# 第 23241 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-0091) 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。 R0045-23241 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0091 ## 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-bounded-search-boundary；資料={"states":["s15","m15","f15","d15"],"alphabet":["a15","b15"],"start_state":"s15","final_states":["f15"],"transitions":[["s15",null,"m15"],["s15",null,"d15"],["m15","a15","m15"],["m15","b15","f15"]],"input":["a15","a15","a15","b15"],"max_steps":1}。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-bounded-search-boundary；資料={"states":["s15","m15","f15","d15"],"alphabet":["a15","b15"],"start_state":"s15","final_states":["f15"],"transitions":[["s15",null,"m15"],["s15",null,"d15"],["m15","a15","m15"],["m15","b15","f15"]],"input":["a15","a15","a15","b15"],"max_steps":1}。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-bounded-search-boundary;original snapshot and mathematics retained 操作後 explicit ordered source output fields=[input, max_steps, found_accepting_path, rejection_claim, boundary] 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]=[s15, null, m15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15] 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]=[s15, null, d15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15] 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]=[m15, a15, m15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15] 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]=[m15, b15, f15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15] 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=[s15, m15, f15, d15],alphabet=[a15, b15],start=s15,final_states=[f15];complete relation=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]] 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 ORIGINAL input=[a15, a15, a15, b15];actual start configuration=[s15, 0];max_steps=1 操作後 depth0 accepted=否;one finite FULL-input final path suffices;dead alternatives do not veto success 條件：無 搜尋步數計transition，EPS雖不耗input仍占一步。起點final可接受空字；有一條完整有限成功路徑即足夠，不需所有分支成功。 步驟識別與依賴規則：nfa_bounded_initial_and_finite_path_acceptance_scope · 依賴：G001, S006 S008 · implies操作前 ORIGINAL depth=0;frontier in ORIGINAL-state/position display order=[[s15, 0]] 操作後 ALL [from,ORIGINAL transition index,next_or_null] rows=[[[s15, 0], 0, [m15, 0]], [[s15, 0], 1, [d15, 0]], [[s15, 0], 2, null], [[s15, 0], 3, null]];next distinct configs=[[m15, 0], [d15, 0]];hit FULL-input final=否 條件：無 逐全部config與原relation列驗一步，按深度去重；input位置依symbol或null只增1或0。有限EPS循環只增加可達路徑，不是一條無窮接受run。 步驟識別與依賴規則：nfa_ALL_nondeterministic_bounded_frontier_not_infinite_cycle · 依賴：G001, S007 S009 · implies操作前 ORIGINAL bound=1;input=[a15, a15, a15, b15];found_accepting_path=否 操作後 rejection_claim fixed No;bounded flag alone supplies no GENERAL rejection certificate;any actual completeness argument must be established separately 條件：無 最後No保留不由有限測試作一般拒絕推論，不否定found=True。某有限模型若另證此界已完整涵蓋configuration，可另判實際拒絕；不能把不同的完整判定偷換成本題found旗標。 步驟識別與依賴規則：nfa_bounded_No_rejection_claim_not_found_negation · 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 ([a15, a15, a15, b15], 1, 否, 否, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs) 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: 389b7eb5e20154353cc60fb23f64e25e429581236e9f2ee6ac3bea4883d604e2 - candidate record SHA-256: ea6ead7426f8c58af52de829b255577ee5c013b024941c9f8ae44cfcc60d3979 - 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-0091.json)