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第 23950 題：NFA到DFA的子集合構造

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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

R0046-23950 · q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0025

原題與已知條件

epsilon-closed start、move後closure與accepting-subset交集規則。操作=subsetdfa-epsilon-start；資料={"states":["s24","m24","f24","e124","e224","d24"],"alphabet":["a24","b24"],"start_state":"s24","final_states":["f24"],"transitions":[["s24",null,"m24"],["s24",null,"d24"],["m24","a24","m24"],["m24","b24","f24"],["f24",null,"e124"],["e124",null,"e224"]]}。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-epsilon-start；資料={"states":["s24","m24","f24","e124","e224","d24"],"alphabet":["a24","b24"],"start_state":"s24","final_states":["f24"],"transitions":[["s24",null,"m24"],["s24",null,"d24"],["m24","a24","m24"],["m24","b24","f24"],["f24",null,"e124"],["e124",null,"e224"]]}。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=subsetdfa-epsilon-start;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate]

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=[s24, m24, f24, e124, e224, d24],alphabet=[a24, b24],start=s24;ORIGINAL transition order=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24], [f24, null, e124], [e124, null, e224]]

操作後

final set normalized UTF16=[f24];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 start:stage=0;seed set=[s24];previous set=[s24]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s24, m24, 是], [1, s24, d24, 是], [4, f24, e124, 否], [5, e124, e224, 否]];next set=[d24, m24, s24];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 start:stage=1;seed set=[s24];previous set=[d24, m24, s24]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s24, m24, 是], [1, s24, d24, 是], [4, f24, e124, 否], [5, e124, e224, 否]];next set=[d24, m24, s24];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

逐步反向核對

  • 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

(s24, [d24, m24, s24], 是, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)

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: 69e95ecacd7df2598777a5e0bddb734a274c26b9b2b24e12e9742fd2a9fa3b33
  • candidate record SHA-256: bc686059e1b7e18dc2c0861a9444f27e13e169308ab616acf64f404a19e19947
  • semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
  • question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON