# 第 22419 題：下推自動機

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本頁是已完成第二對話 AI 數理與中英語義覆核的教學詳解，只支持本題明示的模型與前提。沒有人工審查或新 Lean 證明，也不提升原題或 MKO 的 Canonical 真值狀態。

R0043-22419 · q-20261014-advance-20261110-r07-pushdown-automaton-0069



## 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":["push18","pop18","accept18"],"input_alphabet":["a18","b18"],"stack_alphabet":["Z18","A18"],"start_state":"push18","bottom_symbol":"Z18","final_states":["accept18"],"transitions":[["push18","a18","Z18","push18",["Z18","A18"]],["push18","a18","A18","push18",["A18","A18"]],["push18","b18","A18","pop18",[]],["pop18","b18","A18","pop18",[]],["push18",null,"Z18","accept18",["Z18"]],["pop18",null,"Z18","accept18",["Z18"]]],"input":["a18","a18","b18","b18"],"trace":[["push18",0,["Z18"]],["push18",1,["Z18","A18"]],["push18",2,["Z18","A18","A18"]],["pop18",3,["Z18","A18"]],["pop18",4,["Z18"]],["accept18",4,["Z18"]]]}。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":["push18","pop18","accept18"],"input_alphabet":["a18","b18"],"stack_alphabet":["Z18","A18"],"start_state":"push18","bottom_symbol":"Z18","final_states":["accept18"],"transitions":[["push18","a18","Z18","push18",["Z18","A18"]],["push18","a18","A18","push18",["A18","A18"]],["push18","b18","A18","pop18",[]],["pop18","b18","A18","pop18",[]],["push18",null,"Z18","accept18",["Z18"]],["pop18",null,"Z18","accept18",["Z18"]]],"input":["a18","a18","b18","b18"],"trace":[["push18",0,["Z18"]],["push18",1,["Z18","A18"]],["push18",2,["Z18","A18","A18"]],["pop18",3,["Z18","A18"]],["pop18",4,["Z18"]],["accept18",4,["Z18"]]]}。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](https://ocme.evemisslab.com/zh-Hant/mko/mko-pushdown-automaton/)





## 逐步推導

S001 · implies操作前

source operation=pda-accepting-run-witness;original snapshot and mathematics retained

操作後

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

條件：無

原題歷史介面標記保留，直接列來源有序欄位，不需JavaScript；MVT由原mode選欄序，不補空欄或看答案形狀猜。tag與固定No保原聲稱範圍。

步驟識別與依賴規則：source_ordered_interface_explicit_without_javascript · 依賴：G001





S002 · implies操作前

ORIGINAL transition index=0;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push18, a18, Z18, push18, [Z18, A18]];bottom=Z18

操作後

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

條件：無

逐原transition核state、input與stack角色，不把字母表混用；null是epsilon而不是一個字母。stack bottom-first/top-last，pop一個top再按原replacement順序append；只施來源明示bottom guard，不另加更強慣例。

步驟識別與依賴規則：pda_EVERY_original_transition_roles_epsilon_bottom_guard · 依賴：G001, S001





S003 · implies操作前

ORIGINAL transition index=1;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push18, a18, A18, push18, [A18, A18]];bottom=Z18

操作後

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

條件：無

逐原transition核state、input與stack角色，不把字母表混用；null是epsilon而不是一個字母。stack bottom-first/top-last，pop一個top再按原replacement順序append；只施來源明示bottom guard，不另加更強慣例。

步驟識別與依賴規則：pda_EVERY_original_transition_roles_epsilon_bottom_guard · 依賴：G001, S002





S004 · implies操作前

ORIGINAL transition index=2;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push18, b18, A18, pop18, []];bottom=Z18

操作後

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

條件：無

逐原transition核state、input與stack角色，不把字母表混用；null是epsilon而不是一個字母。stack bottom-first/top-last，pop一個top再按原replacement順序append；只施來源明示bottom guard，不另加更強慣例。

步驟識別與依賴規則：pda_EVERY_original_transition_roles_epsilon_bottom_guard · 依賴：G001, S003





S005 · implies操作前

ORIGINAL transition index=3;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[pop18, b18, A18, pop18, []];bottom=Z18

操作後

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

條件：無

逐原transition核state、input與stack角色，不把字母表混用；null是epsilon而不是一個字母。stack bottom-first/top-last，pop一個top再按原replacement順序append；只施來源明示bottom guard，不另加更強慣例。

步驟識別與依賴規則：pda_EVERY_original_transition_roles_epsilon_bottom_guard · 依賴：G001, S004





S006 · implies操作前

ORIGINAL transition index=4;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push18, null, Z18, accept18, [Z18]];bottom=Z18

操作後

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

條件：無

逐原transition核state、input與stack角色，不把字母表混用；null是epsilon而不是一個字母。stack bottom-first/top-last，pop一個top再按原replacement順序append；只施來源明示bottom guard，不另加更強慣例。

步驟識別與依賴規則：pda_EVERY_original_transition_roles_epsilon_bottom_guard · 依賴：G001, S005





S007 · implies操作前

ORIGINAL transition index=5;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[pop18, null, Z18, accept18, [Z18]];bottom=Z18

操作後

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

條件：無

逐原transition核state、input與stack角色，不把字母表混用；null是epsilon而不是一個字母。stack bottom-first/top-last，pop一個top再按原replacement順序append；只施來源明示bottom guard，不另加更強慣例。

步驟識別與依賴規則：pda_EVERY_original_transition_roles_epsilon_bottom_guard · 依賴：G001, S006





S008 · calculation操作前

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

操作後

PDA complete ORIGINAL transition count = 6

條件：無

依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。

步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S007





S009 · implies操作前

ORIGINAL states=[push18, pop18, accept18],input alphabet=[a18, b18],stack alphabet=[Z18, A18];start=push18,bottom=Z18,final states=[accept18]

操作後

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

條件：無

有限完整定義的資格不等於存在接受run，也不推deterministic或CFG等價。接受時可保非空stack（至少bottom），但必須讀完整input到final。

步驟識別與依賴規則：pda_complete_definition_not_determinism_or_accepting_run · 依賴：G001, S008





S010 · implies操作前

ORIGINAL input=[a18, a18, b18, b18];ALL source configurations=[[push18, 0, [Z18]], [push18, 1, [Z18, A18]], [push18, 2, [Z18, A18, A18]], [pop18, 3, [Z18, A18]], [pop18, 4, [Z18]], [accept18, 4, [Z18]]];declared start=push18,bottom=Z18

操作後

first configuration EXACTLY=[push18, 0, [Z18]];all state/position/stack values legal;this is NOT yet a valid-run or acceptance certificate

條件：無

初始state、input位置0及單一bottom必恰等來源；全部config先核合法，再逐步驗transition。合法config列表不保證相鄰步驟相容。

步驟識別與依賴規則：pda_trace_ALL_configurations_qualified_actual_initial_required · 依賴：G001, S009





S011 · implies操作前

ORIGINAL trace step=0;from=[push18, 0, [Z18]];to=[push18, 1, [Z18, A18]];actual input=[a18, a18, b18, b18]

操作後

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, [push18, 1, [Z18, A18]]], [1, null], [2, null], [3, null], [4, [accept18, 0, [Z18]]], [5, null]];FIRST matching index=0

條件：無

逐原transition順序選第一個真產生nextconfig者；input-or-null決定是否消耗，stack只換top並保replacement次序。失敗保先前partial indices，後面的final state不能修好錯步。

步驟識別與依賴規則：pda_each_actual_stack_update_FIRST_original_transition · 依賴：G001, S010





S012 · implies操作前

ORIGINAL trace step=1;from=[push18, 1, [Z18, A18]];to=[push18, 2, [Z18, A18, A18]];actual input=[a18, a18, b18, b18]

操作後

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, [push18, 2, [Z18, A18, A18]]], [2, null], [3, null], [4, null], [5, null]];FIRST matching index=1

條件：無

逐原transition順序選第一個真產生nextconfig者；input-or-null決定是否消耗，stack只換top並保replacement次序。失敗保先前partial indices，後面的final state不能修好錯步。

步驟識別與依賴規則：pda_each_actual_stack_update_FIRST_original_transition · 依賴：G001, S011





S013 · implies操作前

ORIGINAL trace step=2;from=[push18, 2, [Z18, A18, A18]];to=[pop18, 3, [Z18, A18]];actual input=[a18, a18, b18, b18]

操作後

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, [pop18, 3, [Z18, A18]]], [3, null], [4, null], [5, null]];FIRST matching index=2

條件：無

逐原transition順序選第一個真產生nextconfig者；input-or-null決定是否消耗，stack只換top並保replacement次序。失敗保先前partial indices，後面的final state不能修好錯步。

步驟識別與依賴規則：pda_each_actual_stack_update_FIRST_original_transition · 依賴：G001, S012





S014 · implies操作前

ORIGINAL trace step=3;from=[pop18, 3, [Z18, A18]];to=[pop18, 4, [Z18]];actual input=[a18, a18, b18, b18]

操作後

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, [pop18, 4, [Z18]]], [4, null], [5, null]];FIRST matching index=3

條件：無

逐原transition順序選第一個真產生nextconfig者；input-or-null決定是否消耗，stack只換top並保replacement次序。失敗保先前partial indices，後面的final state不能修好錯步。

步驟識別與依賴規則：pda_each_actual_stack_update_FIRST_original_transition · 依賴：G001, S013





S015 · implies操作前

ORIGINAL trace step=4;from=[pop18, 4, [Z18]];to=[accept18, 4, [Z18]];actual input=[a18, a18, b18, b18]

操作後

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, null], [4, null], [5, [accept18, 4, [Z18]]]];FIRST matching index=5

條件：無

逐原transition順序選第一個真產生nextconfig者；input-or-null決定是否消耗，stack只換top並保replacement次序。失敗保先前partial indices，後面的final state不能修好錯步。

步驟識別與依賴規則：pda_each_actual_stack_update_FIRST_original_transition · 依賴：G001, S014





S016 · implies操作前

ORIGINAL last config=[accept18, 4, [Z18]];input length=4;matched prior indices=[0, 1, 2, 3, 5]

操作後

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

條件：無

接受是合法完整run、讀完input、到final三責任合取；可留bottom或其他stack符號。valid與accepted不同，tag標籤本身不把False變成證書。

步驟識別與依賴規則：pda_run_validity_full_consumption_final_state_SEPARATE · 依賴：G001, S015





S017 · calculation操作前

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

操作後

PDA ORIGINAL full trace length minus one = 5

條件：無

依原始角色、次序與已核中間量精確代值；不改source binding或用浮點近似。

步驟識別與依賴規則：exact_rational_arithmetic · 依賴：G001, S016





S018 · implies操作前

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

操作後

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

條件：無

原step_count按整份trace長度，即使部分步驟失敗也不改成matched次數。只有accepted=True才由完整有限run證接受，不推所有run、最短或搜尋完整。

步驟識別與依賴規則：pda_original_step_count_not_partial_matched_count · 依賴：G001, S017









## 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

([a18, a18, b18, b18], [[push18, 0, [Z18]], [push18, 1, [Z18, A18]], [push18, 2, [Z18, A18, A18]], [pop18, 3, [Z18, A18]], [pop18, 4, [Z18]], [accept18, 4, [Z18]]], [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。

來源雜湊與覆核憑證
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: cdfeb3922f3312a26a45bfe6e7febc9b8ce0f956dcc94547e63f897acc5f3558
- candidate record SHA-256: f58f09b9a280645fda6eb4687e7964e9130694f0bcbedeb9eda1c15225b39797
- semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
- question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
- Lean: NOT_RUN · human review: NOT_PERFORMED

[同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261110-r07-pushdown-automaton-0069.json)