{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-pushdown-automaton",
  "semantic_family_id": "sf-pushdown-automaton",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "下推自動機"
  },
  "summary": {
    "zh-Hant": "nondeterministic PDA以有限控制、input與stack transitions運算，本輪固定讀完input且到final state接受。"
  },
  "statement": {
    "zh-Hant": "PDA tuple含finite Q、input alphabet、stack alphabet、start q0、bottom Z、final states與transition relation。transition序列固定為[from_state,input_symbol_or_null,required_stack_top,to_state,replacement_stack_bottom_to_top]；null表示epsilon且不消耗input。stack序列bottom-first/top-last，bottom不可underflow。接受witness為有限run且讀完input到final state。不混用empty-stack acceptance；bounded search failure不證rejection，也不自動宣稱CFG equivalence或determinism。"
  },
  "formula": {
    "tex": "P(Q,Z)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>P</mi><mo>(</mo><mi>Q</mi><mo>,</mo><mi>Z</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "P",
        "glyph": "P"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "Q",
          "glyph": "Q"
        },
        {
          "type": "symbol",
          "name": "Z",
          "glyph": "Z"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "3f010be84aae23f2acbd934d884b80b8013bbeb0b9837b94810c9ddd27263263"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "states/input/stack alphabets finite、unique且pairwise disjoint。"
    },
    {
      "id": "a2",
      "zh-Hant": "configuration=[state,input_position,stack]；stack bottom-first/top-last，null transition不增加input_position，bottom transition必須保留bottom。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "pushdown_automaton",
      "role_zh": "下推自動機",
      "scope": "representative_formula"
    },
    {
      "token": "Q",
      "role": "finite_state_set",
      "role_zh": "有限狀態集",
      "scope": "representative_formula"
    },
    {
      "token": "Z",
      "role": "bottom_stack_marker",
      "role_zh": "stack底標記",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-computational-model",
      "title_zh": "計算模型",
      "reason_zh": "finite transition model。"
    },
    {
      "id": "mko-finite-automaton",
      "title_zh": "有限自動機",
      "reason_zh": "finite control。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "finite input/stack/run。"
    },
    {
      "id": "mko-context-free-grammar",
      "title_zh": "上下文無關文法",
      "reason_zh": "language context without equivalence claim。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "acceptance convention固定為full input plus final state。",
      "transition tuple、input consumption與bottom-to-top replacement逐欄核對。",
      "finite run證acceptance；未讀完input即final仍不接受，bounded failure不證nonmembership。"
    ],
    "common_misconception_zh": "讓epsilon消耗input、pop掉bottom、混用empty stack與final state，或將搜尋深度不足當reject proof。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪不宣稱一般CFG↔PDA轉換、determinism或language nonmembership。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入CFG-PDA equivalence與pumping lemma。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-pushdown-automaton.md 整理；2026-10-14 脈衝 R07（使用者手動接續 2026-11-10 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}