第 22403 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-accepting-run-witness；資料={"states":["push2","pop2","accept2"],"input_alphabet":["a2","b2"],"stack_alphabet":["Z2","A2"],"start_state":"push2","bottom_symbol":"Z2","final_states":["accept2"],"transitions":[["push2","a2","Z2","push2",["Z2","A2"]],["push2","a2","A2","push2",["A2","A2"]],["push2","b2","A2","pop2",[]],["pop2","b2","A2","pop2",[]],["push2",null,"Z2","accept2",["Z2"]],["pop2",null,"Z2","accept2",["Z2"]]],"input":["a2","a2","b2","b2"],"trace":[["push2",0,["Z2"]],["push2",1,["Z2","A2"]],["push2",2,["Z2","A2","A2"]],["pop2",3,["Z2","A2"]],["pop2",4,["Z2"]],["accept2",4,["Z2"]]]}。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-accepting-run-witness；資料={"states":["push2","pop2","accept2"],"input_alphabet":["a2","b2"],"stack_alphabet":["Z2","A2"],"start_state":"push2","bottom_symbol":"Z2","final_states":["accept2"],"transitions":[["push2","a2","Z2","push2",["Z2","A2"]],["push2","a2","A2","push2",["A2","A2"]],["push2","b2","A2","pop2",[]],["pop2","b2","A2","pop2",[]],["push2",null,"Z2","accept2",["Z2"]],["pop2",null,"Z2","accept2",["Z2"]]],"input":["a2","a2","b2","b2"],"trace":[["push2",0,["Z2"]],["push2",1,["Z2","A2"]],["push2",2,["Z2","A2","A2"]],["pop2",3,["Z2","A2"]],["pop2",4,["Z2"]],["accept2",4,["Z2"]]]}。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-accepting-run-witness;original snapshot and mathematics retained

explicit ordered source output fields=[input, trace, transition_indices, valid_run, accepted, step_count, certificate]

ORIGINAL transition index=0;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push2, a2, Z2, push2, [Z2, A2]];bottom=Z2

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]=[push2, a2, A2, push2, [A2, A2]];bottom=Z2

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]=[push2, b2, A2, pop2, []];bottom=Z2

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]=[pop2, b2, A2, pop2, []];bottom=Z2

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]=[push2, null, Z2, accept2, [Z2]];bottom=Z2

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]=[pop2, null, Z2, accept2, [Z2]];bottom=Z2

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=[push2, pop2, accept2],input alphabet=[a2, b2],stack alphabet=[Z2, A2];start=push2,bottom=Z2,final states=[accept2]

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=[a2, a2, b2, b2];ALL source configurations=[[push2, 0, [Z2]], [push2, 1, [Z2, A2]], [push2, 2, [Z2, A2, A2]], [pop2, 3, [Z2, A2]], [pop2, 4, [Z2]], [accept2, 4, [Z2]]];declared start=push2,bottom=Z2

first configuration EXACTLY=[push2, 0, [Z2]];all state/position/stack values legal;this is NOT yet a valid-run or acceptance certificate

ORIGINAL trace step=0;from=[push2, 0, [Z2]];to=[push2, 1, [Z2, A2]];actual input=[a2, a2, b2, b2]

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, [push2, 1, [Z2, A2]]], [1, null], [2, null], [3, null], [4, [accept2, 0, [Z2]]], [5, null]];FIRST matching index=0

ORIGINAL trace step=1;from=[push2, 1, [Z2, A2]];to=[push2, 2, [Z2, A2, A2]];actual input=[a2, a2, b2, b2]

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, [push2, 2, [Z2, A2, A2]]], [2, null], [3, null], [4, null], [5, null]];FIRST matching index=1

ORIGINAL trace step=2;from=[push2, 2, [Z2, A2, A2]];to=[pop2, 3, [Z2, A2]];actual input=[a2, a2, b2, b2]

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, [pop2, 3, [Z2, A2]]], [3, null], [4, null], [5, null]];FIRST matching index=2

ORIGINAL trace step=3;from=[pop2, 3, [Z2, A2]];to=[pop2, 4, [Z2]];actual input=[a2, a2, b2, b2]

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, [pop2, 4, [Z2]]], [4, null], [5, null]];FIRST matching index=3

ORIGINAL trace step=4;from=[pop2, 4, [Z2]];to=[accept2, 4, [Z2]];actual input=[a2, a2, b2, b2]

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, null], [4, null], [5, [accept2, 4, [Z2]]]];FIRST matching index=5

ORIGINAL last config=[accept2, 4, [Z2]];input length=4;matched prior indices=[0, 1, 2, 3, 5]

valid_run=是;FULL input consumed=是;last state final=是;accepted=是;residual stack may be NONEMPTY

PDA ORIGINAL full trace length minus one := sub(6, 1)

PDA ORIGINAL full trace length minus one = 5

ORIGINAL trace configuration count=6;actual matched transition-index count=5;accepted=是

SOURCE step_count=ORIGINAL trace.length-1=5;does NOT substitute partial matched count;acceptedTrue is a finite witness,not search completeness

逐步反向核對

• S018 · 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。
• S017 · premise_and_result_audit — 以精確有理數重算 PDA ORIGINAL full trace length minus one，不以小數近似替代。
• S016 · 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。
• S015 · 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。
• S014 · 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。
• S013 · 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。
• 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

([a2, a2, b2, b2], [[push2, 0, [Z2]], [push2, 1, [Z2, A2]], [push2, 2, [Z2, A2, A2]], [pop2, 3, [Z2, A2]], [pop2, 4, [Z2]], [accept2, 4, [Z2]]], [0, 1, 2, 3, 5], 是, 是, 5, finite_full_run_witness_proves_acceptance_under_final_state_convention)

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。