# 第 21544 題：上下文無關文法

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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。

R0042-21544 · q-20261014-advance-20261109-r07-context-free-grammar-0044



## 原題與已知條件

finite disjoint nonterminals/terminals、start與complete production set。操作=cfg-derivation-audit；資料={"nonterminals":["S18","A18"],"terminals":["a18","b18"],"start":"S18","productions":[["S18",["a18","A18"]],["A18",["b18","A18"]],["A18",[]]],"allow_epsilon":true,"forms":[["S18"],["a18","A18"],["a18","b18","A18"],["a18","b18","b18","A18"],["a18","b18","b18","b18","A18"],["a18","b18","b18","b18"]]}。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":["S18","A18"],"terminals":["a18","b18"],"start":"S18","productions":[["S18",["a18","A18"]],["A18",["b18","A18"]],["A18",[]]],"allow_epsilon":true,"forms":[["S18"],["a18","A18"],["a18","b18","A18"],["a18","b18","b18","A18"],["a18","b18","b18","b18","A18"],["a18","b18","b18","b18"]]}。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=S18,rhs=[a18, A18];N=[S18, A18],T=[a18, b18],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=A18,rhs=[b18, A18];N=[S18, A18],T=[a18, b18],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=A18,rhs=[];N=[S18, A18],T=[a18, b18],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=[S18, A18],T=[a18, b18];start=S18;complete productions=[[S18, [a18, A18]], [A18, [b18, A18]], [A18, []]];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=[[S18], [a18, A18], [a18, b18, A18], [a18, b18, b18, A18], [a18, b18, b18, b18, A18], [a18, b18, b18, b18]];actual grammar start=S18

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=[S18];to=[a18, A18];original productions=[[S18, [a18, A18]], [A18, [b18, A18]], [A18, []]]

操作後

FIRST matching [position,production_index,lhs,rhs]=[0, 0, S18, [a18, A18]];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=[a18, A18];to=[a18, b18, A18];original productions=[[S18, [a18, A18]], [A18, [b18, A18]], [A18, []]]

操作後

FIRST matching [position,production_index,lhs,rhs]=[1, 1, A18, [b18, A18]];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=[a18, b18, A18];to=[a18, b18, b18, A18];original productions=[[S18, [a18, A18]], [A18, [b18, A18]], [A18, []]]

操作後

FIRST matching [position,production_index,lhs,rhs]=[2, 1, A18, [b18, A18]];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=[a18, b18, b18, A18];to=[a18, b18, b18, b18, A18];original productions=[[S18, [a18, A18]], [A18, [b18, A18]], [A18, []]]

操作後

FIRST matching [position,production_index,lhs,rhs]=[3, 1, A18, [b18, A18]];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=[a18, b18, b18, b18, A18];to=[a18, b18, b18, b18];original productions=[[S18, [a18, A18]], [A18, [b18, A18]], [A18, []]]

操作後

FIRST matching [position,production_index,lhs,rhs]=[4, 2, A18, []];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 actual forms=[[S18], [a18, A18], [a18, b18, A18], [a18, b18, b18, A18], [a18, b18, b18, b18, A18], [a18, b18, b18, b18]];accepted prior witnesses=[[0, 0, S18, [a18, A18]], [1, 1, A18, [b18, A18]], [2, 1, A18, [b18, A18]], [3, 1, A18, [b18, A18]], [4, 2, A18, []]]

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, S011





S013 · implies操作前

ORIGINAL actual final form=[a18, b18, b18, b18];terminals=[a18, b18]

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, S012









## 逐步反向核對


- 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

([[S18], [a18, A18], [a18, b18, A18], [a18, b18, b18, A18], [a18, b18, b18, b18, A18], [a18, b18, b18, b18]], [[0, 0, S18, [a18, A18]], [1, 1, A18, [b18, A18]], [2, 1, A18, [b18, A18]], [3, 1, A18, [b18, A18]], [4, 2, A18, []]], 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: 51bde8c4024211fdafaacf69342524322af91e3adf9801fa69fc0422c7200dc1
- candidate record SHA-256: 5be3acbcaee177455b0cd85afcdd18be58630521dda5a2fbf5fe61d05d85776c
- 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-0044.json)