R0045-23258 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0033
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":["s7","m7","f7","d7"],"alphabet":["a7","b7"],"start_state":"s7","final_states":["f7"],"transitions":[["s7",null,"m7"],["s7",null,"d7"],["m7","a7","m7"],["m7","b7","f7"]],"seed_states":["s7"]}。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":["s7","m7","f7","d7"],"alphabet":["a7","b7"],"start_state":"s7","final_states":["f7"],"transitions":[["s7",null,"m7"],["s7",null,"d7"],["m7","a7","m7"],["m7","b7","f7"]],"seed_states":["s7"]}。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-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。
規則：source_ordered_interface_explicit_without_javascript · 依賴：G001
操作前
ORIGINAL transition index=0;[from,input_or_null,to]=[s7, null, m7];state carrier=[s7, m7, f7, d7];alphabet=[a7, b7]
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]=[s7, null, d7];state carrier=[s7, m7, f7, d7];alphabet=[a7, b7]
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]=[m7, a7, m7];state carrier=[s7, m7, f7, d7];alphabet=[a7, b7]
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]=[m7, b7, f7];state carrier=[s7, m7, f7, d7];alphabet=[a7, b7]
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=[s7, m7, f7, d7],alphabet=[a7, b7],start=s7,final_states=[f7];complete relation=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7]]
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
NFA ORIGINAL EPS closure:ORIGINAL seed list=[s7];complete state carrier=[s7, m7, f7, d7]
操作後
initialize unique closure=[s7];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。
規則：nfa_EPS_closure_contains_ORIGINAL_seeds_duplicates_preserved · 依賴：G001, S006
操作前
NFA ORIGINAL EPS closure:stage=0;ORIGINAL-relation EPS pass from previous set=[s7]
操作後
ALL [ORIGINAL index,from,to,from_in_previous] EPS rows=[[0, s7, m7, 是], [1, s7, d7, 是]];next unique set=[d7, m7, s7];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。
規則：nfa_EVERY_EPS_saturation_stage_least_reachable_set · 依賴：G001, S007
操作前
NFA ORIGINAL EPS closure:stage=1;ORIGINAL-relation EPS pass from previous set=[d7, m7, s7]
操作後
ALL [ORIGINAL index,from,to,from_in_previous] EPS rows=[[0, s7, m7, 是], [1, s7, d7, 是]];next unique set=[d7, m7, s7];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。
規則：nfa_EVERY_EPS_saturation_stage_least_reachable_set · 依賴：G001, S008
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。([s7], [d7, m7, s7], 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。
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: 61d67c77d588f5c812ae7ac8d7f0b65e703ac513f365827272487322c7c26a65candidate record SHA-256: 0a76c84d27fb04a61bc56a25b586299e6583c343c5d15c56d57e45586399cd6esemantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2aLean: NOT_RUN · human review: NOT_PERFORMED