第 22445 題：下推自動機

原題與已知條件

transition=[from,input-or-null,required-top,to,replacement-bottom-to-top]的finite nondeterministic PDA。操作=pda-bounded-search-boundary；資料={"states":["push19","pop19","accept19"],"input_alphabet":["a19","b19"],"stack_alphabet":["Z19","A19"],"start_state":"push19","bottom_symbol":"Z19","final_states":["accept19"],"transitions":[["push19","a19","Z19","push19",["Z19","A19"]],["push19","a19","A19","push19",["A19","A19"]],["push19","b19","A19","pop19",[]],["pop19","b19","A19","pop19",[]],["push19",null,"Z19","accept19",["Z19"]],["pop19",null,"Z19","accept19",["Z19"]]],"input":["a19","a19","a19","a19","b19","b19","b19","b19"],"max_steps":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — transition=[from,input-or-null,required-top,to,replacement-bottom-to-top]的finite nondeterministic PDA。操作=pda-bounded-search-boundary；資料={"states":["push19","pop19","accept19"],"input_alphabet":["a19","b19"],"stack_alphabet":["Z19","A19"],"start_state":"push19","bottom_symbol":"Z19","final_states":["accept19"],"transitions":[["push19","a19","Z19","push19",["Z19","A19"]],["push19","a19","A19","push19",["A19","A19"]],["push19","b19","A19","pop19",[]],["pop19","b19","A19","pop19",[]],["push19",null,"Z19","accept19",["Z19"]],["pop19",null,"Z19","accept19",["Z19"]]],"input":["a19","a19","a19","a19","b19","b19","b19","b19"],"max_steps":1}。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-pushdown-automaton

逐步推導

source operation=pda-bounded-search-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary]

ORIGINAL transition index=0;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push19, a19, Z19, push19, [Z19, A19]];bottom=Z19

states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained

ORIGINAL transition index=1;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push19, a19, A19, push19, [A19, A19]];bottom=Z19

states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained

ORIGINAL transition index=2;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push19, b19, A19, pop19, []];bottom=Z19

states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained

ORIGINAL transition index=3;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[pop19, b19, A19, pop19, []];bottom=Z19

states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained

ORIGINAL transition index=4;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push19, null, Z19, accept19, [Z19]];bottom=Z19

states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained

ORIGINAL transition index=5;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[pop19, null, Z19, accept19, [Z19]];bottom=Z19

states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained

PDA complete ORIGINAL transition count := sum(1, 1, 1, 1, 1, 1)

PDA complete ORIGINAL transition count = 6

ORIGINAL states=[push19, pop19, accept19],input alphabet=[a19, b19],stack alphabet=[Z19, A19];start=push19,bottom=Z19,final states=[accept19]

finite unique pairwise-disjoint alphabets qualified;transition count=6;acceptance convention FULL input in final state,NOT empty stack;qualification alone proves NO run,determinism or CFG equivalence

ORIGINAL input=[a19, a19, a19, a19, b19, b19, b19, b19];initial configuration=[push19, 0, [Z19]];max_steps=1;stack cap=11

depth0 fully-consumed final-state acceptance=否;both STEP and STACK limits apply;not an unbounded rejection test

ORIGINAL search depth=0;ALL frontier configs=[[push19, 0, [Z19]]];stack cap=11

ALL [from,ORIGINAL transition index,next_or_null,within_stack_cap,accepts_IN_THIS_SEARCH] rows=[[[push19, 0, [Z19]], 0, [push19, 1, [Z19, A19]], 是, 否], [[push19, 0, [Z19]], 1, null, 否, 否], [[push19, 0, [Z19]], 2, null, 否, 否], [[push19, 0, [Z19]], 3, null, 否, 否], [[push19, 0, [Z19]], 4, [accept19, 0, [Z19]], 是, 否], [[push19, 0, [Z19]], 5, null, 否, 否]];next distinct frontier=[[accept19, 0, [Z19]], [push19, 1, [Z19, A19]]];hit=否

ORIGINAL input=[a19, a19, a19, a19, b19, b19, b19, b19];step bound=1;stack bound=11;found_accepting_run=否

rejection_or_nonmembership_claim fixed No;longer or larger-stack accepting runs have NOT been excluded

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 — 以精確有理數重算 PDA complete ORIGINAL transition count，不以小數近似替代。
• 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

([a19, a19, a19, a19, b19, b19, b19, b19], 1, 否, 否, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)

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。