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第 23203 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

R0045-23203 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0053

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-accepting-path-witness；資料={"states":["s2","m2","f2","d2"],"alphabet":["a2","b2"],"start_state":"s2","final_states":["f2"],"transitions":[["s2",null,"m2"],["s2",null,"d2"],["m2","a2","m2"],["m2","b2","f2"]],"input":["a2","a2","b2"],"path":[["s2",0],["m2",0],["m2",1],["m2",2],["f2",3]]}。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-accepting-path-witness；資料={"states":["s2","m2","f2","d2"],"alphabet":["a2","b2"],"start_state":"s2","final_states":["f2"],"transitions":[["s2",null,"m2"],["s2",null,"d2"],["m2","a2","m2"],["m2","b2","f2"]],"input":["a2","a2","b2"],"path":[["s2",0],["m2",0],["m2",1],["m2",2],["f2",3]]}。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-accepting-path-witness;original snapshot and mathematics retained

操作後

explicit ordered source output fields=[input, path, transition_indices, valid_path, accepted, 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。

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]=[s2, null, m2];state carrier=[s2, m2, f2, d2];alphabet=[a2, b2]

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]=[s2, null, d2];state carrier=[s2, m2, f2, d2];alphabet=[a2, b2]

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]=[m2, a2, m2];state carrier=[s2, m2, f2, d2];alphabet=[a2, b2]

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]=[m2, b2, f2];state carrier=[s2, m2, f2, d2];alphabet=[a2, b2]

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=[s2, m2, f2, d2],alphabet=[a2, b2],start=s2,final_states=[f2];complete relation=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2]]

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 input=[a2, a2, b2];ALL legal raw path configs=[[s2, 0], [m2, 0], [m2, 1], [m2, 2], [f2, 3]]

操作後

first configuration EXACT actual start=[s2, 0];this is not yet validity or acceptance

條件：無

先核整條path的state／position資格與真initial；合法config列表不保相鄰transition存在。

步驟識別與依賴

規則：nfa_actual_path_initial_and_structural_configurations_required · 依賴：G001, S006

S008 · implies

操作前

ORIGINAL path step=0;from=[s2, 0];to=[m2, 0];input=[a2, a2, b2]

操作後

ALL [ORIGINAL transition index,next_or_null] rows=[[0, [m2, 0]], [1, [d2, 0]], [2, null], [3, null]];FIRST matching index=0

條件：無

按原transition索引取首個真匹配者，包括duplicate edges；null增0，symbol增1且必讀到原input。失敗保partial prior indices，不用後面final state補錯步。

步驟識別與依賴

規則：nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · 依賴：G001, S007

S009 · implies

操作前

ORIGINAL path step=1;from=[m2, 0];to=[m2, 1];input=[a2, a2, b2]

操作後

ALL [ORIGINAL transition index,next_or_null] rows=[[0, null], [1, null], [2, [m2, 1]], [3, null]];FIRST matching index=2

條件：無

按原transition索引取首個真匹配者，包括duplicate edges；null增0，symbol增1且必讀到原input。失敗保partial prior indices，不用後面final state補錯步。

步驟識別與依賴

規則：nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · 依賴：G001, S008

S010 · implies

操作前

ORIGINAL path step=2;from=[m2, 1];to=[m2, 2];input=[a2, a2, b2]

操作後

ALL [ORIGINAL transition index,next_or_null] rows=[[0, null], [1, null], [2, [m2, 2]], [3, null]];FIRST matching index=2

條件：無

按原transition索引取首個真匹配者，包括duplicate edges；null增0，symbol增1且必讀到原input。失敗保partial prior indices，不用後面final state補錯步。

步驟識別與依賴

規則：nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · 依賴：G001, S009

S011 · implies

操作前

ORIGINAL path step=3;from=[m2, 2];to=[f2, 3];input=[a2, a2, b2]

操作後

ALL [ORIGINAL transition index,next_or_null] rows=[[0, null], [1, null], [2, null], [3, [f2, 3]]];FIRST matching index=3

條件：無

按原transition索引取首個真匹配者，包括duplicate edges；null增0，symbol增1且必讀到原input。失敗保partial prior indices，不用後面final state補錯步。

步驟識別與依賴

規則：nfa_EVERY_path_step_FIRST_ORIGINAL_transition_and_input_update · 依賴：G001, S010

S012 · implies

操作前

ORIGINAL last config=[f2, 3];input length=3;matched prior indices=[0, 2, 2, 3]

操作後

valid_path=是;FULL input consumed=是;last final=是;accepted=是;one successful finite path suffices despite dead branches

條件：無

valid與accepted分開，premature final不接受。accepted=True才是這個word的有限存在見證，不推機器deterministic、total或DFA等價；tag不把False變True。

步驟識別與依賴

規則：nfa_valid_path_FULL_input_final_existential_acceptance · 依賴：G001, S011

逐步反向核對

  • S012 · 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。
  • S011 · 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。
  • S010 · 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。
  • 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

([a2, a2, b2], [[s2, 0], [m2, 0], [m2, 1], [m2, 2], [f2, 3]], [0, 2, 2, 3], 是, 是, one_finite_path_consuming_full_input_and_ending_final_proves_membership)

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: 0bf92cc9ff03bea19a12ee1cf20f02da9e5c0c0ce671a371aa44912f0031b894
  • candidate record SHA-256: d5eb1436e3aaeff7c77b3c8090554a51e49b93ed1d04d76339d3de4883f2f4be
  • semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
  • question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON