第 22410 題：下推自動機

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

R0043-22410 · q-20261014-advance-20261110-r07-pushdown-automaton-0060

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":["push9","pop9","accept9"],"input_alphabet":["a9","b9"],"stack_alphabet":["Z9","A9"],"start_state":"push9","bottom_symbol":"Z9","final_states":["accept9"],"transitions":[["push9","a9","Z9","push9",["Z9","A9"]],["push9","a9","A9","push9",["A9","A9"]],["push9","b9","A9","pop9",[]],["pop9","b9","A9","pop9",[]],["push9",null,"Z9","accept9",["Z9"]],["pop9",null,"Z9","accept9",["Z9"]]],"input":["a9","b9"],"trace":[["push9",0,["Z9"]],["push9",1,["Z9","A9"]],["pop9",2,["Z9"]],["accept9",2,["Z9"]]]}。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":["push9","pop9","accept9"],"input_alphabet":["a9","b9"],"stack_alphabet":["Z9","A9"],"start_state":"push9","bottom_symbol":"Z9","final_states":["accept9"],"transitions":[["push9","a9","Z9","push9",["Z9","A9"]],["push9","a9","A9","push9",["A9","A9"]],["push9","b9","A9","pop9",[]],["pop9","b9","A9","pop9",[]],["push9",null,"Z9","accept9",["Z9"]],["pop9",null,"Z9","accept9",["Z9"]]],"input":["a9","b9"],"trace":[["push9",0,["Z9"]],["push9",1,["Z9","A9"]],["pop9",2,["Z9"]],["accept9",2,["Z9"]]]}。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

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]=[push9, a9, Z9, push9, [Z9, A9]];bottom=Z9

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]=[push9, a9, A9, push9, [A9, A9]];bottom=Z9

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

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]=[push9, b9, A9, pop9, []];bottom=Z9

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

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]=[pop9, b9, A9, pop9, []];bottom=Z9

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

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]=[push9, null, Z9, accept9, [Z9]];bottom=Z9

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

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]=[pop9, null, Z9, accept9, [Z9]];bottom=Z9

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

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=[push9, pop9, accept9],input alphabet=[a9, b9],stack alphabet=[Z9, A9];start=push9,bottom=Z9,final states=[accept9]

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=[a9, b9];ALL source configurations=[[push9, 0, [Z9]], [push9, 1, [Z9, A9]], [pop9, 2, [Z9]], [accept9, 2, [Z9]]];declared start=push9,bottom=Z9

操作後

first configuration EXACTLY=[push9, 0, [Z9]];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=[push9, 0, [Z9]];to=[push9, 1, [Z9, A9]];actual input=[a9, b9]

操作後

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, [push9, 1, [Z9, A9]]], [1, null], [2, null], [3, null], [4, [accept9, 0, [Z9]]], [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=[push9, 1, [Z9, A9]];to=[pop9, 2, [Z9]];actual input=[a9, b9]

操作後

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, [pop9, 2, [Z9]]], [3, null], [4, null], [5, null]];FIRST matching index=2

條件：無

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=[pop9, 2, [Z9]];to=[accept9, 2, [Z9]];actual input=[a9, b9]

操作後

ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, null], [4, null], [5, [accept9, 2, [Z9]]]];FIRST matching index=5

條件：無

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 last config=[accept9, 2, [Z9]];input length=2;matched prior indices=[0, 2, 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, S013

S015 · calculation

操作前

PDA ORIGINAL full trace length minus one := sub(4, 1)

操作後

PDA ORIGINAL full trace length minus one = 3

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, S014

S016 · implies

操作前

ORIGINAL trace configuration count=4;actual matched transition-index count=3;accepted=是

操作後

SOURCE step_count=ORIGINAL trace.length-1=3;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, S015

逐步反向核對

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 — 以精確有理數重算 PDA ORIGINAL full trace length minus one，不以小數近似替代。
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

([a9, b9], [[push9, 0, [Z9]], [push9, 1, [Z9, A9]], [pop9, 2, [Z9]], [accept9, 2, [Z9]]], [0, 2, 5], 是, 是, 3, 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: d5ad7a39b12b503887352900971dd11e6b521b20d06c31e7566244d197507be8
candidate record SHA-256: 0825b70364bc6ff3d8c02a641d6cbcb07aca502d24e5641aa27a11959fe37342
semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
Lean: NOT_RUN · human review: NOT_PERFORMED

同源完整 JSON