22","b22"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。 - G001 — finite disjoint nonterminals/terminals、start與complete production set。操作=cfg-membership-witness；資料={"nonterminals":["S22","A22"],"terminals":["a22","b22"],"start":"S22","productions":[["S22",["a22","A22"]],["A22",["b22","A22"]],["A22",[]]],"allow_epsilon":true,"word":["a22","b22","b22"],"forms":[["S22"],["a22","A22"],["a22","b22","A22"],["a22","b22","b22","A22"],["a22","b22","b22"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。 ## 定義與適用範圍 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。 - [mko-context-free-grammar](https://ocme.evemisslab.com/zh-Hant/mko/mko-context-free-grammar/) ## 逐步推導 S001 · implies操作前 source operation=cfg-membership-witness;original snapshot and mathematics retained 操作後 explicit ordered source output fields=[word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate] 條件：無 原題歷史介面標記保留，本頁直接列來源有序欄位，不需JavaScript。tag不是額外證據，固定No保留其原聲稱範圍。 步驟識別與依賴規則：source_ordered_interface_explicit_without_javascript · 依賴：G001 S002 · implies操作前 ORIGINAL production index=0;lhs=S22,rhs=[a22, A22];N=[S22, A22],T=[a22, b22],allow_epsilon=是 操作後 lhs is ONE declared nonterminal;every rhs symbol legal;empty rhs accepted ONLY under the explicit epsilon convention;original production position retained 條件：無 每條production左邊只有一個非終結符；右邊逐symbol核合法，空右邊依明示epsilon慣例。重複production位置可存在，不能排序改變首witness索引。 步驟識別與依賴規則：cfg_each_original_production_single_nonterminal_legal_rhs · 依賴：G001, S001 S003 · implies操作前 ORIGINAL production index=1;lhs=A22,rhs=[b22, A22];N=[S22, A22],T=[a22, b22],allow_epsilon=是 操作後 lhs is ONE declared nonterminal;every rhs symbol legal;empty rhs accepted ONLY under the explicit epsilon convention;original production position retained 條件：無 每條production左邊只有一個非終結符；右邊逐symbol核合法，空右邊依明示epsilon慣例。重複production位置可存在，不能排序改變首witness索引。 步驟識別與依賴規則：cfg_each_original_production_single_nonterminal_legal_rhs · 依賴：G001, S002 S004 · implies操作前 ORIGINAL production index=2;lhs=A22,rhs=[];N=[S22, A22],T=[a22, b22],allow_epsilon=是 操作後 lhs is ONE declared nonterminal;every rhs symbol legal;empty rhs accepted ONLY under the explicit epsilon convention;original production position retained 條件：無 每條production左邊只有一個非終結符；右邊逐symbol核合法，空右邊依明示epsilon慣例。重複production位置可存在，不能排序改變首witness索引。 步驟識別與依賴規則：cfg_each_original_production_single_nonterminal_legal_rhs · 依賴：G001, S003 S005 · implies操作前 ORIGINAL unique disjoint N=[S22, A22],T=[a22, b22];start=S22;complete productions=[[S22, [a22, A22]], [A22, [b22, A22]], [A22, []]];allow_epsilon=是 操作後 start belongs to nonterminals;finite grammar qualified;NO derivation,membership,regularity,unambiguity or PDA equivalence inferred 條件：無 定義合格只是文法的資格，不自動有某個word的推導，也不推regular、unambiguous等額外性質。非終結符／終結符順序保原輸入。 步驟識別與依賴規則：cfg_qualified_alphabets_start_not_language_property · 依賴：G001, S004 S006 · implies操作前 ORIGINAL forms=[[S22], [a22, A22], [a22, b22, A22], [a22, b22, b22, A22], [a22, b22, b22]];actual grammar start=S22 操作後 first form exactly [start]=是;empty forms or wrong start fail even if the last form is terminal 條件：無 推導必須真從[start]開始；單看末項是終結字串不足。空forms不是epsilon推導；epsilon是實際空字串形式[]，仍需要合法production鏈。 步驟識別與依賴規則：cfg_derivation_actual_start_required · 依賴：G001, S005 S007 · implies操作前 ORIGINAL transition index=0;from=[S22];to=[a22, A22];original productions=[[S22, [a22, A22]], [A22, [b22, A22]], [A22, []]] 操作後 FIRST matching [position,production_index,lhs,rhs]=[0, 0, S22, [a22, A22]];replace EXACTLY ONE occurrence,retaining both surrounding substrings 條件：無 按原位置、再原production順序找第一個真替換witness；只換一個lhs occurrence，兩側字串保持。不把一次換全部同名symbol當一步；找不到就保先前partial witnesses並停止。 步驟識別與依賴規則：cfg_each_transition_first_position_then_original_production · 依賴：G001, S006 S008 · implies操作前 ORIGINAL transition index=1;from=[a22, A22];to=[a22, b22, A22];original productions=[[S22, [a22, A22]], [A22, [b22, A22]], [A22, []]] 操作後 FIRST matching [position,production_index,lhs,rhs]=[1, 1, A22, [b22, A22]];replace EXACTLY ONE occurrence,retaining both surrounding substrings 條件：無 按原位置、再原production順序找第一個真替換witness；只換一個lhs occurrence，兩側字串保持。不把一次換全部同名symbol當一步；找不到就保先前partial witnesses並停止。 步驟識別與依賴規則：cfg_each_transition_first_position_then_original_production · 依賴：G001, S007 S009 · implies操作前 ORIGINAL transition index=2;from=[a22, b22, A22];to=[a22, b22, b22, A22];original productions=[[S22, [a22, A22]], [A22, [b22, A22]], [A22, []]] 操作後 FIRST matching [position,production_index,lhs,rhs]=[2, 1, A22, [b22, A22]];replace EXACTLY ONE occurrence,retaining both surrounding substrings 條件：無 按原位置、再原production順序找第一個真替換witness；只換一個lhs occurrence，兩側字串保持。不把一次換全部同名symbol當一步；找不到就保先前partial witnesses並停止。 步驟識別與依賴規則：cfg_each_transition_first_position_then_original_production · 依賴：G001, S008 S010 · implies操作前 ORIGINAL transition index=3;from=[a22, b22, b22, A22];to=[a22, b22, b22];original productions=[[S22, [a22, A22]], [A22, [b22, A22]], [A22, []]] 操作後 FIRST matching [position,production_index,lhs,rhs]=[3, 2, A22, []];replace EXACTLY ONE occurrence,retaining both surrounding substrings 條件：無 按原位置、再原production順序找第一個真替換witness；只換一個lhs occurrence，兩側字串保持。不把一次換全部同名symbol當一步；找不到就保先前partial witnesses並停止。 步驟識別與依賴規則：cfg_each_transition_first_position_then_original_production · 依賴：G001, S009 S011 · implies操作前 ORIGINAL actual forms=[[S22], [a22, A22], [a22, b22, A22], [a22, b22, b22, A22], [a22, b22, b22]];accepted prior witnesses=[[0, 0, S22, [a22, A22]], [1, 1, A22, [b22, A22]], [2, 1, A22, [b22, A22]], [3, 2, A22, []]] 操作後 valid_derivation=是;this is separate from whether the actual final form contains only terminals 條件：無 合法性核完整從start的鏈；錯誤處後面出現terminal字串不能修補前面的錯誤。零步[start]可以合法，但通常尚非terminal word。 步驟識別與依賴規則：cfg_partial_trace_validity_not_final_terminal_flag · 依賴：G001, S010 S012 · implies操作前 ORIGINAL word=[a22, b22, b22];actual last form=[a22, b22, b22];valid chain=是 操作後 terminal_word=是;last equals original word=是;membership certificate=是 條件：無 成員證書需合法start鏈、word全terminal、末形式恰等原word三項同時成立。epsilon需真推到[]，不是把空language誤當含epsilon。 步驟識別與依賴規則：cfg_membership_three_obligations_actual_terminal_target · 依賴：G001, S011 ## 逐步反向核對 - S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 - S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。 ## 結論 ([a22, b22, b22], [[0, 0, S22, [a22, A22]], [1, 1, A22, [b22, A22]], [2, 1, A22, [b22, A22]], [3, 2, A22, []]], 是, 是, 是, derivation_from_start_to_terminal_only_word_proves_membership) 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。 ## 本題覆核與來源綁定 覆核日期：2026-10-26 原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。 來源雜湊與覆核憑證 - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: a7abb36cdc5b64721231fe55d99758fd5dd1236b9c7b1162c77cefa76aba48bc - candidate record SHA-256: 1237f6e687f055c91d4a96142c8ab443dbda06ae9f6c615c950f86f90777cdd2 - semantic review receipt SHA-256: dd44555041dfbb6fe9936aa00b9a5f8c5982c2784ef61dd0cbd5d1a303a49c1a - question-first witness SHA-256: 9a6011263f6ac53b9a1fd57addf2f3e2f6088b54802c539f332d689480dc2d17 - Lean: NOT_RUN · human review: NOT_PERFORMED [同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261109-r07-context-free-grammar-0073.json)