第 22427 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":["push1","pop1","accept1"],"input_alphabet":["a1","b1"],"stack_alphabet":["Z1","A1"],"start_state":"push1","bottom_symbol":"Z1","final_states":["accept1"],"transitions":[["push1","a1","Z1","push1",["Z1","A1"]],["push1","a1","A1","push1",["A1","A1"]],["push1","b1","A1","pop1",[]],["pop1","b1","A1","pop1",[]],["push1",null,"Z1","accept1",["Z1"]],["pop1",null,"Z1","accept1",["Z1"]]],"input":["a1","a1","b1","b1"],"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":["push1","pop1","accept1"],"input_alphabet":["a1","b1"],"stack_alphabet":["Z1","A1"],"start_state":"push1","bottom_symbol":"Z1","final_states":["accept1"],"transitions":[["push1","a1","Z1","push1",["Z1","A1"]],["push1","a1","A1","push1",["A1","A1"]],["push1","b1","A1","pop1",[]],["pop1","b1","A1","pop1",[]],["push1",null,"Z1","accept1",["Z1"]],["pop1",null,"Z1","accept1",["Z1"]]],"input":["a1","a1","b1","b1"],"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]=[push1, a1, Z1, push1, [Z1, A1]];bottom=Z1

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]=[push1, a1, A1, push1, [A1, A1]];bottom=Z1

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]=[push1, b1, A1, pop1, []];bottom=Z1

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]=[pop1, b1, A1, pop1, []];bottom=Z1

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]=[push1, null, Z1, accept1, [Z1]];bottom=Z1

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]=[pop1, null, Z1, accept1, [Z1]];bottom=Z1

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=[push1, pop1, accept1],input alphabet=[a1, b1],stack alphabet=[Z1, A1];start=push1,bottom=Z1,final states=[accept1]

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=[a1, a1, b1, b1];initial configuration=[push1, 0, [Z1]];max_steps=1;stack cap=7

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=[[push1, 0, [Z1]]];stack cap=7

ALL [from,ORIGINAL transition index,next_or_null,within_stack_cap,accepts_IN_THIS_SEARCH] rows=[[[push1, 0, [Z1]], 0, [push1, 1, [Z1, A1]], 是, 否], [[push1, 0, [Z1]], 1, null, 否, 否], [[push1, 0, [Z1]], 2, null, 否, 否], [[push1, 0, [Z1]], 3, null, 否, 否], [[push1, 0, [Z1]], 4, [accept1, 0, [Z1]], 是, 否], [[push1, 0, [Z1]], 5, null, 否, 否]];next distinct frontier=[[accept1, 0, [Z1]], [push1, 1, [Z1, A1]]];hit=否

ORIGINAL input=[a1, a1, b1, b1];step bound=1;stack bound=7;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

([a1, a1, b1, b1], 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。