第 23251 題：非確定有限自動機

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-epsilon-closure；資料={"states":["s0","m0","f0","d0"],"alphabet":["a0","b0"],"start_state":"s0","final_states":["f0"],"transitions":[["s0",null,"m0"],["s0",null,"d0"],["m0","a0","m0"],["m0","b0","f0"]],"seed_states":["s0"]}。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":["s0","m0","f0","d0"],"alphabet":["a0","b0"],"start_state":"s0","final_states":["f0"],"transitions":[["s0",null,"m0"],["s0",null,"d0"],["m0","a0","m0"],["m0","b0","f0"]],"seed_states":["s0"]}。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

逐步推導

source operation=nfa-epsilon-closure;original snapshot and mathematics retained

explicit ordered source output fields=[seed_states, epsilon_closure, closure_certificate, certificate]

ORIGINAL transition index=0;[from,input_or_null,to]=[s0, null, m0];state carrier=[s0, m0, f0, d0];alphabet=[a0, b0]

state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows

ORIGINAL transition index=1;[from,input_or_null,to]=[s0, null, d0];state carrier=[s0, m0, f0, d0];alphabet=[a0, b0]

state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows

ORIGINAL transition index=2;[from,input_or_null,to]=[m0, a0, m0];state carrier=[s0, m0, f0, d0];alphabet=[a0, b0]

state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows

ORIGINAL transition index=3;[from,input_or_null,to]=[m0, b0, f0];state carrier=[s0, m0, f0, d0];alphabet=[a0, b0]

state endpoints legal;symbol either declared input or null EPS consuming0;ORIGINAL row position retained,including duplicate rows

ORIGINAL states=[s0, m0, f0, d0],alphabet=[a0, b0],start=s0,final_states=[f0];complete relation=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0]]

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

NFA ORIGINAL EPS closure:ORIGINAL seed list=[s0];complete state carrier=[s0, m0, f0, d0]

initialize unique closure=[s0];seeds themselves are reachable by zero EPS steps;RAW seed list not changed

NFA ORIGINAL EPS closure:stage=0;ORIGINAL-relation EPS pass from previous set=[s0]

ALL [ORIGINAL index,from,to,from_in_previous] EPS rows=[[0, s0, m0, 是], [1, s0, d0, 是]];next unique set=[d0, m0, s0];stable=否

NFA ORIGINAL EPS closure:stage=1;ORIGINAL-relation EPS pass from previous set=[d0, m0, s0]

ALL [ORIGINAL index,from,to,from_in_previous] EPS rows=[[0, s0, m0, 是], [1, s0, d0, 是]];next unique set=[d0, m0, s0];stable=是

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

([s0], [d0, m0, s0], 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。

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。