# 第 21540 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-0040) 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。 R0042-21540 · q-20261014-advance-20261109-r07-context-free-grammar-0040 ## 原題與已知條件 finite disjoint nonterminals/terminals、start與complete production set。操作=cfg-derivation-audit；資料={"nonterminals":["S14","A14"],"terminals":["a14","b14"],"start":"S14","productions":[["S14",["a14","A14"]],["A14",["b14","A14"]],["A14",[]]],"allow_epsilon":true,"forms":[["S14"],["a14","A14"],["a14","b14","A14"],["a14","b14","b14","A14"],["a14","b14","b14","b14","A14"],["a14","b14","b14","b14","b14","A14"],["a14","b14","b14","b14","b14"]]}。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-derivation-audit；資料={"nonterminals":["S14","A14"],"terminals":["a14","b14"],"start":"S14","productions":[["S14",["a14","A14"]],["A14",["b14","A14"]],["A14",[]]],"allow_epsilon":true,"forms":[["S14"],["a14","A14"],["a14","b14","A14"],["a14","b14","b14","A14"],["a14","b14","b14","b14","A14"],["a14","b14","b14","b14","b14","A14"],["a14","b14","b14","b14","b14"]]}。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-derivation-audit;original snapshot and mathematics retained 操作後 explicit ordered source output fields=[forms, step_witnesses, valid_derivation, final_form_terminal_only, 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=S14,rhs=[a14, A14];N=[S14, A14],T=[a14, b14],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=A14,rhs=[b14, A14];N=[S14, A14],T=[a14, b14],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=A14,rhs=[];N=[S14, A14],T=[a14, b14],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=[S14, A14],T=[a14, b14];start=S14;complete productions=[[S14, [a14, A14]], [A14, [b14, A14]], [A14, []]];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=[[S14], [a14, A14], [a14, b14, A14], [a14, b14, b14, A14], [a14, b14, b14, b14, A14], [a14, b14, b14, b14, b14, A14], [a14, b14, b14, b14, b14]];actual grammar start=S14 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=[S14];to=[a14, A14];original productions=[[S14, [a14, A14]], [A14, [b14, A14]], [A14, []]] 操作後 FIRST matching [position,production_index,lhs,rhs]=[0, 0, S14, [a14, A14]];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=[a14, A14];to=[a14, b14, A14];original productions=[[S14, [a14, A14]], [A14, [b14, A14]], [A14, []]] 操作後 FIRST matching [position,production_index,lhs,rhs]=[1, 1, A14, [b14, A14]];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=[a14, b14, A14];to=[a14, b14, b14, A14];original productions=[[S14, [a14, A14]], [A14, [b14, A14]], [A14, []]] 操作後 FIRST matching [position,production_index,lhs,rhs]=[2, 1, A14, [b14, A14]];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=[a14, b14, b14, A14];to=[a14, b14, b14, b14, A14];original productions=[[S14, [a14, A14]], [A14, [b14, A14]], [A14, []]] 操作後 FIRST matching [position,production_index,lhs,rhs]=[3, 1, A14, [b14, A14]];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 transition index=4;from=[a14, b14, b14, b14, A14];to=[a14, b14, b14, b14, b14, A14];original productions=[[S14, [a14, A14]], [A14, [b14, A14]], [A14, []]] 操作後 FIRST matching [position,production_index,lhs,rhs]=[4, 1, A14, [b14, A14]];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, S010 S012 · implies操作前 ORIGINAL transition index=5;from=[a14, b14, b14, b14, b14, A14];to=[a14, b14, b14, b14, b14];original productions=[[S14, [a14, A14]], [A14, [b14, A14]], [A14, []]] 操作後 FIRST matching [position,production_index,lhs,rhs]=[5, 2, A14, []];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, S011 S013 · implies操作前 ORIGINAL actual forms=[[S14], [a14, A14], [a14, b14, A14], [a14, b14, b14, A14], [a14, b14, b14, b14, A14], [a14, b14, b14, b14, b14, A14], [a14, b14, b14, b14, b14]];accepted prior witnesses=[[0, 0, S14, [a14, A14]], [1, 1, A14, [b14, A14]], [2, 1, A14, [b14, A14]], [3, 1, A14, [b14, A14]], [4, 1, A14, [b14, A14]], [5, 2, A14, []]] 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, S012 S014 · implies操作前 ORIGINAL actual final form=[a14, b14, b14, b14, b14];terminals=[a14, b14] 操作後 final_form_terminal_only=是;valid_derivation remains=是 條件：無 末形式空字串時逐symbol條件vacuous True，但仍須另核整段推導；forms全空則沒有末形式而是False。 步驟識別與依賴規則：cfg_final_terminal_only_audited_separately · 依賴：G001, S013 ## 逐步反向核對 - S014 · 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。 - S013 · 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。 - 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 ([[S14], [a14, A14], [a14, b14, A14], [a14, b14, b14, A14], [a14, b14, b14, b14, A14], [a14, b14, b14, b14, b14, A14], [a14, b14, b14, b14, b14]], [[0, 0, S14, [a14, A14]], [1, 1, A14, [b14, A14]], [2, 1, A14, [b14, A14]], [3, 1, A14, [b14, A14]], [4, 1, A14, [b14, A14]], [5, 2, A14, []]], 是, 是, finite_production_sequence) 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: 16a4de6ee25517726aa6723e3600d3f53a77c810bfbfc5a56cf5b8c23d95dd60 - candidate record SHA-256: cf95663ded9d09dee09772109b52a8767c9c388870d59ca58ac0b2a3375a55e8 - 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-0040.json)