R0045-23219 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0069
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":["s18","m18","f18","d18"],"alphabet":["a18","b18"],"start_state":"s18","final_states":["f18"],"transitions":[["s18",null,"m18"],["s18",null,"d18"],["m18","a18","m18"],["m18","b18","f18"]],"input":["a18","a18","b18"],"path":[["s18",0],["m18",0],["m18",1],["m18",2],["f18",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":["s18","m18","f18","d18"],"alphabet":["a18","b18"],"start_state":"s18","final_states":["f18"],"transitions":[["s18",null,"m18"],["s18",null,"d18"],["m18","a18","m18"],["m18","b18","f18"]],"input":["a18","a18","b18"],"path":[["s18",0],["m18",0],["m18",1],["m18",2],["f18",3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。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。
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。
規則：source_ordered_interface_explicit_without_javascript · 依賴：G001
操作前
ORIGINAL transition index=0;[from,input_or_null,to]=[s18, null, m18];state carrier=[s18, m18, f18, d18];alphabet=[a18, b18]
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。
規則：nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · 依賴：G001, S001
操作前
ORIGINAL transition index=1;[from,input_or_null,to]=[s18, null, d18];state carrier=[s18, m18, f18, d18];alphabet=[a18, b18]
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。
規則：nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · 依賴：G001, S002
操作前
ORIGINAL transition index=2;[from,input_or_null,to]=[m18, a18, m18];state carrier=[s18, m18, f18, d18];alphabet=[a18, b18]
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。
規則：nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · 依賴：G001, S003
操作前
ORIGINAL transition index=3;[from,input_or_null,to]=[m18, b18, f18];state carrier=[s18, m18, f18, d18];alphabet=[a18, b18]
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。
規則：nfa_EVERY_original_relation_row_roles_empty_multi_EPS_legal · 依賴：G001, S004
操作前
ORIGINAL states=[s18, m18, f18, d18],alphabet=[a18, b18],start=s18,final_states=[f18];complete relation=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]]
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。
規則：nfa_qualified_definition_EMPTY_alphabet_finals_edges_allowed · 依賴：G001, S005
操作前
ORIGINAL input=[a18, a18, b18];ALL legal raw path configs=[[s18, 0], [m18, 0], [m18, 1], [m18, 2], [f18, 3]]
操作後
first configuration EXACT actual start=[s18, 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。
規則：nfa_actual_path_initial_and_structural_configurations_required · 依賴：G001, S006
操作前
ORIGINAL path step=0;from=[s18, 0];to=[m18, 0];input=[a18, a18, b18]
操作後
ALL [ORIGINAL transition index,next_or_null] rows=[[0, [m18, 0]], [1, [d18, 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。
規則：nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · 依賴：G001, S007
操作前
ORIGINAL path step=1;from=[m18, 0];to=[m18, 1];input=[a18, a18, b18]
操作後
ALL [ORIGINAL transition index,next_or_null] rows=[[0, null], [1, null], [2, [m18, 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。
規則：nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · 依賴：G001, S008
操作前
ORIGINAL path step=2;from=[m18, 1];to=[m18, 2];input=[a18, a18, b18]
操作後
ALL [ORIGINAL transition index,next_or_null] rows=[[0, null], [1, null], [2, [m18, 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。
規則：nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · 依賴：G001, S009
操作前
ORIGINAL path step=3;from=[m18, 2];to=[f18, 3];input=[a18, a18, b18]
操作後
ALL [ORIGINAL transition index,next_or_null] rows=[[0, null], [1, null], [2, null], [3, [f18, 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。
規則：nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · 依賴：G001, S010
操作前
ORIGINAL last config=[f18, 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。
規則：nfa_valid_path_FULL_input_final_existential_acceptance · 依賴：G001, S011
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。([a18, a18, b18], [[s18, 0], [m18, 0], [m18, 1], [m18, 2], [f18, 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。
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。
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 734ce28146170be851992003638c3fd4331fe889b8a39248313247d117eccbb8candidate record SHA-256: 5430fd08a520229647e7d08145babb83f256a2335bb5668cc8362ed09f1fb277semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2aLean: NOT_RUN · human review: NOT_PERFORMED