R0045-23233 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0083
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":["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"]],"input":["a7","a7","a7","b7"],"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":["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"]],"input":["a7","a7","a7","b7"],"max_steps":1}。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-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。
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
ORIGINAL input=[a7, a7, a7, b7];actual start configuration=[s7, 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。
規則：nfa_bounded_initial_and_finite_path_acceptance_scope · 依賴：G001, S006
操作前
ORIGINAL depth=0;frontier in ORIGINAL-state/position display order=[[s7, 0]]
操作後
ALL [from,ORIGINAL transition index,next_or_null] rows=[[[s7, 0], 0, [m7, 0]], [[s7, 0], 1, [d7, 0]], [[s7, 0], 2, null], [[s7, 0], 3, null]];next distinct configs=[[m7, 0], [d7, 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。
規則：nfa_ALL_nondeterministic_bounded_frontier_not_infinite_cycle · 依賴：G001, S007
操作前
ORIGINAL bound=1;input=[a7, a7, a7, b7];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。
規則：nfa_bounded_No_rejection_claim_not_found_negation · 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。([a7, a7, a7, b7], 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。
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: b3f8e10003e7a534d2fbb4e976d66a2ebdc3d7261f12ee62dd4e69e39b7582adcandidate record SHA-256: 10b5cdd2e70ffddf7104e5221d4e899e7737b89da6792e1988420040923a761bsemantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2aLean: NOT_RUN · human review: NOT_PERFORMED