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第 24722 題：Turing機判定模型

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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

R0048-24722 · q-20261015-advance-20261113-r05-turing-machine-decision-model-0047

原題與已知條件

deterministic single-tape signature、exact configuration semantics與decider total-correctness boundary。操作=tm-exact-step-trace；資料={"states":["q021","q121","qa21","qr21"],"input_alphabet":["021"],"tape_alphabet":["021","X21","_21"],"blank":"_21","start_state":"q021","accept_state":"qa21","reject_state":"qr21","transitions":[["q021","021","q121","X21","R"],["q021","_21","qr21","_21","S"],["q121","_21","qa21","_21","L"]],"input":["021"],"steps":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

  • G001 — deterministic single-tape signature、exact configuration semantics與decider total-correctness boundary。操作=tm-exact-step-trace；資料={"states":["q021","q121","qa21","qr21"],"input_alphabet":["021"],"tape_alphabet":["021","X21","_21"],"blank":"_21","start_state":"q021","accept_state":"qa21","reject_state":"qr21","transitions":[["q021","021","q121","X21","R"],["q021","_21","qr21","_21","S"],["q121","_21","qa21","_21","L"]],"input":["021"],"steps":2}。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 roles = [tm-exact-step-trace]

操作後

derived = [[input, step_budget, configurations, step_outcomes, final_state, local_outcome, global_decider_certificate, boundary]]

條件：無

直接列來源完整欄序，不需JavaScript；原題、布林旗標、數學物件與非聲稱保持分層。

步驟識別與依賴

規則：source_ordered_fields_without_JS · 依賴：G001

S002 · implies

操作前

source roles = [[q021, q121, qa21, qr21], [021], [021, X21, _21], _21, q021, qa21, qr21, [[q021, 021, q121, X21, R], [q021, _21, qr21, _21, S], [q121, _21, qa21, _21, L]]]

操作後

derived = [[q021, q121, qa21, qr21], [021], [021, X21, _21], [[q021, _21, qr21, _21, S], [q021, 021, q121, X21, R], [q121, _21, qa21, _21, L]], 是]

條件：無

三個清單皆非空且唯一；input是blank-free tape子集，但state與字母標籤不要求不相交。每個(from,read)的來源NUL-joined key唯一、halt無出邊。partial table合法，不補reject規則；canonical transition序與UTF16集合序分開。

步驟識別與依賴

規則：TM_complete_signature_partial_transition_table_allowed · 依賴：G001, S001

S003 · implies

操作前

source roles = [[021], q021, _21]

操作後

derived = [[q021, 0, [[0, 021]]]]

條件：無

初始head0、input放0起的整數格，其餘全blank；稀疏記錄僅保非blank。tape可往負index伸展。

步驟識別與依賴

規則：TM_exact_initial_configuration_blank_elsewhere · 依賴：G001, S002

S004 · calculation

操作前

TM exact integer head move0 := sum(0, 1)

操作後

TM exact integer head move0 = 1

條件：無

依原始角色與已證中間量精確代值，分母先核非零，不用浮點近似。

步驟識別與依賴

規則：exact_rational_arithmetic · 依賴：G001, S003

S005 · implies

操作前

source roles = [0, [q021, 0, [[0, 021]]], 021, [q021, 021, q121, X21, R]]

操作後

derived = [step, [q121, 1, [[0, X21]]]]

條件：無

先在舊head寫字（blank刪格），再換state及移head；全部L/R/S按原rule，不混成停機語意。

步驟識別與依賴

規則：TM_source_rule_exact_read_write_state_move · 依賴：G001, S004

S006 · calculation

操作前

TM exact integer head move1 := sum(1, -1)

操作後

TM exact integer head move1 = 0

條件：無

依原始角色與已證中間量精確代值，分母先核非零，不用浮點近似。

步驟識別與依賴

規則：exact_rational_arithmetic · 依賴：G001, S005

S007 · implies

操作前

source roles = [1, [q121, 1, [[0, X21]]], _21, [q121, _21, qa21, _21, L]]

操作後

derived = [step, [qa21, 0, [[0, X21]]]]

條件：無

先在舊head寫字（blank刪格），再換state及移head；全部L/R/S按原rule，不混成停機語意。

步驟識別與依賴

規則：TM_source_rule_exact_read_write_state_move · 依賴：G001, S006

S008 · implies

操作前

source roles = [[021], 2, [[q021, 0, [[0, 021]]], [q121, 1, [[0, X21]]], [qa21, 0, [[0, X21]]]], [step, step]]

操作後

derived = [qa21, accept, 否]

條件：無

已halt才接受／拒絕；budget用盡或stuck不是普通reject，也不證永久不停止。一個input的有限trace不提供all-input decider證書。

步驟識別與依賴

規則：TM_finite_trace_input_budget_scope_NOT_global_decider · 依賴：G001, S007

逐步反向核對

  • 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 — 以精確有理數重算 TM exact integer head move1，不以小數近似替代。
  • 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 — 以精確有理數重算 TM exact integer head move0，不以小數近似替代。
  • 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

([021], 2, [[q021, 0, [[0, 021]]], [q121, 1, [[0, X21]]], [qa21, 0, [[0, X21]]]], [step, step], qa21, accept, 否, finite_trace_is_input_local_and_does_not_certify_a_global_decider)

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: da6606dc3de8c69ab4a9b304093843a06f1f04c454feff5999ef6632366399a4
  • candidate record SHA-256: 645cc7e6b4611e8d39e78ca9f90517709680592d0fcbd8756586f6b9f946664f
  • semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0
  • question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9
  • Lean: NOT_RUN · human review: NOT_PERFORMED
同源完整 JSON