{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0001","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61678421d102170153e68126ba0cb168a1d19366fe2f4e1216e34b2ba812b3c0","family_id":"qf-mod40-cfg-qualification","seed":202611097001,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A0\",\"S0\"],\"productions\":[[\"A0\",[\"b0\",\"A0\"]],[\"A0\",[]],[\"S0\",[\"a0\",\"A0\"]]],\"start\":\"S0\",\"terminals\":[\"a0\",\"b0\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S0, A0], terminals=[a0, b0], start=S0, productions=[[S0, [a0, A0]], [A0, [b0, A0]], [A0, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S0, A0], [a0, b0], S0, [[S0, [a0, A0]], [A0, [b0, A0]], [A0, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S0, A0], [a0, b0], S0, [[S0, [a0, A0]], [A0, [b0, A0]], [A0, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0002","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b61e28b10b5b3a70e492d1e32223e738e34e46dbf3189e335cfc54b13e2acb3","family_id":"qf-mod40-cfg-qualification","seed":202611097002,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A1\",\"S1\"],\"productions\":[[\"A1\",[\"b1\",\"A1\"]],[\"A1\",[]],[\"S1\",[\"a1\",\"A1\"]]],\"start\":\"S1\",\"terminals\":[\"a1\",\"b1\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S1, A1], terminals=[a1, b1], start=S1, productions=[[S1, [a1, A1]], [A1, [b1, A1]], [A1, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S1, A1], [a1, b1], S1, [[S1, [a1, A1]], [A1, [b1, A1]], [A1, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S1, A1], [a1, b1], S1, [[S1, [a1, A1]], [A1, [b1, A1]], [A1, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0003","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f38fda15ccc80b56d2e0cb957fca5ec19f547aec446b7c24979c10920cf2251","family_id":"qf-mod40-cfg-qualification","seed":202611097003,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A2\",\"S2\"],\"productions\":[[\"A2\",[\"b2\",\"A2\"]],[\"A2\",[]],[\"S2\",[\"a2\",\"A2\"]]],\"start\":\"S2\",\"terminals\":[\"a2\",\"b2\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S2, A2], terminals=[a2, b2], start=S2, productions=[[S2, [a2, A2]], [A2, [b2, A2]], [A2, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S2, A2], [a2, b2], S2, [[S2, [a2, A2]], [A2, [b2, A2]], [A2, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S2, A2], [a2, b2], S2, [[S2, [a2, A2]], [A2, [b2, A2]], [A2, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0004","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccd6d2d07ad93cce0880c976182ce17d64916d833355a61aa90b5eae579b64ed","family_id":"qf-mod40-cfg-qualification","seed":202611097004,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A3\",\"S3\"],\"productions\":[[\"A3\",[\"b3\",\"A3\"]],[\"A3\",[]],[\"S3\",[\"a3\",\"A3\"]]],\"start\":\"S3\",\"terminals\":[\"a3\",\"b3\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S3, A3], terminals=[a3, b3], start=S3, productions=[[S3, [a3, A3]], [A3, [b3, A3]], [A3, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S3, A3], [a3, b3], S3, [[S3, [a3, A3]], [A3, [b3, A3]], [A3, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S3, A3], [a3, b3], S3, [[S3, [a3, A3]], [A3, [b3, A3]], [A3, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0005","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18036aaff8ec5fb6afa64d572fc2069b0d9139a2c3f367c0f0ea656865161c79","family_id":"qf-mod40-cfg-qualification","seed":202611097005,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A4\",\"S4\"],\"productions\":[[\"A4\",[\"b4\",\"A4\"]],[\"A4\",[]],[\"S4\",[\"a4\",\"A4\"]]],\"start\":\"S4\",\"terminals\":[\"a4\",\"b4\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S4, A4], terminals=[a4, b4], start=S4, productions=[[S4, [a4, A4]], [A4, [b4, A4]], [A4, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S4, A4], [a4, b4], S4, [[S4, [a4, A4]], [A4, [b4, A4]], [A4, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S4, A4], [a4, b4], S4, [[S4, [a4, A4]], [A4, [b4, A4]], [A4, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0006","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb35b4b5929067e8c7d847d80c587bd82382b122b406869bad48d320e19aac79","family_id":"qf-mod40-cfg-qualification","seed":202611097006,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A5\",\"S5\"],\"productions\":[[\"A5\",[\"b5\",\"A5\"]],[\"A5\",[]],[\"S5\",[\"a5\",\"A5\"]]],\"start\":\"S5\",\"terminals\":[\"a5\",\"b5\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S5, A5], terminals=[a5, b5], start=S5, productions=[[S5, [a5, A5]], [A5, [b5, A5]], [A5, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S5, A5], [a5, b5], S5, [[S5, [a5, A5]], [A5, [b5, A5]], [A5, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S5, A5], [a5, b5], S5, [[S5, [a5, A5]], [A5, [b5, A5]], [A5, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0007","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70856772ad977fe9173de95710b620d6b9e9127e9d52b55412250155a324e6ec","family_id":"qf-mod40-cfg-qualification","seed":202611097007,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A6\",\"S6\"],\"productions\":[[\"A6\",[\"b6\",\"A6\"]],[\"A6\",[]],[\"S6\",[\"a6\",\"A6\"]]],\"start\":\"S6\",\"terminals\":[\"a6\",\"b6\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S6, A6], terminals=[a6, b6], start=S6, productions=[[S6, [a6, A6]], [A6, [b6, A6]], [A6, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S6, A6], [a6, b6], S6, [[S6, [a6, A6]], [A6, [b6, A6]], [A6, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S6, A6], [a6, b6], S6, [[S6, [a6, A6]], [A6, [b6, A6]], [A6, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0008","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f53f906ef94cb08a6b1e71e19cc965c5cc7ca9724c482f270038d241a6f87ba0","family_id":"qf-mod40-cfg-qualification","seed":202611097008,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A7\",\"S7\"],\"productions\":[[\"A7\",[\"b7\",\"A7\"]],[\"A7\",[]],[\"S7\",[\"a7\",\"A7\"]]],\"start\":\"S7\",\"terminals\":[\"a7\",\"b7\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S7, A7], terminals=[a7, b7], start=S7, productions=[[S7, [a7, A7]], [A7, [b7, A7]], [A7, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S7, A7], [a7, b7], S7, [[S7, [a7, A7]], [A7, [b7, A7]], [A7, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S7, A7], [a7, b7], S7, [[S7, [a7, A7]], [A7, [b7, A7]], [A7, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0009","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9522335331016703005fbadd1f7b4b726e39675ba5c4a10b7a83b126ec4c306","family_id":"qf-mod40-cfg-qualification","seed":202611097009,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A8\",\"S8\"],\"productions\":[[\"A8\",[\"b8\",\"A8\"]],[\"A8\",[]],[\"S8\",[\"a8\",\"A8\"]]],\"start\":\"S8\",\"terminals\":[\"a8\",\"b8\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S8, A8], terminals=[a8, b8], start=S8, productions=[[S8, [a8, A8]], [A8, [b8, A8]], [A8, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S8, A8], [a8, b8], S8, [[S8, [a8, A8]], [A8, [b8, A8]], [A8, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S8, A8], [a8, b8], S8, [[S8, [a8, A8]], [A8, [b8, A8]], [A8, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0010","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cd52e7973c0c342190c0541c15ec479201af2362216657c9c8511466b95ff22","family_id":"qf-mod40-cfg-qualification","seed":202611097010,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A9\",\"S9\"],\"productions\":[[\"A9\",[\"b9\",\"A9\"]],[\"A9\",[]],[\"S9\",[\"a9\",\"A9\"]]],\"start\":\"S9\",\"terminals\":[\"a9\",\"b9\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S9, A9], terminals=[a9, b9], start=S9, productions=[[S9, [a9, A9]], [A9, [b9, A9]], [A9, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S9, A9], [a9, b9], S9, [[S9, [a9, A9]], [A9, [b9, A9]], [A9, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S9, A9], [a9, b9], S9, [[S9, [a9, A9]], [A9, [b9, A9]], [A9, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0011","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7182ddddbb8803e32b1b9f110f2a72320dc8ccbc906b3fdb385ea42b888c4a34","family_id":"qf-mod40-cfg-qualification","seed":202611097011,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A10\",\"S10\"],\"productions\":[[\"A10\",[\"b10\",\"A10\"]],[\"A10\",[]],[\"S10\",[\"a10\",\"A10\"]]],\"start\":\"S10\",\"terminals\":[\"a10\",\"b10\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S10, A10], terminals=[a10, b10], start=S10, productions=[[S10, [a10, A10]], [A10, [b10, A10]], [A10, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S10, A10], [a10, b10], S10, [[S10, [a10, A10]], [A10, [b10, A10]], [A10, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S10, A10], [a10, b10], S10, [[S10, [a10, A10]], [A10, [b10, A10]], [A10, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0012","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6283bff74036f6ca63527b67e2f2cd7fb19971b23e8fee57725b335cba82c737","family_id":"qf-mod40-cfg-qualification","seed":202611097012,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A11\",\"S11\"],\"productions\":[[\"A11\",[\"b11\",\"A11\"]],[\"A11\",[]],[\"S11\",[\"a11\",\"A11\"]]],\"start\":\"S11\",\"terminals\":[\"a11\",\"b11\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S11, A11], terminals=[a11, b11], start=S11, productions=[[S11, [a11, A11]], [A11, [b11, A11]], [A11, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S11, A11], [a11, b11], S11, [[S11, [a11, A11]], [A11, [b11, A11]], [A11, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S11, A11], [a11, b11], S11, [[S11, [a11, A11]], [A11, [b11, A11]], [A11, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0013","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31638300525162f19636c0947d0c292fbc09adaf8ffedb5bf2d1f09a00abab85","family_id":"qf-mod40-cfg-qualification","seed":202611097013,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A12\",\"S12\"],\"productions\":[[\"A12\",[\"b12\",\"A12\"]],[\"A12\",[]],[\"S12\",[\"a12\",\"A12\"]]],\"start\":\"S12\",\"terminals\":[\"a12\",\"b12\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S12, A12], terminals=[a12, b12], start=S12, productions=[[S12, [a12, A12]], [A12, [b12, A12]], [A12, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S12, A12], [a12, b12], S12, [[S12, [a12, A12]], [A12, [b12, A12]], [A12, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S12, A12], [a12, b12], S12, [[S12, [a12, A12]], [A12, [b12, A12]], [A12, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0014","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f55462241f4ebc778fc6d08d66a46b430d4c6cca938f9ba575fdf1ce683ccdd0","family_id":"qf-mod40-cfg-qualification","seed":202611097014,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A13\",\"S13\"],\"productions\":[[\"A13\",[\"b13\",\"A13\"]],[\"A13\",[]],[\"S13\",[\"a13\",\"A13\"]]],\"start\":\"S13\",\"terminals\":[\"a13\",\"b13\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S13, A13], terminals=[a13, b13], start=S13, productions=[[S13, [a13, A13]], [A13, [b13, A13]], [A13, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S13, A13], [a13, b13], S13, [[S13, [a13, A13]], [A13, [b13, A13]], [A13, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S13, A13], [a13, b13], S13, [[S13, [a13, A13]], [A13, [b13, A13]], [A13, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0015","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce3ed7c0126478f8ef412cdba62749a9d32f001b9434851207f8aa00c6f326d0","family_id":"qf-mod40-cfg-qualification","seed":202611097015,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A14\",\"S14\"],\"productions\":[[\"A14\",[\"b14\",\"A14\"]],[\"A14\",[]],[\"S14\",[\"a14\",\"A14\"]]],\"start\":\"S14\",\"terminals\":[\"a14\",\"b14\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S14, A14], terminals=[a14, b14], start=S14, productions=[[S14, [a14, A14]], [A14, [b14, A14]], [A14, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S14, A14], [a14, b14], S14, [[S14, [a14, A14]], [A14, [b14, A14]], [A14, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S14, A14], [a14, b14], S14, [[S14, [a14, A14]], [A14, [b14, A14]], [A14, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0016","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92a06eb7c3e7bea500e7e0022110265d89ae3650326a2c87d707de99cbdb7cd1","family_id":"qf-mod40-cfg-qualification","seed":202611097016,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A15\",\"S15\"],\"productions\":[[\"A15\",[\"b15\",\"A15\"]],[\"A15\",[]],[\"S15\",[\"a15\",\"A15\"]]],\"start\":\"S15\",\"terminals\":[\"a15\",\"b15\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S15, A15], terminals=[a15, b15], start=S15, productions=[[S15, [a15, A15]], [A15, [b15, A15]], [A15, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S15, A15], [a15, b15], S15, [[S15, [a15, A15]], [A15, [b15, A15]], [A15, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S15, A15], [a15, b15], S15, [[S15, [a15, A15]], [A15, [b15, A15]], [A15, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0017","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04eb811b62ce7ae6f2b1577998f5c33896da7f2200a59bf2ad4d711d6cf9b8cd","family_id":"qf-mod40-cfg-qualification","seed":202611097017,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A16\",\"S16\"],\"productions\":[[\"A16\",[\"b16\",\"A16\"]],[\"A16\",[]],[\"S16\",[\"a16\",\"A16\"]]],\"start\":\"S16\",\"terminals\":[\"a16\",\"b16\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S16, A16], terminals=[a16, b16], start=S16, productions=[[S16, [a16, A16]], [A16, [b16, A16]], [A16, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S16, A16], [a16, b16], S16, [[S16, [a16, A16]], [A16, [b16, A16]], [A16, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S16, A16], [a16, b16], S16, [[S16, [a16, A16]], [A16, [b16, A16]], [A16, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0018","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef1750128614f1485edfae390716262f0a133ec1d0beddbc838cc60c31102ce8","family_id":"qf-mod40-cfg-qualification","seed":202611097018,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A17\",\"S17\"],\"productions\":[[\"A17\",[\"b17\",\"A17\"]],[\"A17\",[]],[\"S17\",[\"a17\",\"A17\"]]],\"start\":\"S17\",\"terminals\":[\"a17\",\"b17\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S17, A17], terminals=[a17, b17], start=S17, productions=[[S17, [a17, A17]], [A17, [b17, A17]], [A17, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S17, A17], [a17, b17], S17, [[S17, [a17, A17]], [A17, [b17, A17]], [A17, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S17, A17], [a17, b17], S17, [[S17, [a17, A17]], [A17, [b17, A17]], [A17, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0019","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0116cc6ff62ae5d7f1dd4e5d8b88ae625aa2262f378eea29fbf45b6e2078a5f4","family_id":"qf-mod40-cfg-qualification","seed":202611097019,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A18\",\"S18\"],\"productions\":[[\"A18\",[\"b18\",\"A18\"]],[\"A18\",[]],[\"S18\",[\"a18\",\"A18\"]]],\"start\":\"S18\",\"terminals\":[\"a18\",\"b18\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S18, A18], terminals=[a18, b18], start=S18, productions=[[S18, [a18, A18]], [A18, [b18, A18]], [A18, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S18, A18], [a18, b18], S18, [[S18, [a18, A18]], [A18, [b18, A18]], [A18, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S18, A18], [a18, b18], S18, [[S18, [a18, A18]], [A18, [b18, A18]], [A18, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0020","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79aec1e489966860b23625ca9f8bf7dfe38f0841629de503b77755c0b9bfc671","family_id":"qf-mod40-cfg-qualification","seed":202611097020,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A19\",\"S19\"],\"productions\":[[\"A19\",[\"b19\",\"A19\"]],[\"A19\",[]],[\"S19\",[\"a19\",\"A19\"]]],\"start\":\"S19\",\"terminals\":[\"a19\",\"b19\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S19, A19], terminals=[a19, b19], start=S19, productions=[[S19, [a19, A19]], [A19, [b19, A19]], [A19, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S19, A19], [a19, b19], S19, [[S19, [a19, A19]], [A19, [b19, A19]], [A19, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S19, A19], [a19, b19], S19, [[S19, [a19, A19]], [A19, [b19, A19]], [A19, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0021","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5db0c52a397395ba3288ecf38f86d86f98137c765f14c86c7aca5ea42765c89","family_id":"qf-mod40-cfg-qualification","seed":202611097021,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A20\",\"S20\"],\"productions\":[[\"A20\",[\"b20\",\"A20\"]],[\"A20\",[]],[\"S20\",[\"a20\",\"A20\"]]],\"start\":\"S20\",\"terminals\":[\"a20\",\"b20\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S20, A20], terminals=[a20, b20], start=S20, productions=[[S20, [a20, A20]], [A20, [b20, A20]], [A20, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S20, A20], [a20, b20], S20, [[S20, [a20, A20]], [A20, [b20, A20]], [A20, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S20, A20], [a20, b20], S20, [[S20, [a20, A20]], [A20, [b20, A20]], [A20, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0022","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0db105aa7693ae54ac097e06d9cae3fa5d0dbc5d11142d85af89cf62568a768a","family_id":"qf-mod40-cfg-qualification","seed":202611097022,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A21\",\"S21\"],\"productions\":[[\"A21\",[\"b21\",\"A21\"]],[\"A21\",[]],[\"S21\",[\"a21\",\"A21\"]]],\"start\":\"S21\",\"terminals\":[\"a21\",\"b21\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S21, A21], terminals=[a21, b21], start=S21, productions=[[S21, [a21, A21]], [A21, [b21, A21]], [A21, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S21, A21], [a21, b21], S21, [[S21, [a21, A21]], [A21, [b21, A21]], [A21, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S21, A21], [a21, b21], S21, [[S21, [a21, A21]], [A21, [b21, A21]], [A21, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0023","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"822fdf258b2f37c744bcdd124ba763a03583263dbad0f45323012341fab5efb1","family_id":"qf-mod40-cfg-qualification","seed":202611097023,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A22\",\"S22\"],\"productions\":[[\"A22\",[\"b22\",\"A22\"]],[\"A22\",[]],[\"S22\",[\"a22\",\"A22\"]]],\"start\":\"S22\",\"terminals\":[\"a22\",\"b22\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S22, A22], terminals=[a22, b22], start=S22, productions=[[S22, [a22, A22]], [A22, [b22, A22]], [A22, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S22, A22], [a22, b22], S22, [[S22, [a22, A22]], [A22, [b22, A22]], [A22, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S22, A22], [a22, b22], S22, [[S22, [a22, A22]], [A22, [b22, A22]], [A22, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0024","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1770ce767baa2ccb6a7e0bcba7fe1bfbe5a6bcc9ef329b49dce11ac8199b092a","family_id":"qf-mod40-cfg-qualification","seed":202611097024,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A23\",\"S23\"],\"productions\":[[\"A23\",[\"b23\",\"A23\"]],[\"A23\",[]],[\"S23\",[\"a23\",\"A23\"]]],\"start\":\"S23\",\"terminals\":[\"a23\",\"b23\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S23, A23], terminals=[a23, b23], start=S23, productions=[[S23, [a23, A23]], [A23, [b23, A23]], [A23, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S23, A23], [a23, b23], S23, [[S23, [a23, A23]], [A23, [b23, A23]], [A23, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S23, A23], [a23, b23], S23, [[S23, [a23, A23]], [A23, [b23, A23]], [A23, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0025","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd0dc6070919137ba778fe13e19d88371e02d1d471bc6f0e4422f4b20fd2e611","family_id":"qf-mod40-cfg-qualification","seed":202611097025,"semantic_template":"qf-mod40-cfg-qualification/{\"allow_epsilon\":true,\"nonterminals\":[\"A24\",\"S24\"],\"productions\":[[\"A24\",[\"b24\",\"A24\"]],[\"A24\",[]],[\"S24\",[\"a24\",\"A24\"]]],\"start\":\"S24\",\"terminals\":[\"a24\",\"b24\"]}"},"fields":{"title":"Grammar Qualification Requires Complete Alphabets, Start and Productions","stem":"Use the complete finite grammar: nonterminals=[S24, A24], terminals=[a24, b24], start=S24, productions=[[S24, [a24, A24]], [A24, [b24, A24]], [A24, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Give (nonterminals, terminals, start, productions, allow_epsilon, grammar_qualified, certificate).","answer":{"kind":"expression","value":"([S24, A24], [a24, b24], S24, [[S24, [a24, A24]], [A24, [b24, A24]], [A24, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)","explanation":"Report the original alphabets, start, production list and explicit epsilon flag, followed by the qualification certificate. A legal grammar definition alone is not a derivation or a word-membership witness. The exact result, retaining the requested order, is ([S24, A24], [a24, b24], S24, [[S24, [a24, A24]], [A24, [b24, A24]], [A24, []]], Yes, Yes, finite_disjoint_alphabets_single_nonterminal_lhs)."},"skills":["Grammar Qualification Requires Complete Alphabets, Start and Productions"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0026","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acec022aaab6e7b276eab28765b42fbde274867b5f7ecf34688663f91e835582","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097026,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S0\"],[\"a0\",\"A0\"],[\"a0\"]],\"nonterminals\":[\"A0\",\"S0\"],\"productions\":[[\"A0\",[\"b0\",\"A0\"]],[\"A0\",[]],[\"S0\",[\"a0\",\"A0\"]]],\"start\":\"S0\",\"terminals\":[\"a0\",\"b0\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S0, A0], terminals=[a0, b0], start=S0, productions=[[S0, [a0, A0]], [A0, [b0, A0]], [A0, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S0], [a0, A0], [a0]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S0], [a0, A0], [a0]], [[0, 0, S0, [a0, A0]], [1, 2, A0, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S0], [a0, A0], [a0]], [[0, 0, S0, [a0, A0]], [1, 2, A0, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0027","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ef6bbaf4277f18b3639d799d4840df916314fafbb16e35ad70deab1df6fc1bf","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097027,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S1\"],[\"a1\",\"A1\"],[\"a1\",\"b1\",\"A1\"],[\"a1\",\"b1\"]],\"nonterminals\":[\"A1\",\"S1\"],\"productions\":[[\"A1\",[\"b1\",\"A1\"]],[\"A1\",[]],[\"S1\",[\"a1\",\"A1\"]]],\"start\":\"S1\",\"terminals\":[\"a1\",\"b1\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S1, A1], terminals=[a1, b1], start=S1, productions=[[S1, [a1, A1]], [A1, [b1, A1]], [A1, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S1], [a1, A1], [a1, b1, A1], [a1, b1]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S1], [a1, A1], [a1, b1, A1], [a1, b1]], [[0, 0, S1, [a1, A1]], [1, 1, A1, [b1, A1]], [2, 2, A1, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S1], [a1, A1], [a1, b1, A1], [a1, b1]], [[0, 0, S1, [a1, A1]], [1, 1, A1, [b1, A1]], [2, 2, A1, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0028","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e61b7830f02dd3ff6c8afe3dcf8fac650c3cb23d1a71a3556a5ead77095ceea","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097028,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S2\"],[\"a2\",\"A2\"],[\"a2\",\"b2\",\"A2\"],[\"a2\",\"b2\",\"b2\",\"A2\"],[\"a2\",\"b2\",\"b2\"]],\"nonterminals\":[\"A2\",\"S2\"],\"productions\":[[\"A2\",[\"b2\",\"A2\"]],[\"A2\",[]],[\"S2\",[\"a2\",\"A2\"]]],\"start\":\"S2\",\"terminals\":[\"a2\",\"b2\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S2, A2], terminals=[a2, b2], start=S2, productions=[[S2, [a2, A2]], [A2, [b2, A2]], [A2, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S2], [a2, A2], [a2, b2, A2], [a2, b2, b2, A2], [a2, b2, b2]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S2], [a2, A2], [a2, b2, A2], [a2, b2, b2, A2], [a2, b2, b2]], [[0, 0, S2, [a2, A2]], [1, 1, A2, [b2, A2]], [2, 1, A2, [b2, A2]], [3, 2, A2, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S2], [a2, A2], [a2, b2, A2], [a2, b2, b2, A2], [a2, b2, b2]], [[0, 0, S2, [a2, A2]], [1, 1, A2, [b2, A2]], [2, 1, A2, [b2, A2]], [3, 2, A2, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0029","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afa45d5d46b48ed4f539c5f449eabc89d538e84b4d5b70f006ffc2f6ccfb350c","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097029,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S3\"],[\"a3\",\"A3\"],[\"a3\",\"b3\",\"A3\"],[\"a3\",\"b3\",\"b3\",\"A3\"],[\"a3\",\"b3\",\"b3\",\"b3\",\"A3\"],[\"a3\",\"b3\",\"b3\",\"b3\"]],\"nonterminals\":[\"A3\",\"S3\"],\"productions\":[[\"A3\",[\"b3\",\"A3\"]],[\"A3\",[]],[\"S3\",[\"a3\",\"A3\"]]],\"start\":\"S3\",\"terminals\":[\"a3\",\"b3\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S3, A3], terminals=[a3, b3], start=S3, productions=[[S3, [a3, A3]], [A3, [b3, A3]], [A3, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S3], [a3, A3], [a3, b3, A3], [a3, b3, b3, A3], [a3, b3, b3, b3, A3], [a3, b3, b3, b3]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S3], [a3, A3], [a3, b3, A3], [a3, b3, b3, A3], [a3, b3, b3, b3, A3], [a3, b3, b3, b3]], [[0, 0, S3, [a3, A3]], [1, 1, A3, [b3, A3]], [2, 1, A3, [b3, A3]], [3, 1, A3, [b3, A3]], [4, 2, A3, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S3], [a3, A3], [a3, b3, A3], [a3, b3, b3, A3], [a3, b3, b3, b3, A3], [a3, b3, b3, b3]], [[0, 0, S3, [a3, A3]], [1, 1, A3, [b3, A3]], [2, 1, A3, [b3, A3]], [3, 1, A3, [b3, A3]], [4, 2, A3, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0030","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b8ea2f7f073a879f20e46d9a82c2ede71f6e376e2678f25691c87715aaf1a81","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097030,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S4\"],[\"a4\",\"A4\"],[\"a4\",\"b4\",\"A4\"],[\"a4\",\"b4\",\"b4\",\"A4\"],[\"a4\",\"b4\",\"b4\",\"b4\",\"A4\"],[\"a4\",\"b4\",\"b4\",\"b4\",\"b4\",\"A4\"],[\"a4\",\"b4\",\"b4\",\"b4\",\"b4\"]],\"nonterminals\":[\"A4\",\"S4\"],\"productions\":[[\"A4\",[\"b4\",\"A4\"]],[\"A4\",[]],[\"S4\",[\"a4\",\"A4\"]]],\"start\":\"S4\",\"terminals\":[\"a4\",\"b4\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S4, A4], terminals=[a4, b4], start=S4, productions=[[S4, [a4, A4]], [A4, [b4, A4]], [A4, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S4], [a4, A4], [a4, b4, A4], [a4, b4, b4, A4], [a4, b4, b4, b4, A4], [a4, b4, b4, b4, b4, A4], [a4, b4, b4, b4, b4]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S4], [a4, A4], [a4, b4, A4], [a4, b4, b4, A4], [a4, b4, b4, b4, A4], [a4, b4, b4, b4, b4, A4], [a4, b4, b4, b4, b4]], [[0, 0, S4, [a4, A4]], [1, 1, A4, [b4, A4]], [2, 1, A4, [b4, A4]], [3, 1, A4, [b4, A4]], [4, 1, A4, [b4, A4]], [5, 2, A4, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S4], [a4, A4], [a4, b4, A4], [a4, b4, b4, A4], [a4, b4, b4, b4, A4], [a4, b4, b4, b4, b4, A4], [a4, b4, b4, b4, b4]], [[0, 0, S4, [a4, A4]], [1, 1, A4, [b4, A4]], [2, 1, A4, [b4, A4]], [3, 1, A4, [b4, A4]], [4, 1, A4, [b4, A4]], [5, 2, A4, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0031","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b3ebbc4fb6ad3470d0cea5c3d45b2b7f8e329b99755f9111661482330086e09","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097031,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S5\"],[\"a5\",\"A5\"],[\"a5\"]],\"nonterminals\":[\"A5\",\"S5\"],\"productions\":[[\"A5\",[\"b5\",\"A5\"]],[\"A5\",[]],[\"S5\",[\"a5\",\"A5\"]]],\"start\":\"S5\",\"terminals\":[\"a5\",\"b5\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S5, A5], terminals=[a5, b5], start=S5, productions=[[S5, [a5, A5]], [A5, [b5, A5]], [A5, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S5], [a5, A5], [a5]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S5], [a5, A5], [a5]], [[0, 0, S5, [a5, A5]], [1, 2, A5, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S5], [a5, A5], [a5]], [[0, 0, S5, [a5, A5]], [1, 2, A5, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0032","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac3c1147ebfd2449fdf1110b0ac1ea60578c0259007079bab5690378b6f2b0e2","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097032,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S6\"],[\"a6\",\"A6\"],[\"a6\",\"b6\",\"A6\"],[\"a6\",\"b6\"]],\"nonterminals\":[\"A6\",\"S6\"],\"productions\":[[\"A6\",[\"b6\",\"A6\"]],[\"A6\",[]],[\"S6\",[\"a6\",\"A6\"]]],\"start\":\"S6\",\"terminals\":[\"a6\",\"b6\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S6, A6], terminals=[a6, b6], start=S6, productions=[[S6, [a6, A6]], [A6, [b6, A6]], [A6, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S6], [a6, A6], [a6, b6, A6], [a6, b6]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S6], [a6, A6], [a6, b6, A6], [a6, b6]], [[0, 0, S6, [a6, A6]], [1, 1, A6, [b6, A6]], [2, 2, A6, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S6], [a6, A6], [a6, b6, A6], [a6, b6]], [[0, 0, S6, [a6, A6]], [1, 1, A6, [b6, A6]], [2, 2, A6, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0033","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"661c6fe272ca19c75db954e3a1fd84a32c1eca66d056d80a2080da02dfebf560","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097033,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S7\"],[\"a7\",\"A7\"],[\"a7\",\"b7\",\"A7\"],[\"a7\",\"b7\",\"b7\",\"A7\"],[\"a7\",\"b7\",\"b7\"]],\"nonterminals\":[\"A7\",\"S7\"],\"productions\":[[\"A7\",[\"b7\",\"A7\"]],[\"A7\",[]],[\"S7\",[\"a7\",\"A7\"]]],\"start\":\"S7\",\"terminals\":[\"a7\",\"b7\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S7, A7], terminals=[a7, b7], start=S7, productions=[[S7, [a7, A7]], [A7, [b7, A7]], [A7, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S7], [a7, A7], [a7, b7, A7], [a7, b7, b7, A7], [a7, b7, b7]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S7], [a7, A7], [a7, b7, A7], [a7, b7, b7, A7], [a7, b7, b7]], [[0, 0, S7, [a7, A7]], [1, 1, A7, [b7, A7]], [2, 1, A7, [b7, A7]], [3, 2, A7, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S7], [a7, A7], [a7, b7, A7], [a7, b7, b7, A7], [a7, b7, b7]], [[0, 0, S7, [a7, A7]], [1, 1, A7, [b7, A7]], [2, 1, A7, [b7, A7]], [3, 2, A7, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0034","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24cae9e029e3f29d06b748887de24d4dbff30db38fd4ab651ba49aa33821facc","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097034,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S8\"],[\"a8\",\"A8\"],[\"a8\",\"b8\",\"A8\"],[\"a8\",\"b8\",\"b8\",\"A8\"],[\"a8\",\"b8\",\"b8\",\"b8\",\"A8\"],[\"a8\",\"b8\",\"b8\",\"b8\"]],\"nonterminals\":[\"A8\",\"S8\"],\"productions\":[[\"A8\",[\"b8\",\"A8\"]],[\"A8\",[]],[\"S8\",[\"a8\",\"A8\"]]],\"start\":\"S8\",\"terminals\":[\"a8\",\"b8\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S8, A8], terminals=[a8, b8], start=S8, productions=[[S8, [a8, A8]], [A8, [b8, A8]], [A8, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S8], [a8, A8], [a8, b8, A8], [a8, b8, b8, A8], [a8, b8, b8, b8, A8], [a8, b8, b8, b8]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S8], [a8, A8], [a8, b8, A8], [a8, b8, b8, A8], [a8, b8, b8, b8, A8], [a8, b8, b8, b8]], [[0, 0, S8, [a8, A8]], [1, 1, A8, [b8, A8]], [2, 1, A8, [b8, A8]], [3, 1, A8, [b8, A8]], [4, 2, A8, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S8], [a8, A8], [a8, b8, A8], [a8, b8, b8, A8], [a8, b8, b8, b8, A8], [a8, b8, b8, b8]], [[0, 0, S8, [a8, A8]], [1, 1, A8, [b8, A8]], [2, 1, A8, [b8, A8]], [3, 1, A8, [b8, A8]], [4, 2, A8, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0035","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9a9be3960feec2f2f5703ff53e51c5d05a0895475243e44eac41e9084b5b694","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097035,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S9\"],[\"a9\",\"A9\"],[\"a9\",\"b9\",\"A9\"],[\"a9\",\"b9\",\"b9\",\"A9\"],[\"a9\",\"b9\",\"b9\",\"b9\",\"A9\"],[\"a9\",\"b9\",\"b9\",\"b9\",\"b9\",\"A9\"],[\"a9\",\"b9\",\"b9\",\"b9\",\"b9\"]],\"nonterminals\":[\"A9\",\"S9\"],\"productions\":[[\"A9\",[\"b9\",\"A9\"]],[\"A9\",[]],[\"S9\",[\"a9\",\"A9\"]]],\"start\":\"S9\",\"terminals\":[\"a9\",\"b9\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S9, A9], terminals=[a9, b9], start=S9, productions=[[S9, [a9, A9]], [A9, [b9, A9]], [A9, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S9], [a9, A9], [a9, b9, A9], [a9, b9, b9, A9], [a9, b9, b9, b9, A9], [a9, b9, b9, b9, b9, A9], [a9, b9, b9, b9, b9]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S9], [a9, A9], [a9, b9, A9], [a9, b9, b9, A9], [a9, b9, b9, b9, A9], [a9, b9, b9, b9, b9, A9], [a9, b9, b9, b9, b9]], [[0, 0, S9, [a9, A9]], [1, 1, A9, [b9, A9]], [2, 1, A9, [b9, A9]], [3, 1, A9, [b9, A9]], [4, 1, A9, [b9, A9]], [5, 2, A9, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S9], [a9, A9], [a9, b9, A9], [a9, b9, b9, A9], [a9, b9, b9, b9, A9], [a9, b9, b9, b9, b9, A9], [a9, b9, b9, b9, b9]], [[0, 0, S9, [a9, A9]], [1, 1, A9, [b9, A9]], [2, 1, A9, [b9, A9]], [3, 1, A9, [b9, A9]], [4, 1, A9, [b9, A9]], [5, 2, A9, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0036","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1be55cea283693fa209914a171498593fe83fcd37455a032ad8f534854c4f619","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097036,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S10\"],[\"a10\",\"A10\"],[\"a10\"]],\"nonterminals\":[\"A10\",\"S10\"],\"productions\":[[\"A10\",[\"b10\",\"A10\"]],[\"A10\",[]],[\"S10\",[\"a10\",\"A10\"]]],\"start\":\"S10\",\"terminals\":[\"a10\",\"b10\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S10, A10], terminals=[a10, b10], start=S10, productions=[[S10, [a10, A10]], [A10, [b10, A10]], [A10, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S10], [a10, A10], [a10]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S10], [a10, A10], [a10]], [[0, 0, S10, [a10, A10]], [1, 2, A10, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S10], [a10, A10], [a10]], [[0, 0, S10, [a10, A10]], [1, 2, A10, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0037","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c00cafb25fbcd343f30c079a6da86f4880cc8dc49c9b29b4faefe05e49e5603","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097037,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S11\"],[\"a11\",\"A11\"],[\"a11\",\"b11\",\"A11\"],[\"a11\",\"b11\"]],\"nonterminals\":[\"A11\",\"S11\"],\"productions\":[[\"A11\",[\"b11\",\"A11\"]],[\"A11\",[]],[\"S11\",[\"a11\",\"A11\"]]],\"start\":\"S11\",\"terminals\":[\"a11\",\"b11\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S11, A11], terminals=[a11, b11], start=S11, productions=[[S11, [a11, A11]], [A11, [b11, A11]], [A11, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S11], [a11, A11], [a11, b11, A11], [a11, b11]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S11], [a11, A11], [a11, b11, A11], [a11, b11]], [[0, 0, S11, [a11, A11]], [1, 1, A11, [b11, A11]], [2, 2, A11, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S11], [a11, A11], [a11, b11, A11], [a11, b11]], [[0, 0, S11, [a11, A11]], [1, 1, A11, [b11, A11]], [2, 2, A11, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0038","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"153221d459cf4ff7abae49cb090fd847844a88f4d5044d7f7281e6b361492610","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097038,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S12\"],[\"a12\",\"A12\"],[\"a12\",\"b12\",\"A12\"],[\"a12\",\"b12\",\"b12\",\"A12\"],[\"a12\",\"b12\",\"b12\"]],\"nonterminals\":[\"A12\",\"S12\"],\"productions\":[[\"A12\",[\"b12\",\"A12\"]],[\"A12\",[]],[\"S12\",[\"a12\",\"A12\"]]],\"start\":\"S12\",\"terminals\":[\"a12\",\"b12\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S12, A12], terminals=[a12, b12], start=S12, productions=[[S12, [a12, A12]], [A12, [b12, A12]], [A12, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S12], [a12, A12], [a12, b12, A12], [a12, b12, b12, A12], [a12, b12, b12]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S12], [a12, A12], [a12, b12, A12], [a12, b12, b12, A12], [a12, b12, b12]], [[0, 0, S12, [a12, A12]], [1, 1, A12, [b12, A12]], [2, 1, A12, [b12, A12]], [3, 2, A12, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S12], [a12, A12], [a12, b12, A12], [a12, b12, b12, A12], [a12, b12, b12]], [[0, 0, S12, [a12, A12]], [1, 1, A12, [b12, A12]], [2, 1, A12, [b12, A12]], [3, 2, A12, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0039","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"add277c67e0c987e6a8f6599a7cf540cd643fda697701f1ca06233e39c2d0013","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097039,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S13\"],[\"a13\",\"A13\"],[\"a13\",\"b13\",\"A13\"],[\"a13\",\"b13\",\"b13\",\"A13\"],[\"a13\",\"b13\",\"b13\",\"b13\",\"A13\"],[\"a13\",\"b13\",\"b13\",\"b13\"]],\"nonterminals\":[\"A13\",\"S13\"],\"productions\":[[\"A13\",[\"b13\",\"A13\"]],[\"A13\",[]],[\"S13\",[\"a13\",\"A13\"]]],\"start\":\"S13\",\"terminals\":[\"a13\",\"b13\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S13, A13], terminals=[a13, b13], start=S13, productions=[[S13, [a13, A13]], [A13, [b13, A13]], [A13, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S13], [a13, A13], [a13, b13, A13], [a13, b13, b13, A13], [a13, b13, b13, b13, A13], [a13, b13, b13, b13]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S13], [a13, A13], [a13, b13, A13], [a13, b13, b13, A13], [a13, b13, b13, b13, A13], [a13, b13, b13, b13]], [[0, 0, S13, [a13, A13]], [1, 1, A13, [b13, A13]], [2, 1, A13, [b13, A13]], [3, 1, A13, [b13, A13]], [4, 2, A13, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S13], [a13, A13], [a13, b13, A13], [a13, b13, b13, A13], [a13, b13, b13, b13, A13], [a13, b13, b13, b13]], [[0, 0, S13, [a13, A13]], [1, 1, A13, [b13, A13]], [2, 1, A13, [b13, A13]], [3, 1, A13, [b13, A13]], [4, 2, A13, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0040","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16a4de6ee25517726aa6723e3600d3f53a77c810bfbfc5a56cf5b8c23d95dd60","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097040,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S14\"],[\"a14\",\"A14\"],[\"a14\",\"b14\",\"A14\"],[\"a14\",\"b14\",\"b14\",\"A14\"],[\"a14\",\"b14\",\"b14\",\"b14\",\"A14\"],[\"a14\",\"b14\",\"b14\",\"b14\",\"b14\",\"A14\"],[\"a14\",\"b14\",\"b14\",\"b14\",\"b14\"]],\"nonterminals\":[\"A14\",\"S14\"],\"productions\":[[\"A14\",[\"b14\",\"A14\"]],[\"A14\",[]],[\"S14\",[\"a14\",\"A14\"]]],\"start\":\"S14\",\"terminals\":[\"a14\",\"b14\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S14, A14], terminals=[a14, b14], start=S14, productions=[[S14, [a14, A14]], [A14, [b14, A14]], [A14, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S14], [a14, A14], [a14, b14, A14], [a14, b14, b14, A14], [a14, b14, b14, b14, A14], [a14, b14, b14, b14, b14, A14], [a14, b14, b14, b14, b14]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S14], [a14, A14], [a14, b14, A14], [a14, b14, b14, A14], [a14, b14, b14, b14, A14], [a14, b14, b14, b14, b14, A14], [a14, b14, b14, b14, b14]], [[0, 0, S14, [a14, A14]], [1, 1, A14, [b14, A14]], [2, 1, A14, [b14, A14]], [3, 1, A14, [b14, A14]], [4, 1, A14, [b14, A14]], [5, 2, A14, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S14], [a14, A14], [a14, b14, A14], [a14, b14, b14, A14], [a14, b14, b14, b14, A14], [a14, b14, b14, b14, b14, A14], [a14, b14, b14, b14, b14]], [[0, 0, S14, [a14, A14]], [1, 1, A14, [b14, A14]], [2, 1, A14, [b14, A14]], [3, 1, A14, [b14, A14]], [4, 1, A14, [b14, A14]], [5, 2, A14, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0041","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca2ace0ca56645f6b0de3d8166772329b4e89edc1565dc42d1f23192f01729a7","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097041,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S15\"],[\"a15\",\"A15\"],[\"a15\"]],\"nonterminals\":[\"A15\",\"S15\"],\"productions\":[[\"A15\",[\"b15\",\"A15\"]],[\"A15\",[]],[\"S15\",[\"a15\",\"A15\"]]],\"start\":\"S15\",\"terminals\":[\"a15\",\"b15\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S15, A15], terminals=[a15, b15], start=S15, productions=[[S15, [a15, A15]], [A15, [b15, A15]], [A15, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S15], [a15, A15], [a15]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S15], [a15, A15], [a15]], [[0, 0, S15, [a15, A15]], [1, 2, A15, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S15], [a15, A15], [a15]], [[0, 0, S15, [a15, A15]], [1, 2, A15, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0042","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1e8b056735b80a987a94eed7461764c3aa8913e226a2f259a343bcc8593fb92","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097042,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S16\"],[\"a16\",\"A16\"],[\"a16\",\"b16\",\"A16\"],[\"a16\",\"b16\"]],\"nonterminals\":[\"A16\",\"S16\"],\"productions\":[[\"A16\",[\"b16\",\"A16\"]],[\"A16\",[]],[\"S16\",[\"a16\",\"A16\"]]],\"start\":\"S16\",\"terminals\":[\"a16\",\"b16\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S16, A16], terminals=[a16, b16], start=S16, productions=[[S16, [a16, A16]], [A16, [b16, A16]], [A16, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S16], [a16, A16], [a16, b16, A16], [a16, b16]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S16], [a16, A16], [a16, b16, A16], [a16, b16]], [[0, 0, S16, [a16, A16]], [1, 1, A16, [b16, A16]], [2, 2, A16, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S16], [a16, A16], [a16, b16, A16], [a16, b16]], [[0, 0, S16, [a16, A16]], [1, 1, A16, [b16, A16]], [2, 2, A16, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0043","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddbf92a1f6af6bb8f3514dd7785d6678bc04e426fdb02d4dc752de54cc4d37ad","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097043,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S17\"],[\"a17\",\"A17\"],[\"a17\",\"b17\",\"A17\"],[\"a17\",\"b17\",\"b17\",\"A17\"],[\"a17\",\"b17\",\"b17\"]],\"nonterminals\":[\"A17\",\"S17\"],\"productions\":[[\"A17\",[\"b17\",\"A17\"]],[\"A17\",[]],[\"S17\",[\"a17\",\"A17\"]]],\"start\":\"S17\",\"terminals\":[\"a17\",\"b17\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S17, A17], terminals=[a17, b17], start=S17, productions=[[S17, [a17, A17]], [A17, [b17, A17]], [A17, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S17], [a17, A17], [a17, b17, A17], [a17, b17, b17, A17], [a17, b17, b17]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S17], [a17, A17], [a17, b17, A17], [a17, b17, b17, A17], [a17, b17, b17]], [[0, 0, S17, [a17, A17]], [1, 1, A17, [b17, A17]], [2, 1, A17, [b17, A17]], [3, 2, A17, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S17], [a17, A17], [a17, b17, A17], [a17, b17, b17, A17], [a17, b17, b17]], [[0, 0, S17, [a17, A17]], [1, 1, A17, [b17, A17]], [2, 1, A17, [b17, A17]], [3, 2, A17, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0044","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51bde8c4024211fdafaacf69342524322af91e3adf9801fa69fc0422c7200dc1","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097044,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S18\"],[\"a18\",\"A18\"],[\"a18\",\"b18\",\"A18\"],[\"a18\",\"b18\",\"b18\",\"A18\"],[\"a18\",\"b18\",\"b18\",\"b18\",\"A18\"],[\"a18\",\"b18\",\"b18\",\"b18\"]],\"nonterminals\":[\"A18\",\"S18\"],\"productions\":[[\"A18\",[\"b18\",\"A18\"]],[\"A18\",[]],[\"S18\",[\"a18\",\"A18\"]]],\"start\":\"S18\",\"terminals\":[\"a18\",\"b18\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S18, A18], terminals=[a18, b18], start=S18, productions=[[S18, [a18, A18]], [A18, [b18, A18]], [A18, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S18], [a18, A18], [a18, b18, A18], [a18, b18, b18, A18], [a18, b18, b18, b18, A18], [a18, b18, b18, b18]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S18], [a18, A18], [a18, b18, A18], [a18, b18, b18, A18], [a18, b18, b18, b18, A18], [a18, b18, b18, b18]], [[0, 0, S18, [a18, A18]], [1, 1, A18, [b18, A18]], [2, 1, A18, [b18, A18]], [3, 1, A18, [b18, A18]], [4, 2, A18, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S18], [a18, A18], [a18, b18, A18], [a18, b18, b18, A18], [a18, b18, b18, b18, A18], [a18, b18, b18, b18]], [[0, 0, S18, [a18, A18]], [1, 1, A18, [b18, A18]], [2, 1, A18, [b18, A18]], [3, 1, A18, [b18, A18]], [4, 2, A18, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0045","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eea5adfa1907b1962111d7a6713c9c5becf3ba0849ce82c29767931a9b28ad6c","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097045,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S19\"],[\"a19\",\"A19\"],[\"a19\",\"b19\",\"A19\"],[\"a19\",\"b19\",\"b19\",\"A19\"],[\"a19\",\"b19\",\"b19\",\"b19\",\"A19\"],[\"a19\",\"b19\",\"b19\",\"b19\",\"b19\",\"A19\"],[\"a19\",\"b19\",\"b19\",\"b19\",\"b19\"]],\"nonterminals\":[\"A19\",\"S19\"],\"productions\":[[\"A19\",[\"b19\",\"A19\"]],[\"A19\",[]],[\"S19\",[\"a19\",\"A19\"]]],\"start\":\"S19\",\"terminals\":[\"a19\",\"b19\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S19, A19], terminals=[a19, b19], start=S19, productions=[[S19, [a19, A19]], [A19, [b19, A19]], [A19, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S19], [a19, A19], [a19, b19, A19], [a19, b19, b19, A19], [a19, b19, b19, b19, A19], [a19, b19, b19, b19, b19, A19], [a19, b19, b19, b19, b19]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S19], [a19, A19], [a19, b19, A19], [a19, b19, b19, A19], [a19, b19, b19, b19, A19], [a19, b19, b19, b19, b19, A19], [a19, b19, b19, b19, b19]], [[0, 0, S19, [a19, A19]], [1, 1, A19, [b19, A19]], [2, 1, A19, [b19, A19]], [3, 1, A19, [b19, A19]], [4, 1, A19, [b19, A19]], [5, 2, A19, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S19], [a19, A19], [a19, b19, A19], [a19, b19, b19, A19], [a19, b19, b19, b19, A19], [a19, b19, b19, b19, b19, A19], [a19, b19, b19, b19, b19]], [[0, 0, S19, [a19, A19]], [1, 1, A19, [b19, A19]], [2, 1, A19, [b19, A19]], [3, 1, A19, [b19, A19]], [4, 1, A19, [b19, A19]], [5, 2, A19, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0046","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7879ae40dbfd764a5bc3336e44860ddcdf58e3570c6924833306b4a92e48b04","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097046,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S20\"],[\"a20\",\"A20\"],[\"a20\"]],\"nonterminals\":[\"A20\",\"S20\"],\"productions\":[[\"A20\",[\"b20\",\"A20\"]],[\"A20\",[]],[\"S20\",[\"a20\",\"A20\"]]],\"start\":\"S20\",\"terminals\":[\"a20\",\"b20\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S20, A20], terminals=[a20, b20], start=S20, productions=[[S20, [a20, A20]], [A20, [b20, A20]], [A20, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S20], [a20, A20], [a20]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S20], [a20, A20], [a20]], [[0, 0, S20, [a20, A20]], [1, 2, A20, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S20], [a20, A20], [a20]], [[0, 0, S20, [a20, A20]], [1, 2, A20, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0047","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cee0ec7a4de2b18126419aedf2408b681770ce40e401c98715295df0ba305ee4","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097047,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S21\"],[\"a21\",\"A21\"],[\"a21\",\"b21\",\"A21\"],[\"a21\",\"b21\"]],\"nonterminals\":[\"A21\",\"S21\"],\"productions\":[[\"A21\",[\"b21\",\"A21\"]],[\"A21\",[]],[\"S21\",[\"a21\",\"A21\"]]],\"start\":\"S21\",\"terminals\":[\"a21\",\"b21\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S21, A21], terminals=[a21, b21], start=S21, productions=[[S21, [a21, A21]], [A21, [b21, A21]], [A21, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S21], [a21, A21], [a21, b21, A21], [a21, b21]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S21], [a21, A21], [a21, b21, A21], [a21, b21]], [[0, 0, S21, [a21, A21]], [1, 1, A21, [b21, A21]], [2, 2, A21, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S21], [a21, A21], [a21, b21, A21], [a21, b21]], [[0, 0, S21, [a21, A21]], [1, 1, A21, [b21, A21]], [2, 2, A21, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0048","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e00b81e92e28fc8eb5e7268cd4a0a76d3af990381c3c0f7490f01be6535b5c9","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097048,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S22\"],[\"a22\",\"A22\"],[\"a22\",\"b22\",\"A22\"],[\"a22\",\"b22\",\"b22\",\"A22\"],[\"a22\",\"b22\",\"b22\"]],\"nonterminals\":[\"A22\",\"S22\"],\"productions\":[[\"A22\",[\"b22\",\"A22\"]],[\"A22\",[]],[\"S22\",[\"a22\",\"A22\"]]],\"start\":\"S22\",\"terminals\":[\"a22\",\"b22\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S22, A22], terminals=[a22, b22], start=S22, productions=[[S22, [a22, A22]], [A22, [b22, A22]], [A22, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S22], [a22, A22], [a22, b22, A22], [a22, b22, b22, A22], [a22, b22, b22]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S22], [a22, A22], [a22, b22, A22], [a22, b22, b22, A22], [a22, b22, b22]], [[0, 0, S22, [a22, A22]], [1, 1, A22, [b22, A22]], [2, 1, A22, [b22, A22]], [3, 2, A22, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S22], [a22, A22], [a22, b22, A22], [a22, b22, b22, A22], [a22, b22, b22]], [[0, 0, S22, [a22, A22]], [1, 1, A22, [b22, A22]], [2, 1, A22, [b22, A22]], [3, 2, A22, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0049","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2432b4d6e55a9abc05b7bbbd281694b40cd293a19989b66d9d84df344a2437ee","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097049,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S23\"],[\"a23\",\"A23\"],[\"a23\",\"b23\",\"A23\"],[\"a23\",\"b23\",\"b23\",\"A23\"],[\"a23\",\"b23\",\"b23\",\"b23\",\"A23\"],[\"a23\",\"b23\",\"b23\",\"b23\"]],\"nonterminals\":[\"A23\",\"S23\"],\"productions\":[[\"A23\",[\"b23\",\"A23\"]],[\"A23\",[]],[\"S23\",[\"a23\",\"A23\"]]],\"start\":\"S23\",\"terminals\":[\"a23\",\"b23\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S23, A23], terminals=[a23, b23], start=S23, productions=[[S23, [a23, A23]], [A23, [b23, A23]], [A23, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S23], [a23, A23], [a23, b23, A23], [a23, b23, b23, A23], [a23, b23, b23, b23, A23], [a23, b23, b23, b23]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S23], [a23, A23], [a23, b23, A23], [a23, b23, b23, A23], [a23, b23, b23, b23, A23], [a23, b23, b23, b23]], [[0, 0, S23, [a23, A23]], [1, 1, A23, [b23, A23]], [2, 1, A23, [b23, A23]], [3, 1, A23, [b23, A23]], [4, 2, A23, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S23], [a23, A23], [a23, b23, A23], [a23, b23, b23, A23], [a23, b23, b23, b23, A23], [a23, b23, b23, b23]], [[0, 0, S23, [a23, A23]], [1, 1, A23, [b23, A23]], [2, 1, A23, [b23, A23]], [3, 1, A23, [b23, A23]], [4, 2, A23, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0050","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11a01061d3b8301470a36a2102a9d635672b671b099a7464bf2f8546acca164c","family_id":"qf-mod40-cfg-derivation-audit","seed":202611097050,"semantic_template":"qf-mod40-cfg-derivation-audit/{\"allow_epsilon\":true,\"forms\":[[\"S24\"],[\"a24\",\"A24\"],[\"a24\",\"b24\",\"A24\"],[\"a24\",\"b24\",\"b24\",\"A24\"],[\"a24\",\"b24\",\"b24\",\"b24\",\"A24\"],[\"a24\",\"b24\",\"b24\",\"b24\",\"b24\",\"A24\"],[\"a24\",\"b24\",\"b24\",\"b24\",\"b24\"]],\"nonterminals\":[\"A24\",\"S24\"],\"productions\":[[\"A24\",[\"b24\",\"A24\"]],[\"A24\",[]],[\"S24\",[\"a24\",\"A24\"]]],\"start\":\"S24\",\"terminals\":[\"a24\",\"b24\"]}"},"fields":{"title":"Each Derivation Step Replaces One Nonterminal by One Declared Right Side","stem":"Use the complete finite grammar: nonterminals=[S24, A24], terminals=[a24, b24], start=S24, productions=[[S24, [a24, A24]], [A24, [b24, A24]], [A24, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Supplied forms=[[S24], [a24, A24], [a24, b24, A24], [a24, b24, b24, A24], [a24, b24, b24, b24, A24], [a24, b24, b24, b24, b24, A24], [a24, b24, b24, b24, b24]]. Give (forms, step_witnesses, valid_derivation, final_form_terminal_only, certificate).","answer":{"kind":"expression","value":"([[S24], [a24, A24], [a24, b24, A24], [a24, b24, b24, A24], [a24, b24, b24, b24, A24], [a24, b24, b24, b24, b24, A24], [a24, b24, b24, b24, b24]], [[0, 0, S24, [a24, A24]], [1, 1, A24, [b24, A24]], [2, 1, A24, [b24, A24]], [3, 1, A24, [b24, A24]], [4, 1, A24, [b24, A24]], [5, 2, A24, []]], Yes, Yes, finite_production_sequence)","explanation":"A derivation must begin with [start]; each successive form replaces exactly one matching lhs occurrence. Each step witness is [position,production index,lhs,rhs], using the first matching witness in source order. Keep validity separate from whether the last form is terminal-only. The exact result, retaining the requested order, is ([[S24], [a24, A24], [a24, b24, A24], [a24, b24, b24, A24], [a24, b24, b24, b24, A24], [a24, b24, b24, b24, b24, A24], [a24, b24, b24, b24, b24]], [[0, 0, S24, [a24, A24]], [1, 1, A24, [b24, A24]], [2, 1, A24, [b24, A24]], [3, 1, A24, [b24, A24]], [4, 1, A24, [b24, A24]], [5, 2, A24, []]], Yes, Yes, finite_production_sequence)."},"skills":["Each Derivation Step Replaces One Nonterminal by One Declared Right Side"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0051","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8787ddb745e49202df98577be7a0215562851e3c7fb1685a9e74edd9cecbc018","family_id":"qf-mod40-cfg-membership-witness","seed":202611097051,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S0\"],[\"a0\",\"A0\"],[\"a0\"]],\"nonterminals\":[\"A0\",\"S0\"],\"productions\":[[\"A0\",[\"b0\",\"A0\"]],[\"A0\",[]],[\"S0\",[\"a0\",\"A0\"]]],\"start\":\"S0\",\"terminals\":[\"a0\",\"b0\"],\"word\":[\"a0\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S0, A0], terminals=[a0, b0], start=S0, productions=[[S0, [a0, A0]], [A0, [b0, A0]], [A0, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a0], supplied forms=[[S0], [a0, A0], [a0]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a0], [[0, 0, S0, [a0, A0]], [1, 2, A0, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a0], [[0, 0, S0, [a0, A0]], [1, 2, A0, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0052","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b8653d4b37a62c22b0249259be869472523af17195710766027154dcdc54c78","family_id":"qf-mod40-cfg-membership-witness","seed":202611097052,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S1\"],[\"a1\",\"A1\"],[\"a1\",\"b1\",\"A1\"],[\"a1\",\"b1\"]],\"nonterminals\":[\"A1\",\"S1\"],\"productions\":[[\"A1\",[\"b1\",\"A1\"]],[\"A1\",[]],[\"S1\",[\"a1\",\"A1\"]]],\"start\":\"S1\",\"terminals\":[\"a1\",\"b1\"],\"word\":[\"a1\",\"b1\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S1, A1], terminals=[a1, b1], start=S1, productions=[[S1, [a1, A1]], [A1, [b1, A1]], [A1, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a1, b1], supplied forms=[[S1], [a1, A1], [a1, b1, A1], [a1, b1]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a1, b1], [[0, 0, S1, [a1, A1]], [1, 1, A1, [b1, A1]], [2, 2, A1, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a1, b1], [[0, 0, S1, [a1, A1]], [1, 1, A1, [b1, A1]], [2, 2, A1, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0053","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20cf8b4dce318a213eebf686aa8f8cbe65f523cb8b01ee9a101317eb2f3b0ef7","family_id":"qf-mod40-cfg-membership-witness","seed":202611097053,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S2\"],[\"a2\",\"A2\"],[\"a2\",\"b2\",\"A2\"],[\"a2\",\"b2\",\"b2\",\"A2\"],[\"a2\",\"b2\",\"b2\"]],\"nonterminals\":[\"A2\",\"S2\"],\"productions\":[[\"A2\",[\"b2\",\"A2\"]],[\"A2\",[]],[\"S2\",[\"a2\",\"A2\"]]],\"start\":\"S2\",\"terminals\":[\"a2\",\"b2\"],\"word\":[\"a2\",\"b2\",\"b2\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S2, A2], terminals=[a2, b2], start=S2, productions=[[S2, [a2, A2]], [A2, [b2, A2]], [A2, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a2, b2, b2], supplied forms=[[S2], [a2, A2], [a2, b2, A2], [a2, b2, b2, A2], [a2, b2, b2]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a2, b2, b2], [[0, 0, S2, [a2, A2]], [1, 1, A2, [b2, A2]], [2, 1, A2, [b2, A2]], [3, 2, A2, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a2, b2, b2], [[0, 0, S2, [a2, A2]], [1, 1, A2, [b2, A2]], [2, 1, A2, [b2, A2]], [3, 2, A2, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0054","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cc427244bafa1bd8a93c75d3024e479a1a6211af4ae5d7db80becb87c5a0581","family_id":"qf-mod40-cfg-membership-witness","seed":202611097054,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S3\"],[\"a3\",\"A3\"],[\"a3\",\"b3\",\"A3\"],[\"a3\",\"b3\",\"b3\",\"A3\"],[\"a3\",\"b3\",\"b3\",\"b3\",\"A3\"],[\"a3\",\"b3\",\"b3\",\"b3\"]],\"nonterminals\":[\"A3\",\"S3\"],\"productions\":[[\"A3\",[\"b3\",\"A3\"]],[\"A3\",[]],[\"S3\",[\"a3\",\"A3\"]]],\"start\":\"S3\",\"terminals\":[\"a3\",\"b3\"],\"word\":[\"a3\",\"b3\",\"b3\",\"b3\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S3, A3], terminals=[a3, b3], start=S3, productions=[[S3, [a3, A3]], [A3, [b3, A3]], [A3, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a3, b3, b3, b3], supplied forms=[[S3], [a3, A3], [a3, b3, A3], [a3, b3, b3, A3], [a3, b3, b3, b3, A3], [a3, b3, b3, b3]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a3, b3, b3, b3], [[0, 0, S3, [a3, A3]], [1, 1, A3, [b3, A3]], [2, 1, A3, [b3, A3]], [3, 1, A3, [b3, A3]], [4, 2, A3, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a3, b3, b3, b3], [[0, 0, S3, [a3, A3]], [1, 1, A3, [b3, A3]], [2, 1, A3, [b3, A3]], [3, 1, A3, [b3, A3]], [4, 2, A3, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0055","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2425363a2e06ab26d6b194cee64516d0aad8703687f7c2296786d3ae7361634","family_id":"qf-mod40-cfg-membership-witness","seed":202611097055,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S4\"],[\"a4\",\"A4\"],[\"a4\",\"b4\",\"A4\"],[\"a4\",\"b4\",\"b4\",\"A4\"],[\"a4\",\"b4\",\"b4\",\"b4\",\"A4\"],[\"a4\",\"b4\",\"b4\",\"b4\",\"b4\",\"A4\"],[\"a4\",\"b4\",\"b4\",\"b4\",\"b4\"]],\"nonterminals\":[\"A4\",\"S4\"],\"productions\":[[\"A4\",[\"b4\",\"A4\"]],[\"A4\",[]],[\"S4\",[\"a4\",\"A4\"]]],\"start\":\"S4\",\"terminals\":[\"a4\",\"b4\"],\"word\":[\"a4\",\"b4\",\"b4\",\"b4\",\"b4\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S4, A4], terminals=[a4, b4], start=S4, productions=[[S4, [a4, A4]], [A4, [b4, A4]], [A4, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a4, b4, b4, b4, b4], supplied forms=[[S4], [a4, A4], [a4, b4, A4], [a4, b4, b4, A4], [a4, b4, b4, b4, A4], [a4, b4, b4, b4, b4, A4], [a4, b4, b4, b4, b4]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a4, b4, b4, b4, b4], [[0, 0, S4, [a4, A4]], [1, 1, A4, [b4, A4]], [2, 1, A4, [b4, A4]], [3, 1, A4, [b4, A4]], [4, 1, A4, [b4, A4]], [5, 2, A4, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a4, b4, b4, b4, b4], [[0, 0, S4, [a4, A4]], [1, 1, A4, [b4, A4]], [2, 1, A4, [b4, A4]], [3, 1, A4, [b4, A4]], [4, 1, A4, [b4, A4]], [5, 2, A4, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0056","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d42be0ebd3357d767609a628081ca1a01fd1847bd7bbb3571e111e5733f12eab","family_id":"qf-mod40-cfg-membership-witness","seed":202611097056,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S5\"],[\"a5\",\"A5\"],[\"a5\"]],\"nonterminals\":[\"A5\",\"S5\"],\"productions\":[[\"A5\",[\"b5\",\"A5\"]],[\"A5\",[]],[\"S5\",[\"a5\",\"A5\"]]],\"start\":\"S5\",\"terminals\":[\"a5\",\"b5\"],\"word\":[\"a5\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S5, A5], terminals=[a5, b5], start=S5, productions=[[S5, [a5, A5]], [A5, [b5, A5]], [A5, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a5], supplied forms=[[S5], [a5, A5], [a5]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a5], [[0, 0, S5, [a5, A5]], [1, 2, A5, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a5], [[0, 0, S5, [a5, A5]], [1, 2, A5, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0057","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bc575fa9b7933c00f6999fefa8dcdd48cc1e8c3c8f4a6cefb03b54bf6ba4226","family_id":"qf-mod40-cfg-membership-witness","seed":202611097057,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S6\"],[\"a6\",\"A6\"],[\"a6\",\"b6\",\"A6\"],[\"a6\",\"b6\"]],\"nonterminals\":[\"A6\",\"S6\"],\"productions\":[[\"A6\",[\"b6\",\"A6\"]],[\"A6\",[]],[\"S6\",[\"a6\",\"A6\"]]],\"start\":\"S6\",\"terminals\":[\"a6\",\"b6\"],\"word\":[\"a6\",\"b6\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S6, A6], terminals=[a6, b6], start=S6, productions=[[S6, [a6, A6]], [A6, [b6, A6]], [A6, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a6, b6], supplied forms=[[S6], [a6, A6], [a6, b6, A6], [a6, b6]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a6, b6], [[0, 0, S6, [a6, A6]], [1, 1, A6, [b6, A6]], [2, 2, A6, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a6, b6], [[0, 0, S6, [a6, A6]], [1, 1, A6, [b6, A6]], [2, 2, A6, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0058","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"597905c2a371d3108b5d313b3feeee07b9edd7872992f482304d41982300dc41","family_id":"qf-mod40-cfg-membership-witness","seed":202611097058,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S7\"],[\"a7\",\"A7\"],[\"a7\",\"b7\",\"A7\"],[\"a7\",\"b7\",\"b7\",\"A7\"],[\"a7\",\"b7\",\"b7\"]],\"nonterminals\":[\"A7\",\"S7\"],\"productions\":[[\"A7\",[\"b7\",\"A7\"]],[\"A7\",[]],[\"S7\",[\"a7\",\"A7\"]]],\"start\":\"S7\",\"terminals\":[\"a7\",\"b7\"],\"word\":[\"a7\",\"b7\",\"b7\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S7, A7], terminals=[a7, b7], start=S7, productions=[[S7, [a7, A7]], [A7, [b7, A7]], [A7, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a7, b7, b7], supplied forms=[[S7], [a7, A7], [a7, b7, A7], [a7, b7, b7, A7], [a7, b7, b7]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a7, b7, b7], [[0, 0, S7, [a7, A7]], [1, 1, A7, [b7, A7]], [2, 1, A7, [b7, A7]], [3, 2, A7, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a7, b7, b7], [[0, 0, S7, [a7, A7]], [1, 1, A7, [b7, A7]], [2, 1, A7, [b7, A7]], [3, 2, A7, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0059","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2126e803e6cdeefffb5516be362913b5705b8f5b9e759357c5a39e71bdc7e4f1","family_id":"qf-mod40-cfg-membership-witness","seed":202611097059,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S8\"],[\"a8\",\"A8\"],[\"a8\",\"b8\",\"A8\"],[\"a8\",\"b8\",\"b8\",\"A8\"],[\"a8\",\"b8\",\"b8\",\"b8\",\"A8\"],[\"a8\",\"b8\",\"b8\",\"b8\"]],\"nonterminals\":[\"A8\",\"S8\"],\"productions\":[[\"A8\",[\"b8\",\"A8\"]],[\"A8\",[]],[\"S8\",[\"a8\",\"A8\"]]],\"start\":\"S8\",\"terminals\":[\"a8\",\"b8\"],\"word\":[\"a8\",\"b8\",\"b8\",\"b8\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S8, A8], terminals=[a8, b8], start=S8, productions=[[S8, [a8, A8]], [A8, [b8, A8]], [A8, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a8, b8, b8, b8], supplied forms=[[S8], [a8, A8], [a8, b8, A8], [a8, b8, b8, A8], [a8, b8, b8, b8, A8], [a8, b8, b8, b8]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a8, b8, b8, b8], [[0, 0, S8, [a8, A8]], [1, 1, A8, [b8, A8]], [2, 1, A8, [b8, A8]], [3, 1, A8, [b8, A8]], [4, 2, A8, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a8, b8, b8, b8], [[0, 0, S8, [a8, A8]], [1, 1, A8, [b8, A8]], [2, 1, A8, [b8, A8]], [3, 1, A8, [b8, A8]], [4, 2, A8, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0060","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"012c1a6fa7d0c0e3b3d82965375ab6b43115f51f26ed6de76ae667f49eb8679c","family_id":"qf-mod40-cfg-membership-witness","seed":202611097060,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S9\"],[\"a9\",\"A9\"],[\"a9\",\"b9\",\"A9\"],[\"a9\",\"b9\",\"b9\",\"A9\"],[\"a9\",\"b9\",\"b9\",\"b9\",\"A9\"],[\"a9\",\"b9\",\"b9\",\"b9\",\"b9\",\"A9\"],[\"a9\",\"b9\",\"b9\",\"b9\",\"b9\"]],\"nonterminals\":[\"A9\",\"S9\"],\"productions\":[[\"A9\",[\"b9\",\"A9\"]],[\"A9\",[]],[\"S9\",[\"a9\",\"A9\"]]],\"start\":\"S9\",\"terminals\":[\"a9\",\"b9\"],\"word\":[\"a9\",\"b9\",\"b9\",\"b9\",\"b9\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S9, A9], terminals=[a9, b9], start=S9, productions=[[S9, [a9, A9]], [A9, [b9, A9]], [A9, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a9, b9, b9, b9, b9], supplied forms=[[S9], [a9, A9], [a9, b9, A9], [a9, b9, b9, A9], [a9, b9, b9, b9, A9], [a9, b9, b9, b9, b9, A9], [a9, b9, b9, b9, b9]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a9, b9, b9, b9, b9], [[0, 0, S9, [a9, A9]], [1, 1, A9, [b9, A9]], [2, 1, A9, [b9, A9]], [3, 1, A9, [b9, A9]], [4, 1, A9, [b9, A9]], [5, 2, A9, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a9, b9, b9, b9, b9], [[0, 0, S9, [a9, A9]], [1, 1, A9, [b9, A9]], [2, 1, A9, [b9, A9]], [3, 1, A9, [b9, A9]], [4, 1, A9, [b9, A9]], [5, 2, A9, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0061","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae789ca7e70a0561cffc37a0ddfb6189152653aa8a153e81cf067ea95cb49ef2","family_id":"qf-mod40-cfg-membership-witness","seed":202611097061,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S10\"],[\"a10\",\"A10\"],[\"a10\"]],\"nonterminals\":[\"A10\",\"S10\"],\"productions\":[[\"A10\",[\"b10\",\"A10\"]],[\"A10\",[]],[\"S10\",[\"a10\",\"A10\"]]],\"start\":\"S10\",\"terminals\":[\"a10\",\"b10\"],\"word\":[\"a10\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S10, A10], terminals=[a10, b10], start=S10, productions=[[S10, [a10, A10]], [A10, [b10, A10]], [A10, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a10], supplied forms=[[S10], [a10, A10], [a10]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a10], [[0, 0, S10, [a10, A10]], [1, 2, A10, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a10], [[0, 0, S10, [a10, A10]], [1, 2, A10, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0062","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d9a2c3927f2a18e12e3330f141aac60aef94e880ed09eb0ed6cc4561f89fdf1","family_id":"qf-mod40-cfg-membership-witness","seed":202611097062,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S11\"],[\"a11\",\"A11\"],[\"a11\",\"b11\",\"A11\"],[\"a11\",\"b11\"]],\"nonterminals\":[\"A11\",\"S11\"],\"productions\":[[\"A11\",[\"b11\",\"A11\"]],[\"A11\",[]],[\"S11\",[\"a11\",\"A11\"]]],\"start\":\"S11\",\"terminals\":[\"a11\",\"b11\"],\"word\":[\"a11\",\"b11\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S11, A11], terminals=[a11, b11], start=S11, productions=[[S11, [a11, A11]], [A11, [b11, A11]], [A11, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a11, b11], supplied forms=[[S11], [a11, A11], [a11, b11, A11], [a11, b11]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a11, b11], [[0, 0, S11, [a11, A11]], [1, 1, A11, [b11, A11]], [2, 2, A11, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a11, b11], [[0, 0, S11, [a11, A11]], [1, 1, A11, [b11, A11]], [2, 2, A11, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0063","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6d30dd5315f39d8194b60e4f824fcd3d215cc46048298a1ae0cf27ac523eceb","family_id":"qf-mod40-cfg-membership-witness","seed":202611097063,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S12\"],[\"a12\",\"A12\"],[\"a12\",\"b12\",\"A12\"],[\"a12\",\"b12\",\"b12\",\"A12\"],[\"a12\",\"b12\",\"b12\"]],\"nonterminals\":[\"A12\",\"S12\"],\"productions\":[[\"A12\",[\"b12\",\"A12\"]],[\"A12\",[]],[\"S12\",[\"a12\",\"A12\"]]],\"start\":\"S12\",\"terminals\":[\"a12\",\"b12\"],\"word\":[\"a12\",\"b12\",\"b12\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S12, A12], terminals=[a12, b12], start=S12, productions=[[S12, [a12, A12]], [A12, [b12, A12]], [A12, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a12, b12, b12], supplied forms=[[S12], [a12, A12], [a12, b12, A12], [a12, b12, b12, A12], [a12, b12, b12]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a12, b12, b12], [[0, 0, S12, [a12, A12]], [1, 1, A12, [b12, A12]], [2, 1, A12, [b12, A12]], [3, 2, A12, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a12, b12, b12], [[0, 0, S12, [a12, A12]], [1, 1, A12, [b12, A12]], [2, 1, A12, [b12, A12]], [3, 2, A12, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0064","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bcf72bd73a40e96c5188d4df08fb28f018216cd3bff8bf17e9aa495e81ffad4","family_id":"qf-mod40-cfg-membership-witness","seed":202611097064,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S13\"],[\"a13\",\"A13\"],[\"a13\",\"b13\",\"A13\"],[\"a13\",\"b13\",\"b13\",\"A13\"],[\"a13\",\"b13\",\"b13\",\"b13\",\"A13\"],[\"a13\",\"b13\",\"b13\",\"b13\"]],\"nonterminals\":[\"A13\",\"S13\"],\"productions\":[[\"A13\",[\"b13\",\"A13\"]],[\"A13\",[]],[\"S13\",[\"a13\",\"A13\"]]],\"start\":\"S13\",\"terminals\":[\"a13\",\"b13\"],\"word\":[\"a13\",\"b13\",\"b13\",\"b13\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S13, A13], terminals=[a13, b13], start=S13, productions=[[S13, [a13, A13]], [A13, [b13, A13]], [A13, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a13, b13, b13, b13], supplied forms=[[S13], [a13, A13], [a13, b13, A13], [a13, b13, b13, A13], [a13, b13, b13, b13, A13], [a13, b13, b13, b13]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a13, b13, b13, b13], [[0, 0, S13, [a13, A13]], [1, 1, A13, [b13, A13]], [2, 1, A13, [b13, A13]], [3, 1, A13, [b13, A13]], [4, 2, A13, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a13, b13, b13, b13], [[0, 0, S13, [a13, A13]], [1, 1, A13, [b13, A13]], [2, 1, A13, [b13, A13]], [3, 1, A13, [b13, A13]], [4, 2, A13, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0065","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d2947c181bb3878211cdb9d53730409aa09675268699248d483fb67e4cc555b","family_id":"qf-mod40-cfg-membership-witness","seed":202611097065,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S14\"],[\"a14\",\"A14\"],[\"a14\",\"b14\",\"A14\"],[\"a14\",\"b14\",\"b14\",\"A14\"],[\"a14\",\"b14\",\"b14\",\"b14\",\"A14\"],[\"a14\",\"b14\",\"b14\",\"b14\",\"b14\",\"A14\"],[\"a14\",\"b14\",\"b14\",\"b14\",\"b14\"]],\"nonterminals\":[\"A14\",\"S14\"],\"productions\":[[\"A14\",[\"b14\",\"A14\"]],[\"A14\",[]],[\"S14\",[\"a14\",\"A14\"]]],\"start\":\"S14\",\"terminals\":[\"a14\",\"b14\"],\"word\":[\"a14\",\"b14\",\"b14\",\"b14\",\"b14\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S14, A14], terminals=[a14, b14], start=S14, productions=[[S14, [a14, A14]], [A14, [b14, A14]], [A14, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a14, b14, b14, b14, b14], supplied forms=[[S14], [a14, A14], [a14, b14, A14], [a14, b14, b14, A14], [a14, b14, b14, b14, A14], [a14, b14, b14, b14, b14, A14], [a14, b14, b14, b14, b14]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a14, b14, b14, b14, b14], [[0, 0, S14, [a14, A14]], [1, 1, A14, [b14, A14]], [2, 1, A14, [b14, A14]], [3, 1, A14, [b14, A14]], [4, 1, A14, [b14, A14]], [5, 2, A14, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a14, b14, b14, b14, b14], [[0, 0, S14, [a14, A14]], [1, 1, A14, [b14, A14]], [2, 1, A14, [b14, A14]], [3, 1, A14, [b14, A14]], [4, 1, A14, [b14, A14]], [5, 2, A14, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0066","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54cf97352674c2e224c908278026992938fc21fe762d7f35a03b96194d151b10","family_id":"qf-mod40-cfg-membership-witness","seed":202611097066,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S15\"],[\"a15\",\"A15\"],[\"a15\"]],\"nonterminals\":[\"A15\",\"S15\"],\"productions\":[[\"A15\",[\"b15\",\"A15\"]],[\"A15\",[]],[\"S15\",[\"a15\",\"A15\"]]],\"start\":\"S15\",\"terminals\":[\"a15\",\"b15\"],\"word\":[\"a15\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S15, A15], terminals=[a15, b15], start=S15, productions=[[S15, [a15, A15]], [A15, [b15, A15]], [A15, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a15], supplied forms=[[S15], [a15, A15], [a15]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a15], [[0, 0, S15, [a15, A15]], [1, 2, A15, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a15], [[0, 0, S15, [a15, A15]], [1, 2, A15, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0067","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3267c29a6f9d1ba06dbcfd8f078cde3f358dbba4da90adb207cad44172085d0d","family_id":"qf-mod40-cfg-membership-witness","seed":202611097067,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S16\"],[\"a16\",\"A16\"],[\"a16\",\"b16\",\"A16\"],[\"a16\",\"b16\"]],\"nonterminals\":[\"A16\",\"S16\"],\"productions\":[[\"A16\",[\"b16\",\"A16\"]],[\"A16\",[]],[\"S16\",[\"a16\",\"A16\"]]],\"start\":\"S16\",\"terminals\":[\"a16\",\"b16\"],\"word\":[\"a16\",\"b16\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S16, A16], terminals=[a16, b16], start=S16, productions=[[S16, [a16, A16]], [A16, [b16, A16]], [A16, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a16, b16], supplied forms=[[S16], [a16, A16], [a16, b16, A16], [a16, b16]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a16, b16], [[0, 0, S16, [a16, A16]], [1, 1, A16, [b16, A16]], [2, 2, A16, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a16, b16], [[0, 0, S16, [a16, A16]], [1, 1, A16, [b16, A16]], [2, 2, A16, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0068","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e549ef87867a26a90d704fc630fbbe69715b98a5aea1fd33b18cc1fbedacb147","family_id":"qf-mod40-cfg-membership-witness","seed":202611097068,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S17\"],[\"a17\",\"A17\"],[\"a17\",\"b17\",\"A17\"],[\"a17\",\"b17\",\"b17\",\"A17\"],[\"a17\",\"b17\",\"b17\"]],\"nonterminals\":[\"A17\",\"S17\"],\"productions\":[[\"A17\",[\"b17\",\"A17\"]],[\"A17\",[]],[\"S17\",[\"a17\",\"A17\"]]],\"start\":\"S17\",\"terminals\":[\"a17\",\"b17\"],\"word\":[\"a17\",\"b17\",\"b17\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S17, A17], terminals=[a17, b17], start=S17, productions=[[S17, [a17, A17]], [A17, [b17, A17]], [A17, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a17, b17, b17], supplied forms=[[S17], [a17, A17], [a17, b17, A17], [a17, b17, b17, A17], [a17, b17, b17]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a17, b17, b17], [[0, 0, S17, [a17, A17]], [1, 1, A17, [b17, A17]], [2, 1, A17, [b17, A17]], [3, 2, A17, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a17, b17, b17], [[0, 0, S17, [a17, A17]], [1, 1, A17, [b17, A17]], [2, 1, A17, [b17, A17]], [3, 2, A17, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0069","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b23f1b0e2f412e530612670d5f8a585afb392690f7841977c5d8ef6a2954af3c","family_id":"qf-mod40-cfg-membership-witness","seed":202611097069,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S18\"],[\"a18\",\"A18\"],[\"a18\",\"b18\",\"A18\"],[\"a18\",\"b18\",\"b18\",\"A18\"],[\"a18\",\"b18\",\"b18\",\"b18\",\"A18\"],[\"a18\",\"b18\",\"b18\",\"b18\"]],\"nonterminals\":[\"A18\",\"S18\"],\"productions\":[[\"A18\",[\"b18\",\"A18\"]],[\"A18\",[]],[\"S18\",[\"a18\",\"A18\"]]],\"start\":\"S18\",\"terminals\":[\"a18\",\"b18\"],\"word\":[\"a18\",\"b18\",\"b18\",\"b18\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S18, A18], terminals=[a18, b18], start=S18, productions=[[S18, [a18, A18]], [A18, [b18, A18]], [A18, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a18, b18, b18, b18], supplied forms=[[S18], [a18, A18], [a18, b18, A18], [a18, b18, b18, A18], [a18, b18, b18, b18, A18], [a18, b18, b18, b18]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a18, b18, b18, b18], [[0, 0, S18, [a18, A18]], [1, 1, A18, [b18, A18]], [2, 1, A18, [b18, A18]], [3, 1, A18, [b18, A18]], [4, 2, A18, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a18, b18, b18, b18], [[0, 0, S18, [a18, A18]], [1, 1, A18, [b18, A18]], [2, 1, A18, [b18, A18]], [3, 1, A18, [b18, A18]], [4, 2, A18, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0070","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9135604bd2f0f7c6097d6baa8891e18dda351c2e9611c04f0d973ddcece71971","family_id":"qf-mod40-cfg-membership-witness","seed":202611097070,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S19\"],[\"a19\",\"A19\"],[\"a19\",\"b19\",\"A19\"],[\"a19\",\"b19\",\"b19\",\"A19\"],[\"a19\",\"b19\",\"b19\",\"b19\",\"A19\"],[\"a19\",\"b19\",\"b19\",\"b19\",\"b19\",\"A19\"],[\"a19\",\"b19\",\"b19\",\"b19\",\"b19\"]],\"nonterminals\":[\"A19\",\"S19\"],\"productions\":[[\"A19\",[\"b19\",\"A19\"]],[\"A19\",[]],[\"S19\",[\"a19\",\"A19\"]]],\"start\":\"S19\",\"terminals\":[\"a19\",\"b19\"],\"word\":[\"a19\",\"b19\",\"b19\",\"b19\",\"b19\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S19, A19], terminals=[a19, b19], start=S19, productions=[[S19, [a19, A19]], [A19, [b19, A19]], [A19, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a19, b19, b19, b19, b19], supplied forms=[[S19], [a19, A19], [a19, b19, A19], [a19, b19, b19, A19], [a19, b19, b19, b19, A19], [a19, b19, b19, b19, b19, A19], [a19, b19, b19, b19, b19]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a19, b19, b19, b19, b19], [[0, 0, S19, [a19, A19]], [1, 1, A19, [b19, A19]], [2, 1, A19, [b19, A19]], [3, 1, A19, [b19, A19]], [4, 1, A19, [b19, A19]], [5, 2, A19, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a19, b19, b19, b19, b19], [[0, 0, S19, [a19, A19]], [1, 1, A19, [b19, A19]], [2, 1, A19, [b19, A19]], [3, 1, A19, [b19, A19]], [4, 1, A19, [b19, A19]], [5, 2, A19, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0071","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59a8cf0e80d8c595860327f6035ed0d7a81f8c87cfbde4a5b9ea8cb4b275daee","family_id":"qf-mod40-cfg-membership-witness","seed":202611097071,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S20\"],[\"a20\",\"A20\"],[\"a20\"]],\"nonterminals\":[\"A20\",\"S20\"],\"productions\":[[\"A20\",[\"b20\",\"A20\"]],[\"A20\",[]],[\"S20\",[\"a20\",\"A20\"]]],\"start\":\"S20\",\"terminals\":[\"a20\",\"b20\"],\"word\":[\"a20\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S20, A20], terminals=[a20, b20], start=S20, productions=[[S20, [a20, A20]], [A20, [b20, A20]], [A20, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a20], supplied forms=[[S20], [a20, A20], [a20]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a20], [[0, 0, S20, [a20, A20]], [1, 2, A20, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a20], [[0, 0, S20, [a20, A20]], [1, 2, A20, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0072","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49e017b31e43a60762be44b73cf40287e53e6c3db892ca66ce2d7db73f11d292","family_id":"qf-mod40-cfg-membership-witness","seed":202611097072,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S21\"],[\"a21\",\"A21\"],[\"a21\",\"b21\",\"A21\"],[\"a21\",\"b21\"]],\"nonterminals\":[\"A21\",\"S21\"],\"productions\":[[\"A21\",[\"b21\",\"A21\"]],[\"A21\",[]],[\"S21\",[\"a21\",\"A21\"]]],\"start\":\"S21\",\"terminals\":[\"a21\",\"b21\"],\"word\":[\"a21\",\"b21\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S21, A21], terminals=[a21, b21], start=S21, productions=[[S21, [a21, A21]], [A21, [b21, A21]], [A21, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a21, b21], supplied forms=[[S21], [a21, A21], [a21, b21, A21], [a21, b21]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a21, b21], [[0, 0, S21, [a21, A21]], [1, 1, A21, [b21, A21]], [2, 2, A21, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a21, b21], [[0, 0, S21, [a21, A21]], [1, 1, A21, [b21, A21]], [2, 2, A21, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0073","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7abb36cdc5b64721231fe55d99758fd5dd1236b9c7b1162c77cefa76aba48bc","family_id":"qf-mod40-cfg-membership-witness","seed":202611097073,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S22\"],[\"a22\",\"A22\"],[\"a22\",\"b22\",\"A22\"],[\"a22\",\"b22\",\"b22\",\"A22\"],[\"a22\",\"b22\",\"b22\"]],\"nonterminals\":[\"A22\",\"S22\"],\"productions\":[[\"A22\",[\"b22\",\"A22\"]],[\"A22\",[]],[\"S22\",[\"a22\",\"A22\"]]],\"start\":\"S22\",\"terminals\":[\"a22\",\"b22\"],\"word\":[\"a22\",\"b22\",\"b22\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S22, A22], terminals=[a22, b22], start=S22, productions=[[S22, [a22, A22]], [A22, [b22, A22]], [A22, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a22, b22, b22], supplied forms=[[S22], [a22, A22], [a22, b22, A22], [a22, b22, b22, A22], [a22, b22, b22]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a22, b22, b22], [[0, 0, S22, [a22, A22]], [1, 1, A22, [b22, A22]], [2, 1, A22, [b22, A22]], [3, 2, A22, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a22, b22, b22], [[0, 0, S22, [a22, A22]], [1, 1, A22, [b22, A22]], [2, 1, A22, [b22, A22]], [3, 2, A22, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0074","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf15c8c43e4544ff08d108ca36c779e1032965af20f516cc007630c5432b385b","family_id":"qf-mod40-cfg-membership-witness","seed":202611097074,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S23\"],[\"a23\",\"A23\"],[\"a23\",\"b23\",\"A23\"],[\"a23\",\"b23\",\"b23\",\"A23\"],[\"a23\",\"b23\",\"b23\",\"b23\",\"A23\"],[\"a23\",\"b23\",\"b23\",\"b23\"]],\"nonterminals\":[\"A23\",\"S23\"],\"productions\":[[\"A23\",[\"b23\",\"A23\"]],[\"A23\",[]],[\"S23\",[\"a23\",\"A23\"]]],\"start\":\"S23\",\"terminals\":[\"a23\",\"b23\"],\"word\":[\"a23\",\"b23\",\"b23\",\"b23\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S23, A23], terminals=[a23, b23], start=S23, productions=[[S23, [a23, A23]], [A23, [b23, A23]], [A23, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a23, b23, b23, b23], supplied forms=[[S23], [a23, A23], [a23, b23, A23], [a23, b23, b23, A23], [a23, b23, b23, b23, A23], [a23, b23, b23, b23]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a23, b23, b23, b23], [[0, 0, S23, [a23, A23]], [1, 1, A23, [b23, A23]], [2, 1, A23, [b23, A23]], [3, 1, A23, [b23, A23]], [4, 2, A23, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a23, b23, b23, b23], [[0, 0, S23, [a23, A23]], [1, 1, A23, [b23, A23]], [2, 1, A23, [b23, A23]], [3, 1, A23, [b23, A23]], [4, 2, A23, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0075","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba918b41b20857ec863e273a0b6f2136e776d4b81df3879464938bfc3a60dcca","family_id":"qf-mod40-cfg-membership-witness","seed":202611097075,"semantic_template":"qf-mod40-cfg-membership-witness/{\"allow_epsilon\":true,\"forms\":[[\"S24\"],[\"a24\",\"A24\"],[\"a24\",\"b24\",\"A24\"],[\"a24\",\"b24\",\"b24\",\"A24\"],[\"a24\",\"b24\",\"b24\",\"b24\",\"A24\"],[\"a24\",\"b24\",\"b24\",\"b24\",\"b24\",\"A24\"],[\"a24\",\"b24\",\"b24\",\"b24\",\"b24\"]],\"nonterminals\":[\"A24\",\"S24\"],\"productions\":[[\"A24\",[\"b24\",\"A24\"]],[\"A24\",[]],[\"S24\",[\"a24\",\"A24\"]]],\"start\":\"S24\",\"terminals\":[\"a24\",\"b24\"],\"word\":[\"a24\",\"b24\",\"b24\",\"b24\",\"b24\"]}"},"fields":{"title":"A Valid Start-to-Terminal Derivation Proves Word Membership","stem":"Use the complete finite grammar: nonterminals=[S24, A24], terminals=[a24, b24], start=S24, productions=[[S24, [a24, A24]], [A24, [b24, A24]], [A24, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Candidate word=[a24, b24, b24, b24, b24], supplied forms=[[S24], [a24, A24], [a24, b24, A24], [a24, b24, b24, A24], [a24, b24, b24, b24, A24], [a24, b24, b24, b24, b24, A24], [a24, b24, b24, b24, b24]]. Give (word, step_witnesses, valid_derivation, terminal_word, membership_certificate, certificate).","answer":{"kind":"expression","value":"([a24, b24, b24, b24, b24], [[0, 0, S24, [a24, A24]], [1, 1, A24, [b24, A24]], [2, 1, A24, [b24, A24]], [3, 1, A24, [b24, A24]], [4, 1, A24, [b24, A24]], [5, 2, A24, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)","explanation":"Require a valid derivation from [start], a terminal-only word and equality of the final form with that word. Empty word membership needs an actual allowed epsilon derivation; the empty language is different from a language containing the empty word. No ambiguity, regularity or CFG-PDA equivalence follows. The exact result, retaining the requested order, is ([a24, b24, b24, b24, b24], [[0, 0, S24, [a24, A24]], [1, 1, A24, [b24, A24]], [2, 1, A24, [b24, A24]], [3, 1, A24, [b24, A24]], [4, 1, A24, [b24, A24]], [5, 2, A24, []]], Yes, Yes, Yes, derivation_from_start_to_terminal_only_word_proves_membership)."},"skills":["A Valid Start-to-Terminal Derivation Proves Word Membership"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0076","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e7116938da1c62cbe85c84376f219ff13bbe5307709f7977c8b19c3fa0a9008","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097076,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":1,\"nonterminals\":[\"A0\",\"S0\"],\"productions\":[[\"A0\",[\"b0\",\"A0\"]],[\"A0\",[]],[\"S0\",[\"a0\",\"A0\"]]],\"start\":\"S0\",\"terminals\":[\"a0\",\"b0\"],\"word\":[\"a0\",\"b0\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S0, A0], terminals=[a0, b0], start=S0, productions=[[S0, [a0, A0]], [A0, [b0, A0]], [A0, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a0, b0] with max_steps=1. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a0, b0], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a0, b0], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0077","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"019717ec9091289ed4283967aac8de239817542fb247bb0b1222eb0f2ac1532c","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097077,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":2,\"nonterminals\":[\"A1\",\"S1\"],\"productions\":[[\"A1\",[\"b1\",\"A1\"]],[\"A1\",[]],[\"S1\",[\"a1\",\"A1\"]]],\"start\":\"S1\",\"terminals\":[\"a1\",\"b1\"],\"word\":[\"a1\",\"b1\",\"b1\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S1, A1], terminals=[a1, b1], start=S1, productions=[[S1, [a1, A1]], [A1, [b1, A1]], [A1, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a1, b1, b1] with max_steps=2. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a1, b1, b1], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a1, b1, b1], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0078","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"deab1a518bf9de7844792d8acce07148c88928083dc92b17e22f078e89aec14d","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097078,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":1,\"nonterminals\":[\"A2\",\"S2\"],\"productions\":[[\"A2\",[\"b2\",\"A2\"]],[\"A2\",[]],[\"S2\",[\"a2\",\"A2\"]]],\"start\":\"S2\",\"terminals\":[\"a2\",\"b2\"],\"word\":[\"a2\",\"b2\",\"b2\",\"b2\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S2, A2], terminals=[a2, b2], start=S2, productions=[[S2, [a2, A2]], [A2, [b2, A2]], [A2, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a2, b2, b2, b2] with max_steps=1. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a2, b2, b2, b2], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a2, b2, b2, b2], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0079","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d912afd18abcf84c580facce4aabf426590c8192b26be806f8618ce6b2cd9b7","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097079,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":2,\"nonterminals\":[\"A3\",\"S3\"],\"productions\":[[\"A3\",[\"b3\",\"A3\"]],[\"A3\",[]],[\"S3\",[\"a3\",\"A3\"]]],\"start\":\"S3\",\"terminals\":[\"a3\",\"b3\"],\"word\":[\"a3\",\"b3\",\"b3\",\"b3\",\"b3\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S3, A3], terminals=[a3, b3], start=S3, productions=[[S3, [a3, A3]], [A3, [b3, A3]], [A3, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a3, b3, b3, b3, b3] with max_steps=2. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a3, b3, b3, b3, b3], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a3, b3, b3, b3, b3], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0080","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b60e0f3570d0b692531cb1b506565f4246f2a955607e070011c5e08d5ed2c2c3","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097080,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":1,\"nonterminals\":[\"A4\",\"S4\"],\"productions\":[[\"A4\",[\"b4\",\"A4\"]],[\"A4\",[]],[\"S4\",[\"a4\",\"A4\"]]],\"start\":\"S4\",\"terminals\":[\"a4\",\"b4\"],\"word\":[\"a4\",\"b4\",\"b4\",\"b4\",\"b4\",\"b4\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S4, A4], terminals=[a4, b4], start=S4, productions=[[S4, [a4, A4]], [A4, [b4, A4]], [A4, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a4, b4, b4, b4, b4, b4] with max_steps=1. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a4, b4, b4, b4, b4, b4], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a4, b4, b4, b4, b4, b4], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0081","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68e6b5fb5aa29ca70a17dc1b70208ffda9761b0bf4c0f7276db6846d7fb443d2","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097081,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":2,\"nonterminals\":[\"A5\",\"S5\"],\"productions\":[[\"A5\",[\"b5\",\"A5\"]],[\"A5\",[]],[\"S5\",[\"a5\",\"A5\"]]],\"start\":\"S5\",\"terminals\":[\"a5\",\"b5\"],\"word\":[\"a5\",\"b5\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S5, A5], terminals=[a5, b5], start=S5, productions=[[S5, [a5, A5]], [A5, [b5, A5]], [A5, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a5, b5] with max_steps=2. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a5, b5], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a5, b5], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0082","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbbfb18eebcc780c4c96b551690c59c1aae0526f20625abbc649107781a13418","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097082,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":1,\"nonterminals\":[\"A6\",\"S6\"],\"productions\":[[\"A6\",[\"b6\",\"A6\"]],[\"A6\",[]],[\"S6\",[\"a6\",\"A6\"]]],\"start\":\"S6\",\"terminals\":[\"a6\",\"b6\"],\"word\":[\"a6\",\"b6\",\"b6\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S6, A6], terminals=[a6, b6], start=S6, productions=[[S6, [a6, A6]], [A6, [b6, A6]], [A6, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a6, b6, b6] with max_steps=1. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a6, b6, b6], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a6, b6, b6], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0083","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b0061d5b85036d2cebdecc119ba6af986858ac99592e17f9ffbf7f0748091fd","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097083,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":2,\"nonterminals\":[\"A7\",\"S7\"],\"productions\":[[\"A7\",[\"b7\",\"A7\"]],[\"A7\",[]],[\"S7\",[\"a7\",\"A7\"]]],\"start\":\"S7\",\"terminals\":[\"a7\",\"b7\"],\"word\":[\"a7\",\"b7\",\"b7\",\"b7\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S7, A7], terminals=[a7, b7], start=S7, productions=[[S7, [a7, A7]], [A7, [b7, A7]], [A7, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a7, b7, b7, b7] with max_steps=2. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a7, b7, b7, b7], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a7, b7, b7, b7], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0084","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"470b1e2b6af29ddf1a01691bce69f5f50d7e0043bda906ee2c761a21be60b334","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097084,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":1,\"nonterminals\":[\"A8\",\"S8\"],\"productions\":[[\"A8\",[\"b8\",\"A8\"]],[\"A8\",[]],[\"S8\",[\"a8\",\"A8\"]]],\"start\":\"S8\",\"terminals\":[\"a8\",\"b8\"],\"word\":[\"a8\",\"b8\",\"b8\",\"b8\",\"b8\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S8, A8], terminals=[a8, b8], start=S8, productions=[[S8, [a8, A8]], [A8, [b8, A8]], [A8, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a8, b8, b8, b8, b8] with max_steps=1. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a8, b8, b8, b8, b8], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a8, b8, b8, b8, b8], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0085","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a35a43dbfdd84b6d38af00124a473a14fbffb99bf485ff8c922e5859cfb8a3cc","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097085,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":2,\"nonterminals\":[\"A9\",\"S9\"],\"productions\":[[\"A9\",[\"b9\",\"A9\"]],[\"A9\",[]],[\"S9\",[\"a9\",\"A9\"]]],\"start\":\"S9\",\"terminals\":[\"a9\",\"b9\"],\"word\":[\"a9\",\"b9\",\"b9\",\"b9\",\"b9\",\"b9\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S9, A9], terminals=[a9, b9], start=S9, productions=[[S9, [a9, A9]], [A9, [b9, A9]], [A9, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a9, b9, b9, b9, b9, b9] with max_steps=2. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a9, b9, b9, b9, b9, b9], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a9, b9, b9, b9, b9, b9], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0086","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24cc6312247716af31d2c232260cd008c2523b9549ca9c4dd917df17b5df3c8d","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097086,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":1,\"nonterminals\":[\"A10\",\"S10\"],\"productions\":[[\"A10\",[\"b10\",\"A10\"]],[\"A10\",[]],[\"S10\",[\"a10\",\"A10\"]]],\"start\":\"S10\",\"terminals\":[\"a10\",\"b10\"],\"word\":[\"a10\",\"b10\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S10, A10], terminals=[a10, b10], start=S10, productions=[[S10, [a10, A10]], [A10, [b10, A10]], [A10, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a10, b10] with max_steps=1. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a10, b10], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a10, b10], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0087","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba215c5756a8cba4a2e29679aea38bf7db18d40ff815f6cc96e57184fa673781","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097087,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":2,\"nonterminals\":[\"A11\",\"S11\"],\"productions\":[[\"A11\",[\"b11\",\"A11\"]],[\"A11\",[]],[\"S11\",[\"a11\",\"A11\"]]],\"start\":\"S11\",\"terminals\":[\"a11\",\"b11\"],\"word\":[\"a11\",\"b11\",\"b11\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S11, A11], terminals=[a11, b11], start=S11, productions=[[S11, [a11, A11]], [A11, [b11, A11]], [A11, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a11, b11, b11] with max_steps=2. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a11, b11, b11], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a11, b11, b11], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0088","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af0be2f67e9f6e5088a6b71f7c7fef0af79706de9dc243b90913a7ac9bda6a30","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097088,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":1,\"nonterminals\":[\"A12\",\"S12\"],\"productions\":[[\"A12\",[\"b12\",\"A12\"]],[\"A12\",[]],[\"S12\",[\"a12\",\"A12\"]]],\"start\":\"S12\",\"terminals\":[\"a12\",\"b12\"],\"word\":[\"a12\",\"b12\",\"b12\",\"b12\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S12, A12], terminals=[a12, b12], start=S12, productions=[[S12, [a12, A12]], [A12, [b12, A12]], [A12, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a12, b12, b12, b12] with max_steps=1. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a12, b12, b12, b12], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a12, b12, b12, b12], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0089","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ce9c1da7b12ce74287ac7ab15d52cf8081067c05cdeb8290ddd3cd70ee24808","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097089,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":2,\"nonterminals\":[\"A13\",\"S13\"],\"productions\":[[\"A13\",[\"b13\",\"A13\"]],[\"A13\",[]],[\"S13\",[\"a13\",\"A13\"]]],\"start\":\"S13\",\"terminals\":[\"a13\",\"b13\"],\"word\":[\"a13\",\"b13\",\"b13\",\"b13\",\"b13\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S13, A13], terminals=[a13, b13], start=S13, productions=[[S13, [a13, A13]], [A13, [b13, A13]], [A13, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a13, b13, b13, b13, b13] with max_steps=2. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a13, b13, b13, b13, b13], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a13, b13, b13, b13, b13], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0090","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73c393645ec704a34bf368fb5c40c45715016b66fe97d8f0f563ce59ed8be6d2","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097090,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":1,\"nonterminals\":[\"A14\",\"S14\"],\"productions\":[[\"A14\",[\"b14\",\"A14\"]],[\"A14\",[]],[\"S14\",[\"a14\",\"A14\"]]],\"start\":\"S14\",\"terminals\":[\"a14\",\"b14\"],\"word\":[\"a14\",\"b14\",\"b14\",\"b14\",\"b14\",\"b14\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S14, A14], terminals=[a14, b14], start=S14, productions=[[S14, [a14, A14]], [A14, [b14, A14]], [A14, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a14, b14, b14, b14, b14, b14] with max_steps=1. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a14, b14, b14, b14, b14, b14], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a14, b14, b14, b14, b14, b14], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0091","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4297e7f5fa76e78f62c8b07055e9e0c5a051aad0e4ec3a95894205154e0b9421","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097091,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":2,\"nonterminals\":[\"A15\",\"S15\"],\"productions\":[[\"A15\",[\"b15\",\"A15\"]],[\"A15\",[]],[\"S15\",[\"a15\",\"A15\"]]],\"start\":\"S15\",\"terminals\":[\"a15\",\"b15\"],\"word\":[\"a15\",\"b15\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S15, A15], terminals=[a15, b15], start=S15, productions=[[S15, [a15, A15]], [A15, [b15, A15]], [A15, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a15, b15] with max_steps=2. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a15, b15], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a15, b15], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0092","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b35f06e59e592f79ab899d135f0014a34f7999600d5904829d5fc9163243cee7","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097092,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":1,\"nonterminals\":[\"A16\",\"S16\"],\"productions\":[[\"A16\",[\"b16\",\"A16\"]],[\"A16\",[]],[\"S16\",[\"a16\",\"A16\"]]],\"start\":\"S16\",\"terminals\":[\"a16\",\"b16\"],\"word\":[\"a16\",\"b16\",\"b16\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S16, A16], terminals=[a16, b16], start=S16, productions=[[S16, [a16, A16]], [A16, [b16, A16]], [A16, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a16, b16, b16] with max_steps=1. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a16, b16, b16], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a16, b16, b16], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0093","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ebadf9029f6c06cf875b8fbfa63640a63d27ef98f794c6ae8e87470cb55bf23","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097093,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":2,\"nonterminals\":[\"A17\",\"S17\"],\"productions\":[[\"A17\",[\"b17\",\"A17\"]],[\"A17\",[]],[\"S17\",[\"a17\",\"A17\"]]],\"start\":\"S17\",\"terminals\":[\"a17\",\"b17\"],\"word\":[\"a17\",\"b17\",\"b17\",\"b17\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S17, A17], terminals=[a17, b17], start=S17, productions=[[S17, [a17, A17]], [A17, [b17, A17]], [A17, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a17, b17, b17, b17] with max_steps=2. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a17, b17, b17, b17], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a17, b17, b17, b17], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0094","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdfe17b1bcb48435423eb8aeafcdb0dc30d63163dd21f8119fc1b1704abb330e","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097094,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":1,\"nonterminals\":[\"A18\",\"S18\"],\"productions\":[[\"A18\",[\"b18\",\"A18\"]],[\"A18\",[]],[\"S18\",[\"a18\",\"A18\"]]],\"start\":\"S18\",\"terminals\":[\"a18\",\"b18\"],\"word\":[\"a18\",\"b18\",\"b18\",\"b18\",\"b18\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S18, A18], terminals=[a18, b18], start=S18, productions=[[S18, [a18, A18]], [A18, [b18, A18]], [A18, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a18, b18, b18, b18, b18] with max_steps=1. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a18, b18, b18, b18, b18], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a18, b18, b18, b18, b18], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0095","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e93b5ea4245aaaecad52e936f6eff0f37761b4260e07b72e5b40cb742258ea52","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097095,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":2,\"nonterminals\":[\"A19\",\"S19\"],\"productions\":[[\"A19\",[\"b19\",\"A19\"]],[\"A19\",[]],[\"S19\",[\"a19\",\"A19\"]]],\"start\":\"S19\",\"terminals\":[\"a19\",\"b19\"],\"word\":[\"a19\",\"b19\",\"b19\",\"b19\",\"b19\",\"b19\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S19, A19], terminals=[a19, b19], start=S19, productions=[[S19, [a19, A19]], [A19, [b19, A19]], [A19, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a19, b19, b19, b19, b19, b19] with max_steps=2. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a19, b19, b19, b19, b19, b19], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a19, b19, b19, b19, b19, b19], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0096","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c51b31fd6d4ed7fec605ccf301d058034b60f840c01864d52dc8b1849c145c98","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097096,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":1,\"nonterminals\":[\"A20\",\"S20\"],\"productions\":[[\"A20\",[\"b20\",\"A20\"]],[\"A20\",[]],[\"S20\",[\"a20\",\"A20\"]]],\"start\":\"S20\",\"terminals\":[\"a20\",\"b20\"],\"word\":[\"a20\",\"b20\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S20, A20], terminals=[a20, b20], start=S20, productions=[[S20, [a20, A20]], [A20, [b20, A20]], [A20, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a20, b20] with max_steps=1. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a20, b20], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a20, b20], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0097","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1934da3a88aa46d6f320fcba5f240c8fe5a7d289eb0ffd8b131e3be947a4142f","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097097,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":2,\"nonterminals\":[\"A21\",\"S21\"],\"productions\":[[\"A21\",[\"b21\",\"A21\"]],[\"A21\",[]],[\"S21\",[\"a21\",\"A21\"]]],\"start\":\"S21\",\"terminals\":[\"a21\",\"b21\"],\"word\":[\"a21\",\"b21\",\"b21\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S21, A21], terminals=[a21, b21], start=S21, productions=[[S21, [a21, A21]], [A21, [b21, A21]], [A21, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a21, b21, b21] with max_steps=2. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a21, b21, b21], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a21, b21, b21], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0098","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a7678d6a9421cddc20c5bd2a2536864ae569251afd7130e462a5a0010515a17","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097098,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":1,\"nonterminals\":[\"A22\",\"S22\"],\"productions\":[[\"A22\",[\"b22\",\"A22\"]],[\"A22\",[]],[\"S22\",[\"a22\",\"A22\"]]],\"start\":\"S22\",\"terminals\":[\"a22\",\"b22\"],\"word\":[\"a22\",\"b22\",\"b22\",\"b22\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S22, A22], terminals=[a22, b22], start=S22, productions=[[S22, [a22, A22]], [A22, [b22, A22]], [A22, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a22, b22, b22, b22] with max_steps=1. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a22, b22, b22, b22], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a22, b22, b22, b22], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0099","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b374760ef8e94c8a7bd04de7edfcb855db83c02e514ccd0360b9dac4a5b712cc","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097099,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":2,\"nonterminals\":[\"A23\",\"S23\"],\"productions\":[[\"A23\",[\"b23\",\"A23\"]],[\"A23\",[]],[\"S23\",[\"a23\",\"A23\"]]],\"start\":\"S23\",\"terminals\":[\"a23\",\"b23\"],\"word\":[\"a23\",\"b23\",\"b23\",\"b23\",\"b23\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S23, A23], terminals=[a23, b23], start=S23, productions=[[S23, [a23, A23]], [A23, [b23, A23]], [A23, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a23, b23, b23, b23, b23] with max_steps=2. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a23, b23, b23, b23, b23], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a23, b23, b23, b23, b23], 2, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261109-r07-context-free-grammar-0100","batch_id":"batch-20261014-advance-20261109-r07-context-free-grammar-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d84d6bf1452d1cc065e3d3497f57c134936b8ec7f5f2da81d6d5355e9c92d3e","family_id":"qf-mod40-cfg-bounded-search-boundary","seed":202611097100,"semantic_template":"qf-mod40-cfg-bounded-search-boundary/{\"allow_epsilon\":true,\"max_steps\":1,\"nonterminals\":[\"A24\",\"S24\"],\"productions\":[[\"A24\",[\"b24\",\"A24\"]],[\"A24\",[]],[\"S24\",[\"a24\",\"A24\"]]],\"start\":\"S24\",\"terminals\":[\"a24\",\"b24\"],\"word\":[\"a24\",\"b24\",\"b24\",\"b24\",\"b24\",\"b24\"]}"},"fields":{"title":"A Bounded Search No-Hit Is Not a Nonmembership Proof","stem":"Use the complete finite grammar: nonterminals=[S24, A24], terminals=[a24, b24], start=S24, productions=[[S24, [a24, A24]], [A24, [b24, A24]], [A24, []]] and allow_epsilon=true. Alphabets must be unique and disjoint, the start belongs to nonterminals, and each production [lhs,rhs] has exactly one nonterminal lhs and a finite legal-symbol rhs. Empty rhs is permitted only under the explicit epsilon convention. Keep original symbol and production order. Search word=[a24, b24, b24, b24, b24, b24] with max_steps=1. Give (word, max_steps, found_in_bounded_search, nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a24, b24, b24, b24, b24, b24], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)","explanation":"Run the declared finite production search from [start], retaining only intermediate forms of length at most word.length+max_steps+1. Report whether this bounded search found the exact form. The final fixed No makes no nonmembership claim, even on a no-hit; without a completeness bound, a longer derivation may prove membership. A found nonterminal form is not by itself a terminal-word membership certificate. The exact result, retaining the requested order, is ([a24, b24, b24, b24, b24, b24], 1, No, No, bounded_search_failure_cannot_prove_nonmembership_without_complete_bound)."},"skills":["A Bounded Search No-Hit Is Not a Nonmembership Proof"],"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."}}