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第 23213 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

R0045-23213 · q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0063

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":["s12","m12","f12","d12"],"alphabet":["a12","b12"],"start_state":"s12","final_states":["f12"],"transitions":[["s12",null,"m12"],["s12",null,"d12"],["m12","a12","m12"],["m12","b12","f12"]],"input":["b12"],"path":[["s12",0],["m12",0],["f12",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-accepting-path-witness；資料={"states":["s12","m12","f12","d12"],"alphabet":["a12","b12"],"start_state":"s12","final_states":["f12"],"transitions":[["s12",null,"m12"],["s12",null,"d12"],["m12","a12","m12"],["m12","b12","f12"]],"input":["b12"],"path":[["s12",0],["m12",0],["f12",1]]}。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]=[s12, null, m12];state carrier=[s12, m12, f12, d12];alphabet=[a12, b12]

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]=[s12, null, d12];state carrier=[s12, m12, f12, d12];alphabet=[a12, b12]

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]=[m12, a12, m12];state carrier=[s12, m12, f12, d12];alphabet=[a12, b12]

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]=[m12, b12, f12];state carrier=[s12, m12, f12, d12];alphabet=[a12, b12]

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=[s12, m12, f12, d12],alphabet=[a12, b12],start=s12,final_states=[f12];complete relation=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]]

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=[b12];ALL legal raw path configs=[[s12, 0], [m12, 0], [f12, 1]]

操作後

first configuration EXACT actual start=[s12, 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=[s12, 0];to=[m12, 0];input=[b12]

操作後

ALL [ORIGINAL transition index,next_or_null] rows=[[0, [m12, 0]], [1, [d12, 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=[m12, 0];to=[f12, 1];input=[b12]

操作後

ALL [ORIGINAL transition index,next_or_null] rows=[[0, null], [1, null], [2, null], [3, [f12, 1]]];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, S008

S010 · implies

操作前

ORIGINAL last config=[f12, 1];input length=1;matched prior indices=[0, 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, S009

逐步反向核對

  • 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

([b12], [[s12, 0], [m12, 0], [f12, 1]], [0, 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: 7760301effda5002b8e5a717ebcbc1f30827879e5962bed52b8dc038ca035402
  • candidate record SHA-256: 32064c3ac3897d0b56f1f395c68075165b236a221c218d6a123622118ec51a9f
  • semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
  • question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON