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第 23291 題：非確定有限自動機

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  1. 原題與已知條件
  2. 定義與適用範圍
  3. 逐步推導
  4. 逐步反向核對
  5. 結論
  6. 本題覆核與來源綁定

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← 逐題詳解目錄 · 原題批次

R0045-23291 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0016

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-qualification；資料={"states":["s15","m15","f15","d15"],"alphabet":["a15","b15"],"start_state":"s15","final_states":["f15"],"transitions":[["s15",null,"m15"],["s15",null,"d15"],["m15","a15","m15"],["m15","b15","f15"]]}。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-qualification；資料={"states":["s15","m15","f15","d15"],"alphabet":["a15","b15"],"start_state":"s15","final_states":["f15"],"transitions":[["s15",null,"m15"],["s15",null,"d15"],["m15","a15","m15"],["m15","b15","f15"]]}。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。

逐步推導

S001 · implies

操作前

source operation=nfa-qualification;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention]

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]=[s15, null, m15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15]

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]=[s15, null, d15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15]

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]=[m15, a15, m15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15]

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]=[m15, b15, f15];state carrier=[s15, m15, f15, d15];alphabet=[a15, b15]

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=[s15, m15, f15, d15],alphabet=[a15, b15],start=s15,final_states=[f15];complete relation=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]]

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 state order=[s15, m15, f15, d15];ORIGINAL alphabet order=[a15, b15];complete relation=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]]

操作後

ALL [state,symbol,ORIGINAL-order targets] rows=[[s15, a15, []], [s15, b15, []], [m15, a15, [m15]], [m15, b15, [f15]], [f15, a15, []], [f15, b15, []], [d15, a15, []], [d15, b15, []]];EMPTY target rows=[[s15, a15, []], [s15, b15, []], [f15, a15, []], [f15, b15, []], [d15, a15, []], [d15, b15, []]];no missing edge repaired

條件：無

逐原state再原symbol列全部target，保原transition次序與重複；空target不是不合格。空alphabet時沒有這類pair，空final set亦不變成任意accept。

步驟識別與依賴

規則：nfa_ORIGINAL_state_symbol_relation_targets_not_totality_assumption · 依賴：G001, S006

逐步反向核對

  • 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

([s15, m15, f15, d15], [a15, b15], s15, [f15], [[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]], [[s15, a15, []], [s15, b15, []], [f15, a15, []], [f15, b15, []], [d15, a15, []], [d15, b15, []]], 是, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)

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: 77d66b5625713de9058c95a985bbbbfd6b1e0f4ab109f591cd91c64f6f2a6e30
  • candidate record SHA-256: d0b1f48486ec2505615219790f490c6df8044310b9f375da47d7ce6ef404097c
  • semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
  • question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON