{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-finite-automaton-01",
  "mko_id": "mko-finite-automaton",
  "title": {
    "zh-Hant": "完整DFA的1011追蹤",
    "en": "Tracing 1011 through a complete DFA"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-finite-automaton-01",
    "source": {
      "mko_id": "mko-finite-automaton",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "6bae5f7a3af28d2877efade8771bee1a1adcddd1d88f055671859e37e3d0c0c8",
      "english_sha256": "b092929e0950241d81acaec1a1ad62c6c520d910717ea23c2581deea8ae52abe"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "calc"
    ],
    "goal": {
      "zh-Hant": "指定DFA拒絕1011，並接受空字ε。",
      "en": "The specified DFA rejects 1011 and accepts epsilon."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "Q={E,O}，Σ={0,1}，start=E，finals={E}；δ(E,0)=E,δ(E,1)=O,δ(O,0)=O,δ(O,1)=E。word=1011。",
          "en": "Q={E,O}, alphabet={0,1}, start=E, finals={E}; delta(E,0)=E,delta(E,1)=O,delta(O,0)=O,delta(O,1)=E; word=1011."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "Q=\\{E,O\\},\\quad\\Sigma=\\{0,1\\}",
        "after_tex": "\\begin{array}{c|cc}\\delta&0&1\\\\\\hline E&E&O\\\\O&O&E\\end{array}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "四個Q×Σ輸入各恰有一個Q內輸出，完整核total與deterministic。",
          "en": "Each of the four state-symbol pairs has exactly one output in Q, completely checking totality and determinism."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "沒有ε transition，也沒有同格多分支。",
            "en": "There is no epsilon transition or multivalued cell."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "w=1011,\\quad q_0=E",
        "after_tex": "(q,i)=(E,0)",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "初始化為唯一start，位置0表示尚未讀任何符號。",
          "en": "Initialize at the unique start; position zero means no symbol has been read."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "start雖final仍不能略過非空word後續輸入。",
            "en": "An accepting start does not allow ignoring the remaining nonempty input."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S01",
          "S02"
        ],
        "before_tex": "(E,0),\\quad w_1=1",
        "after_tex": "(O,1)",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "讀首個1依δ(E,1)=O，位置增加1。",
          "en": "Reading the first 1 uses delta(E,1)=O and advances position by one."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "只有一個合法後繼，不能選擇留E。",
            "en": "There is only one successor; remaining at E is not a choice."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S01",
          "S03"
        ],
        "before_tex": "(O,1),\\quad w_2=0",
        "after_tex": "(O,2)",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第二符號0使O保持O，仍然確實消耗此符號。",
          "en": "The second symbol 0 keeps O unchanged while still consuming that symbol."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "狀態不變不代表ε move，位置仍從1到2。",
            "en": "An unchanged state is not an epsilon move; the position advances from 1 to 2."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S01",
          "S04"
        ],
        "before_tex": "(O,2),\\quad w_3=1",
        "after_tex": "(E,3)",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "第三個1翻回E，此時尚有一個1未讀。",
          "en": "The third symbol 1 returns to E, with one final 1 still unread."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "中途到final不代表整個word已接受。",
            "en": "Reaching a final state midway does not yet accept the whole word."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S01",
          "S05"
        ],
        "before_tex": "(E,3),\\quad w_4=1",
        "after_tex": "(O,4),\\quad4=|w|",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "末個1送到O，全部四符號依序消耗完。",
          "en": "The final 1 sends E to O, consuming all four symbols in order."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "不能在S05的E提前停住而丟棄末字。",
            "en": "One cannot stop at E in S05 and discard the last symbol."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06"
        ],
        "before_tex": "\\delta^*(E,1011)=O,\\quad F=\\{E\\}",
        "after_tex": "O\\notin F\\Rightarrow1011\\text{ rejected}",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整決定run的末狀態非final，因此拒絕此word。",
          "en": "The unique complete run ends outside the final set, rejecting this word."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "這是指定word的完整證書，不是任意目標language的equivalence證明。",
            "en": "This certifies the specified word, not equivalence with an arbitrary target language."
          }
        },
        "leaf_basis": "mko_definition"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S01",
          "S07"
        ],
        "before_tex": "w=\\varepsilon,\\quad|w|=0",
        "after_tex": "\\delta^*(E,\\varepsilon)=E\\in F\\Rightarrow\\varepsilon\\text{ accepted}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "空字不走轉移，start就是末狀態，故接受。",
          "en": "The empty word makes no transition, so the start is final and epsilon is accepted."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "沒有添加ε邊；空字語義與ε transition是不同概念。",
            "en": "No epsilon edge is added; an empty word differs from an epsilon transition."
          }
        },
        "leaf_basis": "calc"
      }
    ],
    "result": {
      "zh-Hant": "指定DFA拒絕1011，並接受空字ε。",
      "en": "The specified DFA rejects 1011 and accepts epsilon."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "中途E出現便接受1011。",
          "en": "An intermediate E accepts 1011."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "尚有末個1，完整末態是O。",
          "en": "The final 1 remains; the full terminal state is O."
        }
      },
      {
        "claim": {
          "zh-Hant": "讀0不變狀態表示不消耗輸入。",
          "en": "An unchanged state on 0 consumes no input."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "DFA每次讀一個符號，位置仍增加。",
          "en": "Every DFA input transition consumes one symbol and advances position."
        }
      },
      {
        "claim": {
          "zh-Hant": "ε需要額外ε transition才接受。",
          "en": "Accepting epsilon requires an epsilon transition."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "空字不走任何邊，start E已final。",
          "en": "The empty word takes no edge and start E is final."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "calc",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "假設有限集合、有限字串與索引、逐步函數更新及自然數歸納；ε表示長度0字串。",
          "en": "Assume finite sets, finite strings and indices, stepwise function updates and induction on natural numbers; epsilon denotes the length-zero word."
        }
      },
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "完整核 Q×Σ 每一輸入恰有一個合法輸出；依序追蹤完整有限 word。空字被接受，當且僅當初始狀態屬於接受狀態集合。一組有限 trace 本身不能證明與任意未指定目標語言的等價，也不能證明該未指定目標語言的 regularity。",
          "en": "Completely check that each Q×Sigma input has exactly one valid output, then trace each complete finite word in order. The empty word is accepted exactly when the initial state is accepting. One finite trace does not establish language equivalence or regularity of an unspecified target language."
        },
        "source": {
          "mko_id": "mko-finite-automaton",
          "sha256": "6bae5f7a3af28d2877efade8771bee1a1adcddd1d88f055671859e37e3d0c0c8",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整Q×Σ決定總表與有限word trace，不由幾條trace推language equivalence。",
      "en": "A full deterministic total Q-by-alphabet table and finite-word traces, not language equivalence from examples."
    },
    "md_symbols": [
      [
        "\\delta^*",
        "整字迭代轉移",
        "iterated transition on a word"
      ],
      [
        "i",
        "已讀符號數",
        "number of consumed symbols"
      ]
    ],
    "md_error_controls": [
      {
        "title": "總表 / Total table",
        "zh": "四個Q×Σ格皆要合法唯一輸出。",
        "en": "All four state-symbol cells need one legal output."
      },
      {
        "title": "讀完輸入 / Consume all input",
        "zh": "接受檢查只在位置等於word長度後執行。",
        "en": "Check acceptance only after position equals word length."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "c631c60e2976f05f20ff82cd04318bc5c870bf60c51efcfb4bf34202ec0524b1",
    "original_source_sha256": "1b4e2cc34b16b6214d69dfde3f776b2089abe1dec7d7ab3534bf2cf53e8d3aae",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [
      "mko_definition"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-P014"
    ],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}