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第 23981 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

R0046-23981 · q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0031

原題與已知條件

epsilon-closed start、move後closure與accepting-subset交集規則。操作=subsetdfa-symbol-transition；資料={"states":["s5","m5","f5","e15","e25","d5"],"alphabet":["a5","b5"],"start_state":"s5","final_states":["f5"],"transitions":[["s5",null,"m5"],["s5",null,"d5"],["m5","a5","m5"],["m5","b5","f5"],["f5",null,"e15"],["e15",null,"e25"]],"subset":["s5","m5","d5"],"symbol":"b5"}。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-symbol-transition；資料={"states":["s5","m5","f5","e15","e25","d5"],"alphabet":["a5","b5"],"start_state":"s5","final_states":["f5"],"transitions":[["s5",null,"m5"],["s5",null,"d5"],["m5","a5","m5"],["m5","b5","f5"],["f5",null,"e15"],["e15",null,"e25"]],"subset":["s5","m5","d5"],"symbol":"b5"}。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-symbol-transition;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, 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=[s5, m5, f5, e15, e25, d5],alphabet=[a5, b5],start=s5;ORIGINAL transition order=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5], [f5, null, e15], [e15, null, e25]]

操作後

final set normalized UTF16=[f5];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 one-symbol transition:EXACT ORIGINAL supplied S=[d5, m5, s5];symbol=b5;ALL matching ORIGINAL edge rows=[[3, m5, f5]]

操作後

direct symbol targets=[f5];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, S002

S004 · implies

操作前

Subset DFA ORIGINAL one-symbol transition post-symbol EPS closure:stage=0;seed set=[f5];previous set=[f5]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s5, m5, 否], [1, s5, d5, 否], [4, f5, e15, 是], [5, e15, e25, 否]];next set=[e15, f5];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, S003

S005 · implies

操作前

Subset DFA ORIGINAL one-symbol transition post-symbol EPS closure:stage=1;seed set=[f5];previous set=[e15, f5]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s5, m5, 否], [1, s5, d5, 否], [4, f5, e15, 是], [5, e15, e25, 是]];next set=[e15, e25, f5];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, S004

S006 · implies

操作前

Subset DFA ORIGINAL one-symbol transition post-symbol EPS closure:stage=2;seed set=[f5];previous set=[e15, e25, f5]

操作後

ALL [ORIGINAL edge index,from,to,enabled] EPS rows=[[0, s5, m5, 否], [1, s5, d5, 否], [4, f5, e15, 是], [5, e15, e25, 是]];next set=[e15, e25, f5];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

逐步反向核對

  • 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

([d5, m5, s5], b5, [f5], [e15, e25, f5], 否, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)

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: 23ef8a53fa8c2b40ba84430ec7828c1a1ad467eae651095ef790fc0bd9c063b4
  • candidate record SHA-256: 4b81bf63de0ad3907058a03a553006b6e2195663f87d20f3e8e61deb961fd07f
  • semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
  • question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON