第 21561 題：上下文無關文法

原題與已知條件

finite disjoint nonterminals/terminals、start與complete production set。操作=cfg-membership-witness；資料={"nonterminals":["S10","A10"],"terminals":["a10","b10"],"start":"S10","productions":[["S10",["a10","A10"]],["A10",["b10","A10"]],["A10",[]]],"allow_epsilon":true,"word":["a10"],"forms":[["S10"],["a10","A10"],["a10"]]}。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":["S10","A10"],"terminals":["a10","b10"],"start":"S10","productions":[["S10",["a10","A10"]],["A10",["b10","A10"]],["A10",[]]],"allow_epsilon":true,"word":["a10"],"forms":[["S10"],["a10","A10"],["a10"]]}。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

source operation=cfg-membership-witness;original snapshot and mathematics retained

explicit ordered source output fields=[word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate]

ORIGINAL production index=0;lhs=S10,rhs=[a10, A10];N=[S10, A10],T=[a10, b10],allow_epsilon=是

lhs is ONE declared nonterminal;every rhs symbol legal;empty rhs accepted ONLY under the explicit epsilon convention;original production position retained

ORIGINAL production index=1;lhs=A10,rhs=[b10, A10];N=[S10, A10],T=[a10, b10],allow_epsilon=是

lhs is ONE declared nonterminal;every rhs symbol legal;empty rhs accepted ONLY under the explicit epsilon convention;original production position retained

ORIGINAL production index=2;lhs=A10,rhs=[];N=[S10, A10],T=[a10, b10],allow_epsilon=是

lhs is ONE declared nonterminal;every rhs symbol legal;empty rhs accepted ONLY under the explicit epsilon convention;original production position retained

ORIGINAL unique disjoint N=[S10, A10],T=[a10, b10];start=S10;complete productions=[[S10, [a10, A10]], [A10, [b10, A10]], [A10, []]];allow_epsilon=是

start belongs to nonterminals;finite grammar qualified;NO derivation,membership,regularity,unambiguity or PDA equivalence inferred

ORIGINAL forms=[[S10], [a10, A10], [a10]];actual grammar start=S10

first form exactly [start]=是;empty forms or wrong start fail even if the last form is terminal

ORIGINAL transition index=0;from=[S10];to=[a10, A10];original productions=[[S10, [a10, A10]], [A10, [b10, A10]], [A10, []]]

FIRST matching [position,production_index,lhs,rhs]=[0, 0, S10, [a10, A10]];replace EXACTLY ONE occurrence,retaining both surrounding substrings

ORIGINAL transition index=1;from=[a10, A10];to=[a10];original productions=[[S10, [a10, A10]], [A10, [b10, A10]], [A10, []]]

FIRST matching [position,production_index,lhs,rhs]=[1, 2, A10, []];replace EXACTLY ONE occurrence,retaining both surrounding substrings

ORIGINAL actual forms=[[S10], [a10, A10], [a10]];accepted prior witnesses=[[0, 0, S10, [a10, A10]], [1, 2, A10, []]]

valid_derivation=是;this is separate from whether the actual final form contains only terminals

ORIGINAL word=[a10];actual last form=[a10];valid chain=是

terminal_word=是;last equals original word=是;membership certificate=是

逐步反向核對

• 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

([a10], [[0, 0, S10, [a10, A10]], [1, 2, A10, []]], 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。