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第 23298 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

R0045-23298 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0023

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":["s22","m22","f22","d22"],"alphabet":["a22","b22"],"start_state":"s22","final_states":["f22"],"transitions":[["s22",null,"m22"],["s22",null,"d22"],["m22","a22","m22"],["m22","b22","f22"]]}。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":["s22","m22","f22","d22"],"alphabet":["a22","b22"],"start_state":"s22","final_states":["f22"],"transitions":[["s22",null,"m22"],["s22",null,"d22"],["m22","a22","m22"],["m22","b22","f22"]]}。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]=[s22, null, m22];state carrier=[s22, m22, f22, d22];alphabet=[a22, b22]

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]=[s22, null, d22];state carrier=[s22, m22, f22, d22];alphabet=[a22, b22]

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]=[m22, a22, m22];state carrier=[s22, m22, f22, d22];alphabet=[a22, b22]

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]=[m22, b22, f22];state carrier=[s22, m22, f22, d22];alphabet=[a22, b22]

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=[s22, m22, f22, d22],alphabet=[a22, b22],start=s22,final_states=[f22];complete relation=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]]

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=[s22, m22, f22, d22];ORIGINAL alphabet order=[a22, b22];complete relation=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]]

操作後

ALL [state,symbol,ORIGINAL-order targets] rows=[[s22, a22, []], [s22, b22, []], [m22, a22, [m22]], [m22, b22, [f22]], [f22, a22, []], [f22, b22, []], [d22, a22, []], [d22, b22, []]];EMPTY target rows=[[s22, a22, []], [s22, b22, []], [f22, a22, []], [f22, b22, []], [d22, a22, []], [d22, b22, []]];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

([s22, m22, f22, d22], [a22, b22], s22, [f22], [[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]], [[s22, a22, []], [s22, b22, []], [f22, a22, []], [f22, b22, []], [d22, a22, []], [d22, b22, []]], 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: 97045c3ff2c3dcb7f979a63a99079e14f203796edf8448514710a4f9fc57dd22
  • candidate record SHA-256: 9dddadb2c20c45a98d36dd3c325421b7b62c4c739d1a674751b10276aa9b9fed
  • semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
  • question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON