第 23953 題：NFA到DFA的子集合構造

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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

R0046-23953 · q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0078

原題與已知條件

epsilon-closed start、move後closure與accepting-subset交集規則。操作=subsetdfa-full-word-trace-invariant；資料={"states":["s2","m2","f2","e12","e22","d2"],"alphabet":["a2","b2"],"start_state":"s2","final_states":["f2"],"transitions":[["s2",null,"m2"],["s2",null,"d2"],["m2","a2","m2"],["m2","b2","f2"],["f2",null,"e12"],["e12",null,"e22"]],"input":["a2","a2","b2"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

G001 — epsilon-closed start、move後closure與accepting-subset交集規則。操作=subsetdfa-full-word-trace-invariant；資料={"states":["s2","m2","f2","e12","e22","d2"],"alphabet":["a2","b2"],"start_state":"s2","final_states":["f2"],"transitions":[["s2",null,"m2"],["s2",null,"d2"],["m2","a2","m2"],["m2","b2","f2"],["f2",null,"e12"],["e12",null,"e22"]],"input":["a2","a2","b2"]}。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-subset-construction-nfa-to-dfa

逐步推導

S001 · implies

操作前

source operation=subsetdfa-full-word-trace-invariant;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary]

條件：無

保留原題介面快照，本頁直接列來源欄序，不需JavaScript。radius由原mode選6／11欄，不補欄；No與實際否定依來源責任分清。

步驟識別與依賴

規則：source_ordered_interface_explicit_without_javascript · 依賴：G001

S002 · implies

操作前

ORIGINAL states=[s2, m2, f2, e12, e22, d2],alphabet=[a2, b2],start=s2;ORIGINAL transition order=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2], [f2, null, e12], [e12, null, e22]]

操作後

final set normalized UTF16=[f2];states nonempty,alphabet/finals/relation MAYEMPTY;DUPLICATE triples rejected in THIS model

條件：無

此題與前一NFA節點不同：transition triple須唯一，finals須排序。所有subset都是真carrier子集合；不偷用舊wrapper的duplicate／final-order慣例。

步驟識別與依賴

規則：subsetdfa_qualified_NFA_signature_DUPLICATE_edges_REJECTED_here · 依賴：G001, S001

S003 · implies

操作前

Subset DFA ORIGINAL whole-word start:stage=0;seed set=[s2];previous set=[s2]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s2, m2, 是], [1, s2, d2, 是], [4, f2, e12, 否], [5, e12, e22, 否]];next set=[d2, m2, s2];stable=否

條件：無

新增state都有前一層有限EPS路徑，任何含seed的EPS封閉集合都必含每層；有限states使飽和終止，得到完整且最小closure，不消耗input。

步驟識別與依賴

規則：subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · 依賴：G001, S002

S004 · implies

操作前

Subset DFA ORIGINAL whole-word start:stage=1;seed set=[s2];previous set=[d2, m2, s2]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s2, m2, 是], [1, s2, d2, 是], [4, f2, e12, 否], [5, e12, e22, 否]];next set=[d2, m2, s2];stable=是

條件：無

新增state都有前一層有限EPS路徑，任何含seed的EPS封閉集合都必含每層；有限states使飽和終止，得到完整且最小closure，不消耗input。

步驟識別與依賴

規則：subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · 依賴：G001, S003

S005 · implies

操作前

Subset DFA ORIGINAL prefix step0:EXACT ORIGINAL supplied S=[d2, m2, s2];symbol=a2;ALL matching ORIGINAL edge rows=[[2, m2, m2]]

操作後

direct symbol targets=[m2];NO EPS preclosure inserted into S;now close ONLY these targets

條件：無

任意給定S未必已EPS-closed，先preclose會改本題運算；定義明示先move(S,a)再closure。empty move得到合法empty dead state。

步驟識別與依賴

規則：subsetdfa_GIVEN_S_symbol_move_BEFORE_closure_NO_preclose · 依賴：G001, S004

S006 · implies

操作前

Subset DFA ORIGINAL prefix step0 post-symbol EPS closure:stage=0;seed set=[m2];previous set=[m2]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s2, m2, 否], [1, s2, d2, 否], [4, f2, e12, 否], [5, e12, e22, 否]];next set=[m2];stable=是

條件：無

新增state都有前一層有限EPS路徑，任何含seed的EPS封閉集合都必含每層；有限states使飽和終止，得到完整且最小closure，不消耗input。

步驟識別與依賴

規則：subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · 依賴：G001, S005

S007 · implies

操作前

Subset DFA ORIGINAL prefix step1:EXACT ORIGINAL supplied S=[m2];symbol=a2;ALL matching ORIGINAL edge rows=[[2, m2, m2]]

操作後

direct symbol targets=[m2];NO EPS preclosure inserted into S;now close ONLY these targets

條件：無

任意給定S未必已EPS-closed，先preclose會改本題運算；定義明示先move(S,a)再closure。empty move得到合法empty dead state。

步驟識別與依賴

規則：subsetdfa_GIVEN_S_symbol_move_BEFORE_closure_NO_preclose · 依賴：G001, S006

S008 · implies

操作前

Subset DFA ORIGINAL prefix step1 post-symbol EPS closure:stage=0;seed set=[m2];previous set=[m2]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s2, m2, 否], [1, s2, d2, 否], [4, f2, e12, 否], [5, e12, e22, 否]];next set=[m2];stable=是

條件：無

新增state都有前一層有限EPS路徑，任何含seed的EPS封閉集合都必含每層；有限states使飽和終止，得到完整且最小closure，不消耗input。

步驟識別與依賴

規則：subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · 依賴：G001, S007

S009 · implies

操作前

Subset DFA ORIGINAL prefix step2:EXACT ORIGINAL supplied S=[m2];symbol=b2;ALL matching ORIGINAL edge rows=[[3, m2, f2]]

操作後

direct symbol targets=[f2];NO EPS preclosure inserted into S;now close ONLY these targets

條件：無

任意給定S未必已EPS-closed，先preclose會改本題運算；定義明示先move(S,a)再closure。empty move得到合法empty dead state。

步驟識別與依賴

規則：subsetdfa_GIVEN_S_symbol_move_BEFORE_closure_NO_preclose · 依賴：G001, S008

S010 · implies

操作前

Subset DFA ORIGINAL prefix step2 post-symbol EPS closure:stage=0;seed set=[f2];previous set=[f2]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s2, m2, 否], [1, s2, d2, 否], [4, f2, e12, 是], [5, e12, e22, 否]];next set=[e12, f2];stable=否

條件：無

新增state都有前一層有限EPS路徑，任何含seed的EPS封閉集合都必含每層；有限states使飽和終止，得到完整且最小closure，不消耗input。

步驟識別與依賴

規則：subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · 依賴：G001, S009

S011 · implies

操作前

Subset DFA ORIGINAL prefix step2 post-symbol EPS closure:stage=1;seed set=[f2];previous set=[e12, f2]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s2, m2, 否], [1, s2, d2, 否], [4, f2, e12, 是], [5, e12, e22, 是]];next set=[e12, e22, f2];stable=否

條件：無

新增state都有前一層有限EPS路徑，任何含seed的EPS封閉集合都必含每層；有限states使飽和終止，得到完整且最小closure，不消耗input。

步驟識別與依賴

規則：subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · 依賴：G001, S010

S012 · implies

操作前

Subset DFA ORIGINAL prefix step2 post-symbol EPS closure:stage=2;seed set=[f2];previous set=[e12, e22, f2]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s2, m2, 否], [1, s2, d2, 否], [4, f2, e12, 是], [5, e12, e22, 是]];next set=[e12, e22, f2];stable=是

條件：無

新增state都有前一層有限EPS路徑，任何含seed的EPS封閉集合都必含每層；有限states使飽和終止，得到完整且最小closure，不消耗input。

步驟識別與依賴

規則：subsetdfa_ALL_EPS_fixedpoint_reachability_not_input_consumption · 依賴：G001, S011

S013 · implies

操作前

ORIGINAL word=[a2, a2, b2];direct NFA configuration stage=0;previous configs=[[s2, 0]]

操作後

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s2, 0], 0, [m2, 0]], [[s2, 0], 1, [d2, 0]]];next configs=[[s2, 0], [m2, 0], [d2, 0]];stable=否

條件：無

這條直接路徑在完整finite state×input-position圖上求所有reachable，不把一條chosen run當成全狀態，也不把DFA同一函式重跑冒充不同觀察。

步驟識別與依賴

規則：subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · 依賴：G001, S012

S014 · implies

操作前

ORIGINAL word=[a2, a2, b2];direct NFA configuration stage=1;previous configs=[[s2, 0], [m2, 0], [d2, 0]]

操作後

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s2, 0], 0, [m2, 0]], [[s2, 0], 1, [d2, 0]], [[m2, 0], 2, [m2, 1]]];next configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1]];stable=否

條件：無

這條直接路徑在完整finite state×input-position圖上求所有reachable，不把一條chosen run當成全狀態，也不把DFA同一函式重跑冒充不同觀察。

步驟識別與依賴

規則：subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · 依賴：G001, S013

S015 · implies

操作前

ORIGINAL word=[a2, a2, b2];direct NFA configuration stage=2;previous configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1]]

操作後

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s2, 0], 0, [m2, 0]], [[s2, 0], 1, [d2, 0]], [[m2, 0], 2, [m2, 1]], [[m2, 1], 2, [m2, 2]]];next configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2]];stable=否

條件：無

這條直接路徑在完整finite state×input-position圖上求所有reachable，不把一條chosen run當成全狀態，也不把DFA同一函式重跑冒充不同觀察。

步驟識別與依賴

規則：subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · 依賴：G001, S014

S016 · implies

操作前

ORIGINAL word=[a2, a2, b2];direct NFA configuration stage=3;previous configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2]]

操作後

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s2, 0], 0, [m2, 0]], [[s2, 0], 1, [d2, 0]], [[m2, 0], 2, [m2, 1]], [[m2, 1], 2, [m2, 2]], [[m2, 2], 3, [f2, 3]]];next configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2], [f2, 3]];stable=否

條件：無

這條直接路徑在完整finite state×input-position圖上求所有reachable，不把一條chosen run當成全狀態，也不把DFA同一函式重跑冒充不同觀察。

步驟識別與依賴

規則：subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · 依賴：G001, S015

S017 · implies

操作前

ORIGINAL word=[a2, a2, b2];direct NFA configuration stage=4;previous configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2], [f2, 3]]

操作後

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s2, 0], 0, [m2, 0]], [[s2, 0], 1, [d2, 0]], [[m2, 0], 2, [m2, 1]], [[m2, 1], 2, [m2, 2]], [[m2, 2], 3, [f2, 3]], [[f2, 3], 4, [e12, 3]]];next configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2], [f2, 3], [e12, 3]];stable=否

條件：無

這條直接路徑在完整finite state×input-position圖上求所有reachable，不把一條chosen run當成全狀態，也不把DFA同一函式重跑冒充不同觀察。

步驟識別與依賴

規則：subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · 依賴：G001, S016

S018 · implies

操作前

ORIGINAL word=[a2, a2, b2];direct NFA configuration stage=5;previous configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2], [f2, 3], [e12, 3]]

操作後

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s2, 0], 0, [m2, 0]], [[s2, 0], 1, [d2, 0]], [[m2, 0], 2, [m2, 1]], [[m2, 1], 2, [m2, 2]], [[m2, 2], 3, [f2, 3]], [[f2, 3], 4, [e12, 3]], [[e12, 3], 5, [e22, 3]]];next configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2], [f2, 3], [e12, 3], [e22, 3]];stable=否

條件：無

這條直接路徑在完整finite state×input-position圖上求所有reachable，不把一條chosen run當成全狀態，也不把DFA同一函式重跑冒充不同觀察。

步驟識別與依賴

規則：subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · 依賴：G001, S017

S019 · implies

操作前

ORIGINAL word=[a2, a2, b2];direct NFA configuration stage=6;previous configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2], [f2, 3], [e12, 3], [e22, 3]]

操作後

ALL enabled [from,ORIGINAL edge index,next] rows=[[[s2, 0], 0, [m2, 0]], [[s2, 0], 1, [d2, 0]], [[m2, 0], 2, [m2, 1]], [[m2, 1], 2, [m2, 2]], [[m2, 2], 3, [f2, 3]], [[f2, 3], 4, [e12, 3]], [[e12, 3], 5, [e22, 3]]];next configs=[[s2, 0], [m2, 0], [d2, 0], [m2, 1], [m2, 2], [f2, 3], [e12, 3], [e22, 3]];stable=是

條件：無

這條直接路徑在完整finite state×input-position圖上求所有reachable，不把一條chosen run當成全狀態，也不把DFA同一函式重跑冒充不同觀察。

步驟識別與依賴

規則：subsetdfa_DIRECT_all_NFA_word_configurations_independent_fixedpoint · 依賴：G001, S018

S020 · implies

操作前

ORIGINAL word=[a2, a2, b2];ALL DFA prefix subsets=[[d2, m2, s2], [m2], [m2], [e12, e22, f2]];independent direct NFA prefix subsets=[[d2, m2, s2], [m2], [m2], [e12, e22, f2]]

操作後

prefix_invariant=是;accepted=是;base is EPS closure of start;for ANY prefix,append one symbol move then finite EPS suffix to obtain EXACTLY all next reachable states

條件：無

一般等價要對任意prefix歸納並證兩個包含：DFA步驟有NFA路徑見證；任一多讀一字的NFA路徑可切成前prefix、symbol邊、EPS後綴。再以與final交非空對應存在完整接受run。最後固定No只拒絕由有限word tests單獨宣稱一般等價，不否定本invariant、accepted或一般定理。

步驟識別與依賴

規則：subsetdfa_GENERAL_arbitrary_prefix_two_inclusions_acceptance_rule · 依賴：G001, S019

逐步反向核對

S020 · 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。
S019 · 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。
S018 · 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。
S017 · 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。
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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
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 — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
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

([a2, a2, b2], [[d2, m2, s2], [m2], [m2], [e12, e22, f2]], [[d2, m2, s2], [m2], [m2], [e12, e22, f2]], 是, 是, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, 否, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)

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: c0453781916f1700601cc04ea9f318b3097a9b570b494bc5f611a3a86157dcce
candidate record SHA-256: 6b3a0171edae33bd387f380b050b25d86c52cacaf39c0d452a1c8bb8929d6c2c
semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
Lean: NOT_RUN · human review: NOT_PERFORMED

同源完整 JSON