# 第 23267 題：非確定有限自動機 [返回所在分頁 117](https://ocme.evemisslab.com/zh-Hant/questions/page/117/) [← 逐題詳解目錄](https://ocme.evemisslab.com/zh-Hant/questions/) · [原題批次](https://ocme.evemisslab.com/zh-Hant/practice/batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations/#q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0042) 本頁是已完成第二對話 AI 數理與中英語義覆核的教學詳解，只支持本題明示的模型與前提。沒有人工審查或新 Lean 證明，也不提升原題或 MKO 的 Canonical 真值狀態。 R0045-23267 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0042 ## 原題與已知條件 transition=[from,input-or-null,to]且允許empty alphabet/final set的finite epsilon-NFA。操作=nfa-epsilon-closure；資料={"states":["s16","m16","f16","d16"],"alphabet":["a16","b16"],"start_state":"s16","final_states":["f16"],"transitions":[["s16",null,"m16"],["s16",null,"d16"],["m16","a16","m16"],["m16","b16","f16"]],"seed_states":["s16"]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。 - G001 — transition=[from,input-or-null,to]且允許empty alphabet/final set的finite epsilon-NFA。操作=nfa-epsilon-closure；資料={"states":["s16","m16","f16","d16"],"alphabet":["a16","b16"],"start_state":"s16","final_states":["f16"],"transitions":[["s16",null,"m16"],["s16",null,"d16"],["m16","a16","m16"],["m16","b16","f16"]],"seed_states":["s16"]}。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-nondeterministic-finite-automaton](https://ocme.evemisslab.com/zh-Hant/mko/mko-nondeterministic-finite-automaton/) ## 逐步推導 S001 · implies操作前 source operation=nfa-epsilon-closure;original snapshot and mathematics retained 操作後 explicit ordered source output fields=[seed_states, epsilon_closure, closure_certificate, certificate] 條件：無 原題快照保留歷史介面；此處直接列來源欄序，不依賴JavaScript。tag與No保聲稱範圍，固定來源排序不冒充數學證明。 步驟識別與依賴規則：source_ordered_interface_explicit_without_javascript · 依賴：G001 S002 · implies操作前 ORIGINAL transition index=0;[from,input_or_null,to]=[s16, null, m16];state carrier=[s16, m16, f16, d16];alphabet=[a16, b16] 操作後 state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows 條件：無 這是relation不是單值total函數，duplicate／多值／缺邊皆保原資料；null消耗0但不是一個alphabet字串。 步驟識別與依賴規則：nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · 依賴：G001, S001 S003 · implies操作前 ORIGINAL transition index=1;[from,input_or_null,to]=[s16, null, d16];state carrier=[s16, m16, f16, d16];alphabet=[a16, b16] 操作後 state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows 條件：無 這是relation不是單值total函數，duplicate／多值／缺邊皆保原資料；null消耗0但不是一個alphabet字串。 步驟識別與依賴規則：nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · 依賴：G001, S002 S004 · implies操作前 ORIGINAL transition index=2;[from,input_or_null,to]=[m16, a16, m16];state carrier=[s16, m16, f16, d16];alphabet=[a16, b16] 操作後 state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows 條件：無 這是relation不是單值total函數，duplicate／多值／缺邊皆保原資料；null消耗0但不是一個alphabet字串。 步驟識別與依賴規則：nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · 依賴：G001, S003 S005 · implies操作前 ORIGINAL transition index=3;[from,input_or_null,to]=[m16, b16, f16];state carrier=[s16, m16, f16, d16];alphabet=[a16, b16] 操作後 state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows 條件：無 這是relation不是單值total函數，duplicate／多值／缺邊皆保原資料；null消耗0但不是一個alphabet字串。 步驟識別與依賴規則：nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · 依賴：G001, S004 S006 · implies操作前 ORIGINAL states=[s16, m16, f16, d16],alphabet=[a16, b16],start=s16,final_states=[f16];complete relation=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]] 操作後 states nonempty with actual start;alphabet/final set/relation MAY be EMPTY;definition qualification does NOT prove determinism,totality,DFA equivalence or a successful run 條件：無 空final set合法且辨識空語言；空alphabet也合法。定義合格不保存在accepting path，亦不推deterministic或DFA等價。 步驟識別與依賴規則：nfa_qualified_definition_EMPTY_alphabet_finals_edges_allowed · 依賴：G001, S005 S007 · implies操作前 NFA ORIGINAL EPS closure:ORIGINAL seed list=[s16];complete state carrier=[s16, m16, f16, d16] 操作後 initialize unique closure=[s16];seeds themselves are reachable by zero EPS steps;RAW seed list not changed 條件：無 seed可空或重複，輸出原seed列表保留；closure集合去重且包含零步可達的seed，不憑EPS cycle宣稱接受。 步驟識別與依賴規則：nfa_EPS_closure_contains_ORIGINAL_seeds_duplicates_preserved · 依賴：G001, S006 S008 · implies操作前 NFA ORIGINAL EPS closure:stage=0;ORIGINAL-relation EPS pass from previous set=[s16] 操作後 ALL [ORIGINAL index,from,to,from_in_previous] EPS rows=[[0, s16, m16, 是], [1, s16, d16, 是]];next unique set=[d16, m16, s16];stable=否 條件：無 每個新增state有前一集合的EPS前驅，故有有限seed路徑；任何含seed且EPS封閉的集合都須含每一層。有限states只會新增有限次，穩定時即完整且最小closure，不消耗input。 步驟識別與依賴規則：nfa_EVERY_EPS_saturation_stage_least_reachable_set · 依賴：G001, S007 S009 · implies操作前 NFA ORIGINAL EPS closure:stage=1;ORIGINAL-relation EPS pass from previous set=[d16, m16, s16] 操作後 ALL [ORIGINAL index,from,to,from_in_previous] EPS rows=[[0, s16, m16, 是], [1, s16, d16, 是]];next unique set=[d16, m16, s16];stable=是 條件：無 每個新增state有前一集合的EPS前驅，故有有限seed路徑；任何含seed且EPS封閉的集合都須含每一層。有限states只會新增有限次，穩定時即完整且最小closure，不消耗input。 步驟識別與依賴規則：nfa_EVERY_EPS_saturation_stage_least_reachable_set · 依賴：G001, S008 ## 逐步反向核對 - 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 ([s16], [d16, m16, s16], 是, finite_epsilon_reachability_without_input_consumption) 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: a644c50980e96590f4656f8d1029f280edf9db1ca03728b0f0a51b58405eff07 - candidate record SHA-256: 78422f7ce72629a885ebc86b350a36218482b70f0c0ca979dfbda63e9e4d3a1a - semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438 - question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a - Lean: NOT_RUN · human review: NOT_PERFORMED [同源完整 JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0042.json)