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第 23946 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

R0046-23946 · q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0021

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":["s20","m20","f20","e120","e220","d20"],"alphabet":["a20","b20"],"start_state":"s20","final_states":["f20"],"transitions":[["s20",null,"m20"],["s20",null,"d20"],["m20","a20","m20"],["m20","b20","f20"],["f20",null,"e120"],["e120",null,"e220"]]}。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":["s20","m20","f20","e120","e220","d20"],"alphabet":["a20","b20"],"start_state":"s20","final_states":["f20"],"transitions":[["s20",null,"m20"],["s20",null,"d20"],["m20","a20","m20"],["m20","b20","f20"],["f20",null,"e120"],["e120",null,"e220"]]}。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=[s20, m20, f20, e120, e220, d20],alphabet=[a20, b20],start=s20;ORIGINAL transition order=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20], [f20, null, e120], [e120, null, e220]]

操作後

final set normalized UTF16=[f20];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=[s20];previous set=[s20]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s20, m20, 是], [1, s20, d20, 是], [4, f20, e120, 否], [5, e120, e220, 否]];next set=[d20, m20, s20];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=[s20];previous set=[d20, m20, s20]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s20, m20, 是], [1, s20, d20, 是], [4, f20, e120, 否], [5, e120, e220, 否]];next set=[d20, m20, s20];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

(s20, [d20, m20, s20], 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: 0e6ae8f52e248e368b79057ee3e6ac5e78d2f38886772cb4a7e40b3599637f04
  • candidate record SHA-256: 110ce003732ad32f2215a6e69cdffbd3650659fae88da97d8a4495e9c86c90fe
  • semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
  • question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON