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第 23949 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

R0046-23949 · q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0024

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":["s23","m23","f23","e123","e223","d23"],"alphabet":["a23","b23"],"start_state":"s23","final_states":["f23"],"transitions":[["s23",null,"m23"],["s23",null,"d23"],["m23","a23","m23"],["m23","b23","f23"],["f23",null,"e123"],["e123",null,"e223"]]}。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":["s23","m23","f23","e123","e223","d23"],"alphabet":["a23","b23"],"start_state":"s23","final_states":["f23"],"transitions":[["s23",null,"m23"],["s23",null,"d23"],["m23","a23","m23"],["m23","b23","f23"],["f23",null,"e123"],["e123",null,"e223"]]}。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=[s23, m23, f23, e123, e223, d23],alphabet=[a23, b23],start=s23;ORIGINAL transition order=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23], [f23, null, e123], [e123, null, e223]]

操作後

final set normalized UTF16=[f23];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=[s23];previous set=[s23]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s23, m23, 是], [1, s23, d23, 是], [4, f23, e123, 否], [5, e123, e223, 否]];next set=[d23, m23, s23];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=[s23];previous set=[d23, m23, s23]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s23, m23, 是], [1, s23, d23, 是], [4, f23, e123, 否], [5, e123, e223, 否]];next set=[d23, m23, s23];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

(s23, [d23, m23, s23], 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: 8bd0194677528c324cc7ddb2e12d81a8661b6022b9982073095e37ff7a846942
  • candidate record SHA-256: 4c21a2947c7af76611e443d57bec6cb2c421200e80de699d89148f1205cd93c0
  • semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
  • question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON