# 第 23203 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-0053) 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。 R0045-23203 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0053 ## 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-accepting-path-witness；資料={"states":["s2","m2","f2","d2"],"alphabet":["a2","b2"],"start_state":"s2","final_states":["f2"],"transitions":[["s2",null,"m2"],["s2",null,"d2"],["m2","a2","m2"],["m2","b2","f2"]],"input":["a2","a2","b2"],"path":[["s2",0],["m2",0],["m2",1],["m2",2],["f2",3]]}。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-accepting-path-witness；資料={"states":["s2","m2","f2","d2"],"alphabet":["a2","b2"],"start_state":"s2","final_states":["f2"],"transitions":[["s2",null,"m2"],["s2",null,"d2"],["m2","a2","m2"],["m2","b2","f2"]],"input":["a2","a2","b2"],"path":[["s2",0],["m2",0],["m2",1],["m2",2],["f2",3]]}。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-accepting-path-witness;original snapshot and mathematics retained 操作後 explicit ordered source output fields=[input, path, transition_indices, valid_path, accepted, 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]=[s2, null, m2];state carrier=[s2, m2, f2, d2];alphabet=[a2, b2] 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]=[s2, null, d2];state carrier=[s2, m2, f2, d2];alphabet=[a2, b2] 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]=[m2, a2, m2];state carrier=[s2, m2, f2, d2];alphabet=[a2, b2] 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]=[m2, b2, f2];state carrier=[s2, m2, f2, d2];alphabet=[a2, b2] 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=[s2, m2, f2, d2],alphabet=[a2, b2],start=s2,final_states=[f2];complete relation=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2]] 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=[a2, a2, b2];ALL legal raw path configs=[[s2, 0], [m2, 0], [m2, 1], [m2, 2], [f2, 3]] 操作後 first configuration EXACT actual start=[s2, 0];this is not yet validity or acceptance 條件：無 先核整條path的state／position資格與真initial；合法config列表不保相鄰transition存在。 步驟識別與依賴規則：nfa_actual_path_initial_and_structural_configurations_required · 依賴：G001, S006 S008 · implies操作前 ORIGINAL path step=0;from=[s2, 0];to=[m2, 0];input=[a2, a2, b2] 操作後 ALL [ORIGINAL transition index,next_or_null] rows=[[0, [m2, 0]], [1, [d2, 0]], [2, null], [3, null]];FIRST matching index=0 條件：無 按原transition索引取首個真匹配者，包括duplicate edges；null增0，symbol增1且必讀到原input。失敗保partial prior indices，不用後面final state補錯步。 步驟識別與依賴規則：nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · 依賴：G001, S007 S009 · implies操作前 ORIGINAL path step=1;from=[m2, 0];to=[m2, 1];input=[a2, a2, b2] 操作後 ALL [ORIGINAL transition index,next_or_null] rows=[[0, null], [1, null], [2, [m2, 1]], [3, null]];FIRST matching index=2 條件：無 按原transition索引取首個真匹配者，包括duplicate edges；null增0，symbol增1且必讀到原input。失敗保partial prior indices，不用後面final state補錯步。 步驟識別與依賴規則：nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · 依賴：G001, S008 S010 · implies操作前 ORIGINAL path step=2;from=[m2, 1];to=[m2, 2];input=[a2, a2, b2] 操作後 ALL [ORIGINAL transition index,next_or_null] rows=[[0, null], [1, null], [2, [m2, 2]], [3, null]];FIRST matching index=2 條件：無 按原transition索引取首個真匹配者，包括duplicate edges；null增0，symbol增1且必讀到原input。失敗保partial prior indices，不用後面final state補錯步。 步驟識別與依賴規則：nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · 依賴：G001, S009 S011 · implies操作前 ORIGINAL path step=3;from=[m2, 2];to=[f2, 3];input=[a2, a2, b2] 操作後 ALL [ORIGINAL transition index,next_or_null] rows=[[0, null], [1, null], [2, null], [3, [f2, 3]]];FIRST matching index=3 條件：無 按原transition索引取首個真匹配者，包括duplicate edges；null增0，symbol增1且必讀到原input。失敗保partial prior indices，不用後面final state補錯步。 步驟識別與依賴規則：nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · 依賴：G001, S010 S012 · implies操作前 ORIGINAL last config=[f2, 3];input length=3;matched prior indices=[0, 2, 2, 3] 操作後 valid_path=是;FULL input consumed=是;last final=是;accepted=是;one successful finite path suffices despite dead branches 條件：無 valid與accepted分開，premature final不接受。accepted=True才是這個word的有限存在見證，不推機器deterministic、total或DFA等價；tag不把False變True。 步驟識別與依賴規則：nfa_valid_path_FULL_input_final_existential_acceptance · 依賴：G001, S011 ## 逐步反向核對 - S012 · 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。 - S011 · 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。 - S010 · 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。 - 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 ([a2, a2, b2], [[s2, 0], [m2, 0], [m2, 1], [m2, 2], [f2, 3]], [0, 2, 2, 3], 是, 是, one_finite_path_consuming_full_input_and_ending_final_proves_membership) 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: 0bf92cc9ff03bea19a12ee1cf20f02da9e5c0c0ce671a371aa44912f0031b894 - candidate record SHA-256: d5eb1436e3aaeff7c77b3c8090554a51e49b93ed1d04d76339d3de4883f2f4be - 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-0053.json)