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第 22471 題：下推自動機

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  1. 原題與已知條件
  2. 定義與適用範圍
  3. 逐步推導
  4. 逐步反向核對
  5. 結論
  6. 本題覆核與來源綁定

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← 逐題詳解目錄 · 原題批次

R0043-22471 · q-20261014-advance-20261110-r07-pushdown-automaton-0021

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-qualification；資料={"states":["push20","pop20","accept20"],"input_alphabet":["a20","b20"],"stack_alphabet":["Z20","A20"],"start_state":"push20","bottom_symbol":"Z20","final_states":["accept20"],"transitions":[["push20","a20","Z20","push20",["Z20","A20"]],["push20","a20","A20","push20",["A20","A20"]],["push20","b20","A20","pop20",[]],["pop20","b20","A20","pop20",[]],["push20",null,"Z20","accept20",["Z20"]],["pop20",null,"Z20","accept20",["Z20"]]]}。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-qualification；資料={"states":["push20","pop20","accept20"],"input_alphabet":["a20","b20"],"stack_alphabet":["Z20","A20"],"start_state":"push20","bottom_symbol":"Z20","final_states":["accept20"],"transitions":[["push20","a20","Z20","push20",["Z20","A20"]],["push20","a20","A20","push20",["A20","A20"]],["push20","b20","A20","pop20",[]],["pop20","b20","A20","pop20",[]],["push20",null,"Z20","accept20",["Z20"]],["pop20",null,"Z20","accept20",["Z20"]]]}。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。

逐步推導

S001 · implies

操作前

source operation=pda-qualification;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention]

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]=[push20, a20, Z20, push20, [Z20, A20]];bottom=Z20

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]=[push20, a20, A20, push20, [A20, A20]];bottom=Z20

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]=[push20, b20, A20, pop20, []];bottom=Z20

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]=[pop20, b20, A20, pop20, []];bottom=Z20

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]=[push20, null, Z20, accept20, [Z20]];bottom=Z20

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]=[pop20, null, Z20, accept20, [Z20]];bottom=Z20

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=[push20, pop20, accept20],input alphabet=[a20, b20],stack alphabet=[Z20, A20];start=push20,bottom=Z20,final states=[accept20]

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

逐步反向核對

  • 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

([push20, pop20, accept20], [a20, b20], [Z20, A20], push20, Z20, [accept20], 6, 是, accept_only_after_full_input_in_final_state_bottom_marker_preserved)

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: 68be3fc5554e2c1a1c40152fa250072970c7e6c491ebc5b445c894fa16b5581d
  • candidate record SHA-256: 15b272fcb2a7958d922d8b436c09d65c84553b005cd5971e78f36f409270f1e8
  • semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
  • question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON