# 第 22442 題：下推自動機 [返回所在分頁 113](https://ocme.evemisslab.com/zh-Hant/questions/page/113/) [← 逐題詳解目錄](https://ocme.evemisslab.com/zh-Hant/questions/) · [原題批次](https://ocme.evemisslab.com/zh-Hant/practice/batch-20261014-advance-20261110-r07-pushdown-automaton-foundations/#q-20261014-advance-20261110-r07-pushdown-automaton-0092) 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。 R0043-22442 · q-20261014-advance-20261110-r07-pushdown-automaton-0092 ## 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":["push16","pop16","accept16"],"input_alphabet":["a16","b16"],"stack_alphabet":["Z16","A16"],"start_state":"push16","bottom_symbol":"Z16","final_states":["accept16"],"transitions":[["push16","a16","Z16","push16",["Z16","A16"]],["push16","a16","A16","push16",["A16","A16"]],["push16","b16","A16","pop16",[]],["pop16","b16","A16","pop16",[]],["push16",null,"Z16","accept16",["Z16"]],["pop16",null,"Z16","accept16",["Z16"]]],"input":["a16","b16"],"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":["push16","pop16","accept16"],"input_alphabet":["a16","b16"],"stack_alphabet":["Z16","A16"],"start_state":"push16","bottom_symbol":"Z16","final_states":["accept16"],"transitions":[["push16","a16","Z16","push16",["Z16","A16"]],["push16","a16","A16","push16",["A16","A16"]],["push16","b16","A16","pop16",[]],["pop16","b16","A16","pop16",[]],["push16",null,"Z16","accept16",["Z16"]],["pop16",null,"Z16","accept16",["Z16"]]],"input":["a16","b16"],"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](https://ocme.evemisslab.com/zh-Hant/mko/mko-pushdown-automaton/) ## 逐步推導 S001 · implies操作前 source operation=pda-bounded-search-boundary;original snapshot and mathematics retained 操作後 explicit ordered source output fields=[input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary] 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]=[push16, a16, Z16, push16, [Z16, A16]];bottom=Z16 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]=[push16, a16, A16, push16, [A16, A16]];bottom=Z16 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]=[push16, b16, A16, pop16, []];bottom=Z16 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]=[pop16, b16, A16, pop16, []];bottom=Z16 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]=[push16, null, Z16, accept16, [Z16]];bottom=Z16 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]=[pop16, null, Z16, accept16, [Z16]];bottom=Z16 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=[push16, pop16, accept16],input alphabet=[a16, b16],stack alphabet=[Z16, A16];start=push16,bottom=Z16,final states=[accept16] 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=[a16, b16];initial configuration=[push16, 0, [Z16]];max_steps=1;stack cap=5 操作後 depth0 fully-consumed final-state acceptance=否;both STEP and STACK limits apply;not an unbounded rejection test 條件：無 起點也可能已接受空input；搜尋同時限制步數與stack長度，不以empty stack替代final-state慣例。 步驟識別與依賴規則：pda_bounded_initial_acceptance_and_TWO_search_limits · 依賴：G001, S009 S011 · implies操作前 ORIGINAL search depth=0;ALL frontier configs=[[push16, 0, [Z16]]];stack cap=5 操作後 ALL [from,ORIGINAL transition index,next_or_null,within_stack_cap,accepts_IN_THIS_SEARCH] rows=[[[push16, 0, [Z16]], 0, [push16, 1, [Z16, A16]], 是, 否], [[push16, 0, [Z16]], 1, null, 否, 否], [[push16, 0, [Z16]], 2, null, 否, 否], [[push16, 0, [Z16]], 3, null, 否, 否], [[push16, 0, [Z16]], 4, [accept16, 0, [Z16]], 是, 否], [[push16, 0, [Z16]], 5, null, 否, 否]];next distinct frontier=[[accept16, 0, [Z16]], [push16, 1, [Z16, A16]]];hit=否 條件：無 逐全部config與原transition嘗試；先按來源stack cap丟棄超長候選，才核本次搜尋接受。可展示整層不靠隱藏剪枝；同層去重不改存在性。 步驟識別與依賴規則：pda_ALL_nondeterministic_frontier_steps_stack_filter_BEFORE_acceptance · 依賴：G001, S010 S012 · implies操作前 ORIGINAL input=[a16, b16];step bound=1;stack bound=5;found_accepting_run=否 操作後 rejection_or_nonmembership_claim fixed No;longer or larger-stack accepting runs have NOT been excluded 條件：無 找到只對此搜尋給有限witness，找不到不證所有run皆拒絕；最後No也不否認前面found=True。未推determinism或CFG等價。 步驟識別與依賴規則：pda_bounded_nohit_NOT_rejection_or_nonmembership · 依賴：G001, S011 ## 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 ([a16, b16], 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。 來源雜湊與覆核憑證 - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 4825b0dc775fbc7076d44fa466a5ff326ce029a081cc30bddc3cdfea4102b4b8 - candidate record SHA-256: 0335834736eaafbf48e80ff48817083871da30fae879d52c22cd4580b7b070d4 - 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-0092.json)