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第 22500 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

R0043-22500 · q-20261014-advance-20261110-r07-pushdown-automaton-0050

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-trace-audit；資料={"states":["push24","pop24","accept24"],"input_alphabet":["a24","b24"],"stack_alphabet":["Z24","A24"],"start_state":"push24","bottom_symbol":"Z24","final_states":["accept24"],"transitions":[["push24","a24","Z24","push24",["Z24","A24"]],["push24","a24","A24","push24",["A24","A24"]],["push24","b24","A24","pop24",[]],["pop24","b24","A24","pop24",[]],["push24",null,"Z24","accept24",["Z24"]],["pop24",null,"Z24","accept24",["Z24"]]],"input":[],"trace":[["push24",0,["Z24"]],["accept24",0,["Z24"]]]}。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-trace-audit；資料={"states":["push24","pop24","accept24"],"input_alphabet":["a24","b24"],"stack_alphabet":["Z24","A24"],"start_state":"push24","bottom_symbol":"Z24","final_states":["accept24"],"transitions":[["push24","a24","Z24","push24",["Z24","A24"]],["push24","a24","A24","push24",["A24","A24"]],["push24","b24","A24","pop24",[]],["pop24","b24","A24","pop24",[]],["push24",null,"Z24","accept24",["Z24"]],["pop24",null,"Z24","accept24",["Z24"]]],"input":[],"trace":[["push24",0,["Z24"]],["accept24",0,["Z24"]]]}。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-trace-audit;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[input, trace, transition_indices, valid_run, accepted, 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]=[push24, a24, Z24, push24, [Z24, A24]];bottom=Z24

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]=[push24, a24, A24, push24, [A24, A24]];bottom=Z24

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]=[push24, b24, A24, pop24, []];bottom=Z24

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]=[pop24, b24, A24, pop24, []];bottom=Z24

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]=[push24, null, Z24, accept24, [Z24]];bottom=Z24

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]=[pop24, null, Z24, accept24, [Z24]];bottom=Z24

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=[push24, pop24, accept24],input alphabet=[a24, b24],stack alphabet=[Z24, A24];start=push24,bottom=Z24,final states=[accept24]

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=[];ALL source configurations=[[push24, 0, [Z24]], [accept24, 0, [Z24]]];declared start=push24,bottom=Z24

操作後

first configuration EXACTLY=[push24, 0, [Z24]];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=[push24, 0, [Z24]];to=[accept24, 0, [Z24]];actual input=[]

操作後

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, null], [4, [accept24, 0, [Z24]]], [5, null]];FIRST matching index=4

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 last config=[accept24, 0, [Z24]];input length=0;matched prior indices=[4]

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, 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

([], [[push24, 0, [Z24]], [accept24, 0, [Z24]]], [4], 是, 是, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)

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: 2f10df3d92dbb9e5cc425dd7b4937d1c7b72f1c2cd0fd8c6c829e5842744fb63
  • candidate record SHA-256: 20bfb9121e47f2933cfce0b60501fb26421c555ce869738c2499e670827945ce
  • semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
  • question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON