第 21568 題：上下文無關文法

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原題與已知條件
定義與適用範圍
逐步推導
逐步反向核對
結論
本題覆核與來源綁定

R0042-21568 · q-20261014-advance-20261109-r07-context-free-grammar-0068

原題與已知條件

finite disjoint nonterminals/terminals、start與complete production set。操作=cfg-membership-witness；資料={"nonterminals":["S17","A17"],"terminals":["a17","b17"],"start":"S17","productions":[["S17",["a17","A17"]],["A17",["b17","A17"]],["A17",[]]],"allow_epsilon":true,"word":["a17","b17","b17"],"forms":[["S17"],["a17","A17"],["a17","b17","A17"],["a17","b17","b17","A17"],["a17","b17","b17"]]}。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":["S17","A17"],"terminals":["a17","b17"],"start":"S17","productions":[["S17",["a17","A17"]],["A17",["b17","A17"]],["A17",[]]],"allow_epsilon":true,"word":["a17","b17","b17"],"forms":[["S17"],["a17","A17"],["a17","b17","A17"],["a17","b17","b17","A17"],["a17","b17","b17"]]}。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

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=S17,rhs=[a17, A17];N=[S17, A17],T=[a17, b17],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=A17,rhs=[b17, A17];N=[S17, A17],T=[a17, b17],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=A17,rhs=[];N=[S17, A17],T=[a17, b17],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=[S17, A17],T=[a17, b17];start=S17;complete productions=[[S17, [a17, A17]], [A17, [b17, A17]], [A17, []]];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=[[S17], [a17, A17], [a17, b17, A17], [a17, b17, b17, A17], [a17, b17, b17]];actual grammar start=S17

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=[S17];to=[a17, A17];original productions=[[S17, [a17, A17]], [A17, [b17, A17]], [A17, []]]

操作後

FIRST matching [position,production_index,lhs,rhs]=[0, 0, S17, [a17, A17]];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=[a17, A17];to=[a17, b17, A17];original productions=[[S17, [a17, A17]], [A17, [b17, A17]], [A17, []]]

操作後

FIRST matching [position,production_index,lhs,rhs]=[1, 1, A17, [b17, A17]];replace EXACTLY ONE occurrence,retaining both surrounding substrings

條件：無

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=[a17, b17, A17];to=[a17, b17, b17, A17];original productions=[[S17, [a17, A17]], [A17, [b17, A17]], [A17, []]]

操作後

FIRST matching [position,production_index,lhs,rhs]=[2, 1, A17, [b17, A17]];replace EXACTLY ONE occurrence,retaining both surrounding substrings

條件：無

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=[a17, b17, b17, A17];to=[a17, b17, b17];original productions=[[S17, [a17, A17]], [A17, [b17, A17]], [A17, []]]

操作後

FIRST matching [position,production_index,lhs,rhs]=[3, 2, A17, []];replace EXACTLY ONE occurrence,retaining both surrounding substrings

條件：無

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=[[S17], [a17, A17], [a17, b17, A17], [a17, b17, b17, A17], [a17, b17, b17]];accepted prior witnesses=[[0, 0, S17, [a17, A17]], [1, 1, A17, [b17, A17]], [2, 1, A17, [b17, A17]], [3, 2, A17, []]]

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=[a17, b17, b17];actual last form=[a17, b17, b17];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

([a17, b17, b17], [[0, 0, S17, [a17, A17]], [1, 1, A17, [b17, A17]], [2, 1, A17, [b17, A17]], [3, 2, A17, []]], 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: e549ef87867a26a90d704fc630fbbe69715b98a5aea1fd33b18cc1fbedacb147
candidate record SHA-256: e478543165f1ae806488c7586439abdcb90ff0154648a25baf8d77d0e2c361d9
semantic review receipt SHA-256: dd44555041dfbb6fe9936aa00b9a5f8c5982c2784ef61dd0cbd5d1a303a49c1a
question-first witness SHA-256: 9a6011263f6ac53b9a1fd57addf2f3e2f6088b54802c539f332d689480dc2d17
Lean: NOT_RUN · human review: NOT_PERFORMED

同源完整 JSON