{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0020", "pilot_number": "R0053-27395", "number": 27395, "family_id": "qf-mod49-pump-quantifier-audit", "title": { "zh-Hant": "第 27395 題：正規語言Pumping引理", "en": "Question 27395: Pumping Quantifiers Place the Word before the Split and the Pump Count after It" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0020", "batch_id": "batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-pumping-lemma-regular-languages" ], "title_zh": "正規語言Pumping引理", "stem_zh": "exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={\"p\":20,\"word\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],\"x_length\":19,\"y_length\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "answer": { "kind": "expression", "value": "(20, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 21, 19, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)", "explanation_zh": "Wave6第十二層核對=(20, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 21, 19, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)；依完整明示資料獨立重算；positive witness由完整DFA loop支撐全部i，a^p b^p反例涵蓋全部合法splits；pumping property只必要不充分；單一split或bounded words不能證regular。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "正規語言Pumping引理" ] }, "variant": { "family_id": "qf-mod49-pump-quantifier-audit", "seed": 202611187020, "parameters": { "p": 20, "word": [ "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "b" ], "x_length": 19, "y_length": 1, "domain": "Exact_regular_pumping_lemma", "operation": "pump-quantifier-audit", "semantic_template": "pump-quantifier-audit|{p:20,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:19,y_length:1}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_or_symbolic_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成一般limit theorem、coverage、number-theory或language theorem證明、人工審定、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-15", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261015-advance-20261118-r07-pumping-lemma-regular-languages-0020", "batch_id": "batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "126839ac1e999b5082932482a0268d6e5eb215a45df9024d56e069a5218dfd0d", "family_id": "qf-mod49-pump-quantifier-audit", "seed": 202611187020, "semantic_template": "pump-quantifier-audit|{p:20,word:[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],x_length:19,y_length:1}" }, "fields": { "title": "Pumping Quantifiers Place the Word before the Split and the Pump Count after It", "stem": "Use positive p=20, word=[a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], |x|=19, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order).", "answer": { "kind": "expression", "value": "(20, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 21, 19, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)", "explanation": "Check |w|≥p, |xy|≤p, |y|>0 and the remaining z length. The required statement is exists p, for every sufficiently long member word, there exists a split, for every i≥0. A legal split alone does not establish its all-i language property or regularity. The exact result, retaining the requested order, is (20, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 21, 19, 1, 1, Yes, Yes, Yes, exists_p_forall_w_exists_split_forall_i)." }, "skills": [ "Pumping Quantifiers Place the Word before the Split and the Pump Count after It" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "exists p forall w exists split forall i量詞與split constraints責任。操作=pump-quantifier-audit；資料={\"p\":20,\"word\":[\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"a\",\"b\"],\"x_length\":19,\"y_length\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。", "en": "Use positive p=20, word=[a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], |x|=19, |y|=1; w=xyz. Give (p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "source roles = [pump-quantifier-audit]", "after": "derived = [[p, word, word_length, x_length, y_length, z_length, word_long_enough, xy_within_p, y_nonempty, quantifier_order]]", "relation": "implies", "rule_id": "source_ordered_fields_without_JS", "conditions": [], "explanation": { "zh-Hant": "此頁直接列完整來源欄序與mode特定角色，不靠JavaScript或由答案外形猜介面；No、undefined、infinity各保原意。", "en": "List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "source roles = [20, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 19, 1]", "after": "derived = [21, 1, 是, 是, 是, exists p forall MEMBER w exists split forall i>=0]", "relation": "implies", "rule_id": "Pumping_ORIGINAL_token_word_split_lengths_AND_quantifier_order", "conditions": [], "explanation": { "zh-Hant": "word是symbol array，長度按tokens計，空字串token也不是自動刪掉。先核合法長度；regular pumping量詞是先p，再每個足夠長的member word，再exists split，最後所有i。此步不由合法split反推regular或已證所有i會員。", "en": "The word is an array of symbols;length counts tokens,including empty-string tokens.First check legal lengths.The pumping order is p,then every sufficiently long MEMBER word,then a split,then ALL i.A legal split alone proves neither regularity nor all-i membership." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。", "en": "Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct." } }, "arithmetic": null } ], "math_model": { "result": [ 20, [ "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "a", "b" ], 21, 19, 1, 1, true, true, true, "exists_p_forall_w_exists_split_forall_i" ] }, "conclusion": { "canonical_value": "(20, [a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, a, b], 21, 19, 1, 1, 是, 是, 是, exists_p_forall_w_exists_split_forall_i)", "explanation": { "zh-Hant": "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。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "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。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations", "batch_path": "public/data/questions/batches/batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations/questions.jsonl", "batch_sha256": "657cec46a009582d6773652bd01d9e8b1a552cf28a6e0706bdf7b5813871f2ec", "original_line": 20, "original_line_sha256": "4470df22548b95b17307ae734d9433bea907a7f7ca87ff4c4e9bf407224f5d5e", "question_compact_sha256": "126839ac1e999b5082932482a0268d6e5eb215a45df9024d56e069a5218dfd0d", "english_row_compact_sha256": "459eddf856fe64bd1a734fd046400cc3195ebfb474a8c8785b0cd0e8229d38d4", "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261118-r07-pumping-lemma-regular-languages-foundations.en.jsonl" }, "mko_bindings": { "mko-pumping-lemma-regular-languages": { "sha256": "7466612ae549a8cda880e2921d8c9163ab7681371f543eb905240583b66e6ed4", "english_sha256": "b98a8e7b0fb61714b4bea18947593dabe418b1c724d8240eaba1b40c826bd9ec", "original_path": "public/data/mko/pumping-lemma-regular-languages.json", "english_path": "public/data/i18n/editions/mko-pumping-lemma-regular-languages.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a54409c0bc3f68730e1072c6d3d772970462c1bc7ffc203b4030a6136a610283", "original_candidate_sha256": "97f58fae4fd48cc00838a0e04235a298e67a1b2138fe251cd0686916737f53f3", "original_candidate_line_sha256": "97f58fae4fd48cc00838a0e04235a298e67a1b2138fe251cd0686916737f53f3" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-010-p54", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62", "question_first_witness_sha256": "79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 2, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "2d9c2308c48874c44f47ef7e8fe29bfa4334c910fa6f4df35244ad6d2381b3e9", "arithmetic_steps": 0 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }