# 第 22427 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-0077) 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。 R0043-22427 · q-20261014-advance-20261110-r07-pushdown-automaton-0077 ## 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](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]=[push1, a1, Z1, push1, [Z1, A1]];bottom=Z1 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]=[push1, a1, A1, push1, [A1, A1]];bottom=Z1 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]=[push1, b1, A1, pop1, []];bottom=Z1 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]=[pop1, b1, A1, pop1, []];bottom=Z1 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]=[push1, null, Z1, accept1, [Z1]];bottom=Z1 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]=[pop1, null, Z1, accept1, [Z1]];bottom=Z1 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=[push1, pop1, accept1],input alphabet=[a1, b1],stack alphabet=[Z1, A1];start=push1,bottom=Z1,final states=[accept1] 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=[a1, a1, b1, b1];initial configuration=[push1, 0, [Z1]];max_steps=1;stack cap=7 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=[[push1, 0, [Z1]]];stack cap=7 操作後 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 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=[a1, a1, b1, b1];step bound=1;stack bound=7;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 ([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。 來源雜湊與覆核憑證 - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 1f64319b91a2c1f03a32bcdfbcd6a0aa8b9b2a92d454624009e35a0ab0adb309 - candidate record SHA-256: ec5d278434b48877e47c22f26b768b97290c024be9780356ef4c6f29c03581db - 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-0077.json)