R0042-21531 · q-20261014-advance-20261109-r07-context-free-grammar-0031
finite disjoint nonterminals/terminals、start與complete production set。操作=cfg-derivation-audit；資料={"nonterminals":["S5","A5"],"terminals":["a5","b5"],"start":"S5","productions":[["S5",["a5","A5"]],["A5",["b5","A5"]],["A5",[]]],"allow_epsilon":true,"forms":[["S5"],["a5","A5"],["a5"]]}。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":["S5","A5"],"terminals":["a5","b5"],"start":"S5","productions":[["S5",["a5","A5"]],["A5",["b5","A5"]],["A5",[]]],"allow_epsilon":true,"forms":[["S5"],["a5","A5"],["a5"]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。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。
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。
規則：source_ordered_interface_explicit_without_javascript · 依賴：G001
操作前
ORIGINAL production index=0;lhs=S5,rhs=[a5, A5];N=[S5, A5],T=[a5, b5],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。
規則：cfg_each_original_production_single_nonterminal_legal_rhs · 依賴：G001, S001
操作前
ORIGINAL production index=1;lhs=A5,rhs=[b5, A5];N=[S5, A5],T=[a5, b5],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。
規則：cfg_each_original_production_single_nonterminal_legal_rhs · 依賴：G001, S002
操作前
ORIGINAL production index=2;lhs=A5,rhs=[];N=[S5, A5],T=[a5, b5],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。
規則：cfg_each_original_production_single_nonterminal_legal_rhs · 依賴：G001, S003
操作前
ORIGINAL unique disjoint N=[S5, A5],T=[a5, b5];start=S5;complete productions=[[S5, [a5, A5]], [A5, [b5, A5]], [A5, []]];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。
規則：cfg_qualified_alphabets_start_not_language_property · 依賴：G001, S004
操作前
ORIGINAL forms=[[S5], [a5, A5], [a5]];actual grammar start=S5
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。
規則：cfg_derivation_actual_start_required · 依賴：G001, S005
操作前
ORIGINAL transition index=0;from=[S5];to=[a5, A5];original productions=[[S5, [a5, A5]], [A5, [b5, A5]], [A5, []]]
操作後
FIRST matching [position,production_index,lhs,rhs]=[0, 0, S5, [a5, A5]];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。
規則：cfg_each_transition_first_position_then_original_production · 依賴：G001, S006
操作前
ORIGINAL transition index=1;from=[a5, A5];to=[a5];original productions=[[S5, [a5, A5]], [A5, [b5, A5]], [A5, []]]
操作後
FIRST matching [position,production_index,lhs,rhs]=[1, 2, A5, []];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。
規則：cfg_each_transition_first_position_then_original_production · 依賴：G001, S007
操作前
ORIGINAL actual forms=[[S5], [a5, A5], [a5]];accepted prior witnesses=[[0, 0, S5, [a5, A5]], [1, 2, A5, []]]
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。
規則：cfg_partial_trace_validity_not_final_terminal_flag · 依賴：G001, S008
操作前
ORIGINAL actual final form=[a5];terminals=[a5, b5]
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。
規則：cfg_final_terminal_only_audited_separately · 依賴：G001, S009
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。([[S5], [a5, A5], [a5]], [[0, 0, S5, [a5, A5]], [1, 2, A5, []]], 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。
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。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 2b3ebbc4fb6ad3470d0cea5c3d45b2b7f8e329b99755f9111661482330086e09candidate record SHA-256: c7ce2df0b2592f973e7d1ed421d1076da0d0be10af6ac01ba013419d8def8385semantic review receipt SHA-256: dd44555041dfbb6fe9936aa00b9a5f8c5982c2784ef61dd0cbd5d1a303a49c1aquestion-first witness SHA-256: 9a6011263f6ac53b9a1fd57addf2f3e2f6088b54802c539f332d689480dc2d17Lean: NOT_RUN · human review: NOT_PERFORMED