{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-0001","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba7e257e2227311d364a9c2a036f5fe96d32a92288e1106f2d9b83d2de5d611d","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127001,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a0\",\"b0\"],\"final_states\":[\"f0\"],\"start_state\":\"s0\",\"states\":[\"d0\",\"e10\",\"e20\",\"f0\",\"m0\",\"s0\"],\"transitions\":[[\"e10\",null,\"e20\"],[\"f0\",null,\"e10\"],[\"m0\",\"a0\",\"m0\"],[\"m0\",\"b0\",\"f0\"],[\"s0\",null,\"d0\"],[\"s0\",null,\"m0\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s0, m0, f0, e10, e20, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0], [f0, null, e10], [e10, null, e20]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s0, [d0, m0, s0], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s0, [d0, m0, s0], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0002","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"714195d86ffc4fa11c439f21464e6d8964a40b08added33f9c800cec740b2970","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127002,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a1\",\"b1\"],\"final_states\":[\"f1\"],\"start_state\":\"s1\",\"states\":[\"d1\",\"e11\",\"e21\",\"f1\",\"m1\",\"s1\"],\"transitions\":[[\"e11\",null,\"e21\"],[\"f1\",null,\"e11\"],[\"m1\",\"a1\",\"m1\"],[\"m1\",\"b1\",\"f1\"],[\"s1\",null,\"d1\"],[\"s1\",null,\"m1\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s1, m1, f1, e11, e21, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1], [f1, null, e11], [e11, null, e21]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s1, [d1, m1, s1], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s1, [d1, m1, s1], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0003","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"659c5cedc8a7db6b34aa68eda01c3277e0dad47db8fc8cc39a7526720d4036fd","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127003,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a2\",\"b2\"],\"final_states\":[\"f2\"],\"start_state\":\"s2\",\"states\":[\"d2\",\"e12\",\"e22\",\"f2\",\"m2\",\"s2\"],\"transitions\":[[\"e12\",null,\"e22\"],[\"f2\",null,\"e12\"],[\"m2\",\"a2\",\"m2\"],[\"m2\",\"b2\",\"f2\"],[\"s2\",null,\"d2\"],[\"s2\",null,\"m2\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s2, m2, f2, e12, e22, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2], [f2, null, e12], [e12, null, e22]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s2, [d2, m2, s2], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s2, [d2, m2, s2], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0004","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59118a0d1df77a0ea8cb58e8fe55aaf8da3916815e161f781c91d5b6603ea4f8","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127004,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a3\",\"b3\"],\"final_states\":[\"f3\"],\"start_state\":\"s3\",\"states\":[\"d3\",\"e13\",\"e23\",\"f3\",\"m3\",\"s3\"],\"transitions\":[[\"e13\",null,\"e23\"],[\"f3\",null,\"e13\"],[\"m3\",\"a3\",\"m3\"],[\"m3\",\"b3\",\"f3\"],[\"s3\",null,\"d3\"],[\"s3\",null,\"m3\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s3, m3, f3, e13, e23, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3], [f3, null, e13], [e13, null, e23]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s3, [d3, m3, s3], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s3, [d3, m3, s3], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0005","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e86ed5559b3b90a31d5e42d8b8576e8369dd9ec96a5f8f2059d1c4b0f392c27","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127005,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a4\",\"b4\"],\"final_states\":[\"f4\"],\"start_state\":\"s4\",\"states\":[\"d4\",\"e14\",\"e24\",\"f4\",\"m4\",\"s4\"],\"transitions\":[[\"e14\",null,\"e24\"],[\"f4\",null,\"e14\"],[\"m4\",\"a4\",\"m4\"],[\"m4\",\"b4\",\"f4\"],[\"s4\",null,\"d4\"],[\"s4\",null,\"m4\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s4, m4, f4, e14, e24, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4], [f4, null, e14], [e14, null, e24]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s4, [d4, m4, s4], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s4, [d4, m4, s4], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0006","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b20ca3d9341190435acaa1b30f1c354fb5d40a5651abeffbc4f9b82d1fcfde73","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127006,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a5\",\"b5\"],\"final_states\":[\"f5\"],\"start_state\":\"s5\",\"states\":[\"d5\",\"e15\",\"e25\",\"f5\",\"m5\",\"s5\"],\"transitions\":[[\"e15\",null,\"e25\"],[\"f5\",null,\"e15\"],[\"m5\",\"a5\",\"m5\"],[\"m5\",\"b5\",\"f5\"],[\"s5\",null,\"d5\"],[\"s5\",null,\"m5\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s5, m5, f5, e15, e25, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5], [f5, null, e15], [e15, null, e25]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s5, [d5, m5, s5], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s5, [d5, m5, s5], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0007","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ae714185b1e4a6d4fcf0411867441b55a94de42a75da47450d4fb572cca0404","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127007,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a6\",\"b6\"],\"final_states\":[\"f6\"],\"start_state\":\"s6\",\"states\":[\"d6\",\"e16\",\"e26\",\"f6\",\"m6\",\"s6\"],\"transitions\":[[\"e16\",null,\"e26\"],[\"f6\",null,\"e16\"],[\"m6\",\"a6\",\"m6\"],[\"m6\",\"b6\",\"f6\"],[\"s6\",null,\"d6\"],[\"s6\",null,\"m6\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s6, m6, f6, e16, e26, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6], [f6, null, e16], [e16, null, e26]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s6, [d6, m6, s6], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s6, [d6, m6, s6], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0008","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61f04f618f2a9cb4dd104a2188a9417a80e1061c5d32d3e37115e748505d9f85","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127008,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a7\",\"b7\"],\"final_states\":[\"f7\"],\"start_state\":\"s7\",\"states\":[\"d7\",\"e17\",\"e27\",\"f7\",\"m7\",\"s7\"],\"transitions\":[[\"e17\",null,\"e27\"],[\"f7\",null,\"e17\"],[\"m7\",\"a7\",\"m7\"],[\"m7\",\"b7\",\"f7\"],[\"s7\",null,\"d7\"],[\"s7\",null,\"m7\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s7, m7, f7, e17, e27, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7], [f7, null, e17], [e17, null, e27]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s7, [d7, m7, s7], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s7, [d7, m7, s7], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0009","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0af19ece5ffed3de365c3085f43f3cbe84d0e23bb2a0425cce084904db2824ce","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127009,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a8\",\"b8\"],\"final_states\":[\"f8\"],\"start_state\":\"s8\",\"states\":[\"d8\",\"e18\",\"e28\",\"f8\",\"m8\",\"s8\"],\"transitions\":[[\"e18\",null,\"e28\"],[\"f8\",null,\"e18\"],[\"m8\",\"a8\",\"m8\"],[\"m8\",\"b8\",\"f8\"],[\"s8\",null,\"d8\"],[\"s8\",null,\"m8\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s8, m8, f8, e18, e28, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8], [f8, null, e18], [e18, null, e28]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s8, [d8, m8, s8], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s8, [d8, m8, s8], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0010","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cedc3dcb658cece2aece51227662df4d32394197bd58fdbf4e1c8afa84e644cb","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127010,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a9\",\"b9\"],\"final_states\":[\"f9\"],\"start_state\":\"s9\",\"states\":[\"d9\",\"e19\",\"e29\",\"f9\",\"m9\",\"s9\"],\"transitions\":[[\"e19\",null,\"e29\"],[\"f9\",null,\"e19\"],[\"m9\",\"a9\",\"m9\"],[\"m9\",\"b9\",\"f9\"],[\"s9\",null,\"d9\"],[\"s9\",null,\"m9\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s9, m9, f9, e19, e29, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9], [f9, null, e19], [e19, null, e29]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s9, [d9, m9, s9], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s9, [d9, m9, s9], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0011","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"005c042e99b0e7ba36783d92cef1c56110f52c142e4afc390ecd2cf86ca517b8","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127011,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a10\",\"b10\"],\"final_states\":[\"f10\"],\"start_state\":\"s10\",\"states\":[\"d10\",\"e110\",\"e210\",\"f10\",\"m10\",\"s10\"],\"transitions\":[[\"e110\",null,\"e210\"],[\"f10\",null,\"e110\"],[\"m10\",\"a10\",\"m10\"],[\"m10\",\"b10\",\"f10\"],[\"s10\",null,\"d10\"],[\"s10\",null,\"m10\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s10, m10, f10, e110, e210, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10], [f10, null, e110], [e110, null, e210]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s10, [d10, m10, s10], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s10, [d10, m10, s10], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0012","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d59a4a954920f7eb9e50fa09e5b4bea0e0db511272a15a04b3c11f0d958aa8f8","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127012,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a11\",\"b11\"],\"final_states\":[\"f11\"],\"start_state\":\"s11\",\"states\":[\"d11\",\"e111\",\"e211\",\"f11\",\"m11\",\"s11\"],\"transitions\":[[\"e111\",null,\"e211\"],[\"f11\",null,\"e111\"],[\"m11\",\"a11\",\"m11\"],[\"m11\",\"b11\",\"f11\"],[\"s11\",null,\"d11\"],[\"s11\",null,\"m11\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s11, m11, f11, e111, e211, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11], [f11, null, e111], [e111, null, e211]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s11, [d11, m11, s11], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s11, [d11, m11, s11], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0013","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c94a98fd59fceea20c44b5b601bcec3139b4b03271d25315d7d96830c692c325","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127013,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a12\",\"b12\"],\"final_states\":[\"f12\"],\"start_state\":\"s12\",\"states\":[\"d12\",\"e112\",\"e212\",\"f12\",\"m12\",\"s12\"],\"transitions\":[[\"e112\",null,\"e212\"],[\"f12\",null,\"e112\"],[\"m12\",\"a12\",\"m12\"],[\"m12\",\"b12\",\"f12\"],[\"s12\",null,\"d12\"],[\"s12\",null,\"m12\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s12, m12, f12, e112, e212, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12], [f12, null, e112], [e112, null, e212]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s12, [d12, m12, s12], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s12, [d12, m12, s12], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0014","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea6eca4bb7d8cc0c682d484f7f24bf5d9bb3b99e18083f95875a41616f78fd29","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127014,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a13\",\"b13\"],\"final_states\":[\"f13\"],\"start_state\":\"s13\",\"states\":[\"d13\",\"e113\",\"e213\",\"f13\",\"m13\",\"s13\"],\"transitions\":[[\"e113\",null,\"e213\"],[\"f13\",null,\"e113\"],[\"m13\",\"a13\",\"m13\"],[\"m13\",\"b13\",\"f13\"],[\"s13\",null,\"d13\"],[\"s13\",null,\"m13\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s13, m13, f13, e113, e213, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13], [f13, null, e113], [e113, null, e213]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s13, [d13, m13, s13], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s13, [d13, m13, s13], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0015","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fc2ad8e97ee93c70d749835dd229c22b2567c99b936051def8b622b7a5a6ba1","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127015,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a14\",\"b14\"],\"final_states\":[\"f14\"],\"start_state\":\"s14\",\"states\":[\"d14\",\"e114\",\"e214\",\"f14\",\"m14\",\"s14\"],\"transitions\":[[\"e114\",null,\"e214\"],[\"f14\",null,\"e114\"],[\"m14\",\"a14\",\"m14\"],[\"m14\",\"b14\",\"f14\"],[\"s14\",null,\"d14\"],[\"s14\",null,\"m14\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s14, m14, f14, e114, e214, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14], [f14, null, e114], [e114, null, e214]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s14, [d14, m14, s14], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s14, [d14, m14, s14], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0016","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3af046dc8db837db38ac5c3e1944bce9131aaedc35759e74bab02aa45d220019","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127016,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a15\",\"b15\"],\"final_states\":[\"f15\"],\"start_state\":\"s15\",\"states\":[\"d15\",\"e115\",\"e215\",\"f15\",\"m15\",\"s15\"],\"transitions\":[[\"e115\",null,\"e215\"],[\"f15\",null,\"e115\"],[\"m15\",\"a15\",\"m15\"],[\"m15\",\"b15\",\"f15\"],[\"s15\",null,\"d15\"],[\"s15\",null,\"m15\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s15, m15, f15, e115, e215, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15], [f15, null, e115], [e115, null, e215]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s15, [d15, m15, s15], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s15, [d15, m15, s15], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0017","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47db65e5e3febe3eda8c69ed2e9147e40c6ded96e7a57b40c0900570626aa392","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127017,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a16\",\"b16\"],\"final_states\":[\"f16\"],\"start_state\":\"s16\",\"states\":[\"d16\",\"e116\",\"e216\",\"f16\",\"m16\",\"s16\"],\"transitions\":[[\"e116\",null,\"e216\"],[\"f16\",null,\"e116\"],[\"m16\",\"a16\",\"m16\"],[\"m16\",\"b16\",\"f16\"],[\"s16\",null,\"d16\"],[\"s16\",null,\"m16\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s16, m16, f16, e116, e216, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16], [f16, null, e116], [e116, null, e216]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s16, [d16, m16, s16], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s16, [d16, m16, s16], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0018","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a9cee717f45e737697ec7c8d27d1e2f181565ddee77fa7d41dada7b74360565","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127018,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a17\",\"b17\"],\"final_states\":[\"f17\"],\"start_state\":\"s17\",\"states\":[\"d17\",\"e117\",\"e217\",\"f17\",\"m17\",\"s17\"],\"transitions\":[[\"e117\",null,\"e217\"],[\"f17\",null,\"e117\"],[\"m17\",\"a17\",\"m17\"],[\"m17\",\"b17\",\"f17\"],[\"s17\",null,\"d17\"],[\"s17\",null,\"m17\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s17, m17, f17, e117, e217, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17], [f17, null, e117], [e117, null, e217]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s17, [d17, m17, s17], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s17, [d17, m17, s17], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0019","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54763724768c8da2601e0545757e5db5bebaf7d0e84f4ead70e00d67a6f5e1a0","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127019,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a18\",\"b18\"],\"final_states\":[\"f18\"],\"start_state\":\"s18\",\"states\":[\"d18\",\"e118\",\"e218\",\"f18\",\"m18\",\"s18\"],\"transitions\":[[\"e118\",null,\"e218\"],[\"f18\",null,\"e118\"],[\"m18\",\"a18\",\"m18\"],[\"m18\",\"b18\",\"f18\"],[\"s18\",null,\"d18\"],[\"s18\",null,\"m18\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s18, m18, f18, e118, e218, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18], [f18, null, e118], [e118, null, e218]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s18, [d18, m18, s18], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s18, [d18, m18, s18], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0020","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6112c968f290edd7ecdc6388d4417623a585e258405b3648b43e5a0bfc0110b8","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127020,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a19\",\"b19\"],\"final_states\":[\"f19\"],\"start_state\":\"s19\",\"states\":[\"d19\",\"e119\",\"e219\",\"f19\",\"m19\",\"s19\"],\"transitions\":[[\"e119\",null,\"e219\"],[\"f19\",null,\"e119\"],[\"m19\",\"a19\",\"m19\"],[\"m19\",\"b19\",\"f19\"],[\"s19\",null,\"d19\"],[\"s19\",null,\"m19\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s19, m19, f19, e119, e219, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19], [f19, null, e119], [e119, null, e219]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s19, [d19, m19, s19], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s19, [d19, m19, s19], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0021","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e6ae8f52e248e368b79057ee3e6ac5e78d2f38886772cb4a7e40b3599637f04","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127021,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a20\",\"b20\"],\"final_states\":[\"f20\"],\"start_state\":\"s20\",\"states\":[\"d20\",\"e120\",\"e220\",\"f20\",\"m20\",\"s20\"],\"transitions\":[[\"e120\",null,\"e220\"],[\"f20\",null,\"e120\"],[\"m20\",\"a20\",\"m20\"],[\"m20\",\"b20\",\"f20\"],[\"s20\",null,\"d20\"],[\"s20\",null,\"m20\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s20, m20, f20, e120, e220, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20], [f20, null, e120], [e120, null, e220]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s20, [d20, m20, s20], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s20, [d20, m20, s20], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0022","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1ee61ee3575127f1585c2518256cfe64bf2f9016e2340c81b3d5881ff0ce50a","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127022,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a21\",\"b21\"],\"final_states\":[\"f21\"],\"start_state\":\"s21\",\"states\":[\"d21\",\"e121\",\"e221\",\"f21\",\"m21\",\"s21\"],\"transitions\":[[\"e121\",null,\"e221\"],[\"f21\",null,\"e121\"],[\"m21\",\"a21\",\"m21\"],[\"m21\",\"b21\",\"f21\"],[\"s21\",null,\"d21\"],[\"s21\",null,\"m21\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s21, m21, f21, e121, e221, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s21, [d21, m21, s21], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s21, [d21, m21, s21], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0023","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a1a83d08dcf9c710608aaa63e63db201338230e9f8087738220b991507222a3","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127023,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a22\",\"b22\"],\"final_states\":[\"f22\"],\"start_state\":\"s22\",\"states\":[\"d22\",\"e122\",\"e222\",\"f22\",\"m22\",\"s22\"],\"transitions\":[[\"e122\",null,\"e222\"],[\"f22\",null,\"e122\"],[\"m22\",\"a22\",\"m22\"],[\"m22\",\"b22\",\"f22\"],[\"s22\",null,\"d22\"],[\"s22\",null,\"m22\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s22, m22, f22, e122, e222, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22], [f22, null, e122], [e122, null, e222]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s22, [d22, m22, s22], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s22, [d22, m22, s22], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0024","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bd0194677528c324cc7ddb2e12d81a8661b6022b9982073095e37ff7a846942","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127024,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a23\",\"b23\"],\"final_states\":[\"f23\"],\"start_state\":\"s23\",\"states\":[\"d23\",\"e123\",\"e223\",\"f23\",\"m23\",\"s23\"],\"transitions\":[[\"e123\",null,\"e223\"],[\"f23\",null,\"e123\"],[\"m23\",\"a23\",\"m23\"],[\"m23\",\"b23\",\"f23\"],[\"s23\",null,\"d23\"],[\"s23\",null,\"m23\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s23, m23, f23, e123, e223, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23], [f23, null, e123], [e123, null, e223]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s23, [d23, m23, s23], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s23, [d23, m23, s23], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0025","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69e95ecacd7df2598777a5e0bddb734a274c26b9b2b24e12e9742fd2a9fa3b33","family_id":"qf-mod43-subsetdfa-epsilon-start","seed":202611127025,"semantic_template":"qf-mod43-subsetdfa-epsilon-start/{\"alphabet\":[\"a24\",\"b24\"],\"final_states\":[\"f24\"],\"start_state\":\"s24\",\"states\":[\"d24\",\"e124\",\"e224\",\"f24\",\"m24\",\"s24\"],\"transitions\":[[\"e124\",null,\"e224\"],[\"f24\",null,\"e124\"],[\"m24\",\"a24\",\"m24\"],[\"m24\",\"b24\",\"f24\"],[\"s24\",null,\"d24\"],[\"s24\",null,\"m24\"]]}"},"fields":{"title":"The DFA Start State Is the Epsilon Closure of the NFA Start Singleton","stem":"Use complete epsilon-NFA states=[s24, m24, f24, e124, e224, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24], [f24, null, e124], [e124, null, e224]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Give (NFA_start, DFA_start_subset, epsilon_closed_certificate, certificate).","answer":{"kind":"expression","value":"(s24, [d24, m24, s24], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)","explanation":"Compute all epsilon-reachable start states, including the original start, without consuming input. This closed set is the DFA start, not the unclosed singleton alone. The exact result, retaining the requested order, is (s24, [d24, m24, s24], Yes, DFA_start_is_epsilon_closure_of_the_NFA_start_singleton)."},"skills":["The DFA Start State Is the Epsilon Closure of the NFA Start Singleton"],"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-20261112-r07-subset-construction-nfa-to-dfa-0026","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab83cb7e36a9f25742ae346558c008c81861c2e58ffb821a50fdd54d59755413","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127026,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a0\",\"b0\"],\"final_states\":[\"f0\"],\"start_state\":\"s0\",\"states\":[\"d0\",\"e10\",\"e20\",\"f0\",\"m0\",\"s0\"],\"subset\":[\"d0\",\"m0\",\"s0\"],\"symbol\":\"a0\",\"transitions\":[[\"e10\",null,\"e20\"],[\"f0\",null,\"e10\"],[\"m0\",\"a0\",\"m0\"],[\"m0\",\"b0\",\"f0\"],[\"s0\",null,\"d0\"],[\"s0\",null,\"m0\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s0, m0, f0, e10, e20, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0], [f0, null, e10], [e10, null, e20]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s0, m0, d0], input symbol=a0. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d0, m0, s0], a0, [m0], [m0], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d0, m0, s0], a0, [m0], [m0], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0027","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9f3831a20f16e4cf9604a8b59d81bf751874cc14f7d22ad961aca876c9439d6","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127027,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a1\",\"b1\"],\"final_states\":[\"f1\"],\"start_state\":\"s1\",\"states\":[\"d1\",\"e11\",\"e21\",\"f1\",\"m1\",\"s1\"],\"subset\":[\"d1\",\"m1\",\"s1\"],\"symbol\":\"b1\",\"transitions\":[[\"e11\",null,\"e21\"],[\"f1\",null,\"e11\"],[\"m1\",\"a1\",\"m1\"],[\"m1\",\"b1\",\"f1\"],[\"s1\",null,\"d1\"],[\"s1\",null,\"m1\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s1, m1, f1, e11, e21, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1], [f1, null, e11], [e11, null, e21]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s1, m1, d1], input symbol=b1. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d1, m1, s1], b1, [f1], [e11, e21, f1], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d1, m1, s1], b1, [f1], [e11, e21, f1], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0028","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cce0d8dffe8e2750516c70d176584e45362ac43392c18727de7ece14e0a1dd82","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127028,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a2\",\"b2\"],\"final_states\":[\"f2\"],\"start_state\":\"s2\",\"states\":[\"d2\",\"e12\",\"e22\",\"f2\",\"m2\",\"s2\"],\"subset\":[\"d2\",\"m2\",\"s2\"],\"symbol\":\"a2\",\"transitions\":[[\"e12\",null,\"e22\"],[\"f2\",null,\"e12\"],[\"m2\",\"a2\",\"m2\"],[\"m2\",\"b2\",\"f2\"],[\"s2\",null,\"d2\"],[\"s2\",null,\"m2\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s2, m2, f2, e12, e22, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2], [f2, null, e12], [e12, null, e22]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s2, m2, d2], input symbol=a2. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d2, m2, s2], a2, [m2], [m2], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d2, m2, s2], a2, [m2], [m2], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0029","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76856e96e0047ae594d1ddbb2096538e7612c061fcb2190dc2edf2cd893dfe7a","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127029,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a3\",\"b3\"],\"final_states\":[\"f3\"],\"start_state\":\"s3\",\"states\":[\"d3\",\"e13\",\"e23\",\"f3\",\"m3\",\"s3\"],\"subset\":[\"d3\",\"m3\",\"s3\"],\"symbol\":\"b3\",\"transitions\":[[\"e13\",null,\"e23\"],[\"f3\",null,\"e13\"],[\"m3\",\"a3\",\"m3\"],[\"m3\",\"b3\",\"f3\"],[\"s3\",null,\"d3\"],[\"s3\",null,\"m3\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s3, m3, f3, e13, e23, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3], [f3, null, e13], [e13, null, e23]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s3, m3, d3], input symbol=b3. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d3, m3, s3], b3, [f3], [e13, e23, f3], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d3, m3, s3], b3, [f3], [e13, e23, f3], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0030","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"100a2b0fbcb3e43971e0ae96f76df4c42b4a475ce4e9f6a8910a9ce78d239596","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127030,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a4\",\"b4\"],\"final_states\":[\"f4\"],\"start_state\":\"s4\",\"states\":[\"d4\",\"e14\",\"e24\",\"f4\",\"m4\",\"s4\"],\"subset\":[\"d4\",\"m4\",\"s4\"],\"symbol\":\"a4\",\"transitions\":[[\"e14\",null,\"e24\"],[\"f4\",null,\"e14\"],[\"m4\",\"a4\",\"m4\"],[\"m4\",\"b4\",\"f4\"],[\"s4\",null,\"d4\"],[\"s4\",null,\"m4\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s4, m4, f4, e14, e24, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4], [f4, null, e14], [e14, null, e24]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s4, m4, d4], input symbol=a4. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d4, m4, s4], a4, [m4], [m4], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d4, m4, s4], a4, [m4], [m4], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0031","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23ef8a53fa8c2b40ba84430ec7828c1a1ad467eae651095ef790fc0bd9c063b4","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127031,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a5\",\"b5\"],\"final_states\":[\"f5\"],\"start_state\":\"s5\",\"states\":[\"d5\",\"e15\",\"e25\",\"f5\",\"m5\",\"s5\"],\"subset\":[\"d5\",\"m5\",\"s5\"],\"symbol\":\"b5\",\"transitions\":[[\"e15\",null,\"e25\"],[\"f5\",null,\"e15\"],[\"m5\",\"a5\",\"m5\"],[\"m5\",\"b5\",\"f5\"],[\"s5\",null,\"d5\"],[\"s5\",null,\"m5\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s5, m5, f5, e15, e25, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5], [f5, null, e15], [e15, null, e25]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s5, m5, d5], input symbol=b5. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d5, m5, s5], b5, [f5], [e15, e25, f5], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d5, m5, s5], b5, [f5], [e15, e25, f5], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0032","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef40915fc459e4a1a06781875c0d3bade8568c54d7ef07616df67c1d229eaa54","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127032,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a6\",\"b6\"],\"final_states\":[\"f6\"],\"start_state\":\"s6\",\"states\":[\"d6\",\"e16\",\"e26\",\"f6\",\"m6\",\"s6\"],\"subset\":[\"d6\",\"m6\",\"s6\"],\"symbol\":\"a6\",\"transitions\":[[\"e16\",null,\"e26\"],[\"f6\",null,\"e16\"],[\"m6\",\"a6\",\"m6\"],[\"m6\",\"b6\",\"f6\"],[\"s6\",null,\"d6\"],[\"s6\",null,\"m6\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s6, m6, f6, e16, e26, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6], [f6, null, e16], [e16, null, e26]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s6, m6, d6], input symbol=a6. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d6, m6, s6], a6, [m6], [m6], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d6, m6, s6], a6, [m6], [m6], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0033","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0d68db238bce043fdc0be9f1f5cd18d7363c086ccd0339120cb88270095d785","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127033,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a7\",\"b7\"],\"final_states\":[\"f7\"],\"start_state\":\"s7\",\"states\":[\"d7\",\"e17\",\"e27\",\"f7\",\"m7\",\"s7\"],\"subset\":[\"d7\",\"m7\",\"s7\"],\"symbol\":\"b7\",\"transitions\":[[\"e17\",null,\"e27\"],[\"f7\",null,\"e17\"],[\"m7\",\"a7\",\"m7\"],[\"m7\",\"b7\",\"f7\"],[\"s7\",null,\"d7\"],[\"s7\",null,\"m7\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s7, m7, f7, e17, e27, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7], [f7, null, e17], [e17, null, e27]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s7, m7, d7], input symbol=b7. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d7, m7, s7], b7, [f7], [e17, e27, f7], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d7, m7, s7], b7, [f7], [e17, e27, f7], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0034","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47eb543b3eee9b496e0df2d2a06c87a1a5f9b805f456bd25299dea6ab056b77e","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127034,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a8\",\"b8\"],\"final_states\":[\"f8\"],\"start_state\":\"s8\",\"states\":[\"d8\",\"e18\",\"e28\",\"f8\",\"m8\",\"s8\"],\"subset\":[\"d8\",\"m8\",\"s8\"],\"symbol\":\"a8\",\"transitions\":[[\"e18\",null,\"e28\"],[\"f8\",null,\"e18\"],[\"m8\",\"a8\",\"m8\"],[\"m8\",\"b8\",\"f8\"],[\"s8\",null,\"d8\"],[\"s8\",null,\"m8\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s8, m8, f8, e18, e28, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8], [f8, null, e18], [e18, null, e28]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s8, m8, d8], input symbol=a8. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d8, m8, s8], a8, [m8], [m8], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d8, m8, s8], a8, [m8], [m8], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0035","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0774f40c1f44e1d73bf9278651a75b6921d8bae48527ad93dc9fdf5704a4d4a6","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127035,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a9\",\"b9\"],\"final_states\":[\"f9\"],\"start_state\":\"s9\",\"states\":[\"d9\",\"e19\",\"e29\",\"f9\",\"m9\",\"s9\"],\"subset\":[\"d9\",\"m9\",\"s9\"],\"symbol\":\"b9\",\"transitions\":[[\"e19\",null,\"e29\"],[\"f9\",null,\"e19\"],[\"m9\",\"a9\",\"m9\"],[\"m9\",\"b9\",\"f9\"],[\"s9\",null,\"d9\"],[\"s9\",null,\"m9\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s9, m9, f9, e19, e29, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9], [f9, null, e19], [e19, null, e29]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s9, m9, d9], input symbol=b9. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d9, m9, s9], b9, [f9], [e19, e29, f9], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d9, m9, s9], b9, [f9], [e19, e29, f9], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0036","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fee5044685e3ef054057b094ab75a4a9b8549e99263b7b6a1b0eeeb4cbad32b6","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127036,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a10\",\"b10\"],\"final_states\":[\"f10\"],\"start_state\":\"s10\",\"states\":[\"d10\",\"e110\",\"e210\",\"f10\",\"m10\",\"s10\"],\"subset\":[\"d10\",\"m10\",\"s10\"],\"symbol\":\"a10\",\"transitions\":[[\"e110\",null,\"e210\"],[\"f10\",null,\"e110\"],[\"m10\",\"a10\",\"m10\"],[\"m10\",\"b10\",\"f10\"],[\"s10\",null,\"d10\"],[\"s10\",null,\"m10\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s10, m10, f10, e110, e210, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10], [f10, null, e110], [e110, null, e210]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s10, m10, d10], input symbol=a10. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d10, m10, s10], a10, [m10], [m10], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d10, m10, s10], a10, [m10], [m10], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0037","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0d2eeb3cb18ee929bfd4bf91f5581356efeda9824677096e4b30c4f317ca13c","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127037,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a11\",\"b11\"],\"final_states\":[\"f11\"],\"start_state\":\"s11\",\"states\":[\"d11\",\"e111\",\"e211\",\"f11\",\"m11\",\"s11\"],\"subset\":[\"d11\",\"m11\",\"s11\"],\"symbol\":\"b11\",\"transitions\":[[\"e111\",null,\"e211\"],[\"f11\",null,\"e111\"],[\"m11\",\"a11\",\"m11\"],[\"m11\",\"b11\",\"f11\"],[\"s11\",null,\"d11\"],[\"s11\",null,\"m11\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s11, m11, f11, e111, e211, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11], [f11, null, e111], [e111, null, e211]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s11, m11, d11], input symbol=b11. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d11, m11, s11], b11, [f11], [e111, e211, f11], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d11, m11, s11], b11, [f11], [e111, e211, f11], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0038","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"800674fce0aec106a619f4590470f11097c815066c2d910e1da17f98aabfb4db","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127038,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a12\",\"b12\"],\"final_states\":[\"f12\"],\"start_state\":\"s12\",\"states\":[\"d12\",\"e112\",\"e212\",\"f12\",\"m12\",\"s12\"],\"subset\":[\"d12\",\"m12\",\"s12\"],\"symbol\":\"a12\",\"transitions\":[[\"e112\",null,\"e212\"],[\"f12\",null,\"e112\"],[\"m12\",\"a12\",\"m12\"],[\"m12\",\"b12\",\"f12\"],[\"s12\",null,\"d12\"],[\"s12\",null,\"m12\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s12, m12, f12, e112, e212, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12], [f12, null, e112], [e112, null, e212]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s12, m12, d12], input symbol=a12. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d12, m12, s12], a12, [m12], [m12], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d12, m12, s12], a12, [m12], [m12], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0039","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfac95854378cf7ee020fb70555c3aafdfc5fe6005dceaecd46b8c3628f2cb26","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127039,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a13\",\"b13\"],\"final_states\":[\"f13\"],\"start_state\":\"s13\",\"states\":[\"d13\",\"e113\",\"e213\",\"f13\",\"m13\",\"s13\"],\"subset\":[\"d13\",\"m13\",\"s13\"],\"symbol\":\"b13\",\"transitions\":[[\"e113\",null,\"e213\"],[\"f13\",null,\"e113\"],[\"m13\",\"a13\",\"m13\"],[\"m13\",\"b13\",\"f13\"],[\"s13\",null,\"d13\"],[\"s13\",null,\"m13\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s13, m13, f13, e113, e213, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13], [f13, null, e113], [e113, null, e213]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s13, m13, d13], input symbol=b13. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d13, m13, s13], b13, [f13], [e113, e213, f13], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d13, m13, s13], b13, [f13], [e113, e213, f13], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0040","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba790a61116c4cba850c06a4ce1ee39fd642bc25cb4b86c1eef585d1ecd517cc","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127040,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a14\",\"b14\"],\"final_states\":[\"f14\"],\"start_state\":\"s14\",\"states\":[\"d14\",\"e114\",\"e214\",\"f14\",\"m14\",\"s14\"],\"subset\":[\"d14\",\"m14\",\"s14\"],\"symbol\":\"a14\",\"transitions\":[[\"e114\",null,\"e214\"],[\"f14\",null,\"e114\"],[\"m14\",\"a14\",\"m14\"],[\"m14\",\"b14\",\"f14\"],[\"s14\",null,\"d14\"],[\"s14\",null,\"m14\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s14, m14, f14, e114, e214, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14], [f14, null, e114], [e114, null, e214]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s14, m14, d14], input symbol=a14. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d14, m14, s14], a14, [m14], [m14], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d14, m14, s14], a14, [m14], [m14], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0041","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2fa231345961d90e31ba2d5f08e5937cf7688efa1dabe33272ada37fdcb403c","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127041,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a15\",\"b15\"],\"final_states\":[\"f15\"],\"start_state\":\"s15\",\"states\":[\"d15\",\"e115\",\"e215\",\"f15\",\"m15\",\"s15\"],\"subset\":[\"d15\",\"m15\",\"s15\"],\"symbol\":\"b15\",\"transitions\":[[\"e115\",null,\"e215\"],[\"f15\",null,\"e115\"],[\"m15\",\"a15\",\"m15\"],[\"m15\",\"b15\",\"f15\"],[\"s15\",null,\"d15\"],[\"s15\",null,\"m15\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s15, m15, f15, e115, e215, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15], [f15, null, e115], [e115, null, e215]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s15, m15, d15], input symbol=b15. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d15, m15, s15], b15, [f15], [e115, e215, f15], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d15, m15, s15], b15, [f15], [e115, e215, f15], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0042","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"660ec5e23ad14da2ed439fc3e932a0cc9b6703b78bff8c2052256f048f5bbc0c","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127042,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a16\",\"b16\"],\"final_states\":[\"f16\"],\"start_state\":\"s16\",\"states\":[\"d16\",\"e116\",\"e216\",\"f16\",\"m16\",\"s16\"],\"subset\":[\"d16\",\"m16\",\"s16\"],\"symbol\":\"a16\",\"transitions\":[[\"e116\",null,\"e216\"],[\"f16\",null,\"e116\"],[\"m16\",\"a16\",\"m16\"],[\"m16\",\"b16\",\"f16\"],[\"s16\",null,\"d16\"],[\"s16\",null,\"m16\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s16, m16, f16, e116, e216, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16], [f16, null, e116], [e116, null, e216]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s16, m16, d16], input symbol=a16. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d16, m16, s16], a16, [m16], [m16], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d16, m16, s16], a16, [m16], [m16], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0043","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2c5906bbe363440d97cc0efdcb6d9edd2b1fbf4efd3f4690cd0ef4fc5253e13","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127043,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a17\",\"b17\"],\"final_states\":[\"f17\"],\"start_state\":\"s17\",\"states\":[\"d17\",\"e117\",\"e217\",\"f17\",\"m17\",\"s17\"],\"subset\":[\"d17\",\"m17\",\"s17\"],\"symbol\":\"b17\",\"transitions\":[[\"e117\",null,\"e217\"],[\"f17\",null,\"e117\"],[\"m17\",\"a17\",\"m17\"],[\"m17\",\"b17\",\"f17\"],[\"s17\",null,\"d17\"],[\"s17\",null,\"m17\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s17, m17, f17, e117, e217, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17], [f17, null, e117], [e117, null, e217]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s17, m17, d17], input symbol=b17. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d17, m17, s17], b17, [f17], [e117, e217, f17], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d17, m17, s17], b17, [f17], [e117, e217, f17], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0044","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b406c7812f8f36e52a388a04cd90211dc283e8eb189515a5065362b235fab97f","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127044,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a18\",\"b18\"],\"final_states\":[\"f18\"],\"start_state\":\"s18\",\"states\":[\"d18\",\"e118\",\"e218\",\"f18\",\"m18\",\"s18\"],\"subset\":[\"d18\",\"m18\",\"s18\"],\"symbol\":\"a18\",\"transitions\":[[\"e118\",null,\"e218\"],[\"f18\",null,\"e118\"],[\"m18\",\"a18\",\"m18\"],[\"m18\",\"b18\",\"f18\"],[\"s18\",null,\"d18\"],[\"s18\",null,\"m18\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s18, m18, f18, e118, e218, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18], [f18, null, e118], [e118, null, e218]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s18, m18, d18], input symbol=a18. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d18, m18, s18], a18, [m18], [m18], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d18, m18, s18], a18, [m18], [m18], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0045","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ad7015a7d49daddad56e54be402b451f19ac59ceb157d1fd63411f4e4e3edec","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127045,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a19\",\"b19\"],\"final_states\":[\"f19\"],\"start_state\":\"s19\",\"states\":[\"d19\",\"e119\",\"e219\",\"f19\",\"m19\",\"s19\"],\"subset\":[\"d19\",\"m19\",\"s19\"],\"symbol\":\"b19\",\"transitions\":[[\"e119\",null,\"e219\"],[\"f19\",null,\"e119\"],[\"m19\",\"a19\",\"m19\"],[\"m19\",\"b19\",\"f19\"],[\"s19\",null,\"d19\"],[\"s19\",null,\"m19\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s19, m19, f19, e119, e219, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19], [f19, null, e119], [e119, null, e219]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s19, m19, d19], input symbol=b19. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d19, m19, s19], b19, [f19], [e119, e219, f19], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d19, m19, s19], b19, [f19], [e119, e219, f19], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0046","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c425cea308be6a715ee5d038a4b44fa5d08197048dd26513ff8723d86ad3c8e9","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127046,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a20\",\"b20\"],\"final_states\":[\"f20\"],\"start_state\":\"s20\",\"states\":[\"d20\",\"e120\",\"e220\",\"f20\",\"m20\",\"s20\"],\"subset\":[\"d20\",\"m20\",\"s20\"],\"symbol\":\"a20\",\"transitions\":[[\"e120\",null,\"e220\"],[\"f20\",null,\"e120\"],[\"m20\",\"a20\",\"m20\"],[\"m20\",\"b20\",\"f20\"],[\"s20\",null,\"d20\"],[\"s20\",null,\"m20\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s20, m20, f20, e120, e220, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20], [f20, null, e120], [e120, null, e220]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s20, m20, d20], input symbol=a20. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d20, m20, s20], a20, [m20], [m20], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d20, m20, s20], a20, [m20], [m20], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0047","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b42652dc44b09a3e259e5b608e0c17a7eb9f7b8226befa38b9e3337075da85a7","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127047,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a21\",\"b21\"],\"final_states\":[\"f21\"],\"start_state\":\"s21\",\"states\":[\"d21\",\"e121\",\"e221\",\"f21\",\"m21\",\"s21\"],\"subset\":[\"d21\",\"m21\",\"s21\"],\"symbol\":\"b21\",\"transitions\":[[\"e121\",null,\"e221\"],[\"f21\",null,\"e121\"],[\"m21\",\"a21\",\"m21\"],[\"m21\",\"b21\",\"f21\"],[\"s21\",null,\"d21\"],[\"s21\",null,\"m21\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s21, m21, f21, e121, e221, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s21, m21, d21], input symbol=b21. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d21, m21, s21], b21, [f21], [e121, e221, f21], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d21, m21, s21], b21, [f21], [e121, e221, f21], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0048","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31ccfabc4eba972fe3eb998814bd17644404deef658c4c4a33da7aabf62d618f","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127048,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a22\",\"b22\"],\"final_states\":[\"f22\"],\"start_state\":\"s22\",\"states\":[\"d22\",\"e122\",\"e222\",\"f22\",\"m22\",\"s22\"],\"subset\":[\"d22\",\"m22\",\"s22\"],\"symbol\":\"a22\",\"transitions\":[[\"e122\",null,\"e222\"],[\"f22\",null,\"e122\"],[\"m22\",\"a22\",\"m22\"],[\"m22\",\"b22\",\"f22\"],[\"s22\",null,\"d22\"],[\"s22\",null,\"m22\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s22, m22, f22, e122, e222, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22], [f22, null, e122], [e122, null, e222]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s22, m22, d22], input symbol=a22. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d22, m22, s22], a22, [m22], [m22], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d22, m22, s22], a22, [m22], [m22], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0049","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c815cc14591c4e3fc9ecd891712f6262e3a73be2eab72fc437e17adfb0201046","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127049,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a23\",\"b23\"],\"final_states\":[\"f23\"],\"start_state\":\"s23\",\"states\":[\"d23\",\"e123\",\"e223\",\"f23\",\"m23\",\"s23\"],\"subset\":[\"d23\",\"m23\",\"s23\"],\"symbol\":\"b23\",\"transitions\":[[\"e123\",null,\"e223\"],[\"f23\",null,\"e123\"],[\"m23\",\"a23\",\"m23\"],[\"m23\",\"b23\",\"f23\"],[\"s23\",null,\"d23\"],[\"s23\",null,\"m23\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s23, m23, f23, e123, e223, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23], [f23, null, e123], [e123, null, e223]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s23, m23, d23], input symbol=b23. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d23, m23, s23], b23, [f23], [e123, e223, f23], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d23, m23, s23], b23, [f23], [e123, e223, f23], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0050","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56381e699bb2e4f106a0a19b74053d2271f3a09f5301c114bbbb1c3ad30e3f07","family_id":"qf-mod43-subsetdfa-symbol-transition","seed":202611127050,"semantic_template":"qf-mod43-subsetdfa-symbol-transition/{\"alphabet\":[\"a24\",\"b24\"],\"final_states\":[\"f24\"],\"start_state\":\"s24\",\"states\":[\"d24\",\"e124\",\"e224\",\"f24\",\"m24\",\"s24\"],\"subset\":[\"d24\",\"m24\",\"s24\"],\"symbol\":\"a24\",\"transitions\":[[\"e124\",null,\"e224\"],[\"f24\",null,\"e124\"],[\"m24\",\"a24\",\"m24\"],[\"m24\",\"b24\",\"f24\"],[\"s24\",null,\"d24\"],[\"s24\",null,\"m24\"]]}"},"fields":{"title":"A DFA Subset Transition Applies Symbol Move before Epsilon Closure","stem":"Use complete epsilon-NFA states=[s24, m24, f24, e124, e224, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24], [f24, null, e124], [e124, null, e224]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[s24, m24, d24], input symbol=a24. Give (subset, symbol, symbol_targets, epsilon_closed_targets, empty_dead_state, certificate).","answer":{"kind":"expression","value":"([d24, m24, s24], a24, [m24], [m24], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)","explanation":"Compute direct symbol targets from this exact S, then their epsilon closure. Do not silently pre-close an arbitrary supplied S; the prefix invariant concerns reachable states from the closed start. Empty targets close to the legal empty dead state. The exact result, retaining the requested order, is ([d24, m24, s24], a24, [m24], [m24], No, delta_D_equals_epsilon_closure_after_symbol_move_and_empty_subset_is_a_legal_dead_state)."},"skills":["A DFA Subset Transition Applies Symbol Move before Epsilon Closure"],"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-20261112-r07-subset-construction-nfa-to-dfa-0051","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6f495c2f1a272a3676e97fab805aef3bba91c822ef7ce2c661e79da10c3c736","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127051,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a0\",\"b0\"],\"final_states\":[\"f0\"],\"start_state\":\"s0\",\"states\":[\"d0\",\"e10\",\"e20\",\"f0\",\"m0\",\"s0\"],\"subset\":[\"f0\",\"m0\"],\"transitions\":[[\"e10\",null,\"e20\"],[\"f0\",null,\"e10\"],[\"m0\",\"a0\",\"m0\"],[\"m0\",\"b0\",\"f0\"],[\"s0\",null,\"d0\"],[\"s0\",null,\"m0\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s0, m0, f0, e10, e20, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0], [f0, null, e10], [e10, null, e20]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f0, m0]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([f0, m0], [f0], [f0], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f0, m0], [f0], [f0], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0052","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c43a6a5ca6c08ed244f52eef1e020c59a8cb747aaeb3838fdcf14c79beb5f48b","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127052,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a1\",\"b1\"],\"final_states\":[\"f1\"],\"start_state\":\"s1\",\"states\":[\"d1\",\"e11\",\"e21\",\"f1\",\"m1\",\"s1\"],\"subset\":[\"d1\",\"m1\"],\"transitions\":[[\"e11\",null,\"e21\"],[\"f1\",null,\"e11\"],[\"m1\",\"a1\",\"m1\"],[\"m1\",\"b1\",\"f1\"],[\"s1\",null,\"d1\"],[\"s1\",null,\"m1\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s1, m1, f1, e11, e21, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1], [f1, null, e11], [e11, null, e21]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m1, d1]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([d1, m1], [f1], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d1, m1], [f1], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0053","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1c385312cf98eda66146fd8999e83ea69eec27155aac8ac1b0172326094a6f5","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127053,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a2\",\"b2\"],\"final_states\":[\"f2\"],\"start_state\":\"s2\",\"states\":[\"d2\",\"e12\",\"e22\",\"f2\",\"m2\",\"s2\"],\"subset\":[\"f2\",\"m2\"],\"transitions\":[[\"e12\",null,\"e22\"],[\"f2\",null,\"e12\"],[\"m2\",\"a2\",\"m2\"],[\"m2\",\"b2\",\"f2\"],[\"s2\",null,\"d2\"],[\"s2\",null,\"m2\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s2, m2, f2, e12, e22, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2], [f2, null, e12], [e12, null, e22]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f2, m2]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([f2, m2], [f2], [f2], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f2, m2], [f2], [f2], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0054","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0db22e142e2f53ed63eabc1a79bd2bbc8b54bd0a6413812101fc65a3159e080","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127054,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a3\",\"b3\"],\"final_states\":[\"f3\"],\"start_state\":\"s3\",\"states\":[\"d3\",\"e13\",\"e23\",\"f3\",\"m3\",\"s3\"],\"subset\":[\"d3\",\"m3\"],\"transitions\":[[\"e13\",null,\"e23\"],[\"f3\",null,\"e13\"],[\"m3\",\"a3\",\"m3\"],[\"m3\",\"b3\",\"f3\"],[\"s3\",null,\"d3\"],[\"s3\",null,\"m3\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s3, m3, f3, e13, e23, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3], [f3, null, e13], [e13, null, e23]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m3, d3]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([d3, m3], [f3], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d3, m3], [f3], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0055","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9db37964dc46e535f5f79aac4df7b81818a0f5d3c0f22b1f3683b5cbfe39d961","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127055,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a4\",\"b4\"],\"final_states\":[\"f4\"],\"start_state\":\"s4\",\"states\":[\"d4\",\"e14\",\"e24\",\"f4\",\"m4\",\"s4\"],\"subset\":[\"f4\",\"m4\"],\"transitions\":[[\"e14\",null,\"e24\"],[\"f4\",null,\"e14\"],[\"m4\",\"a4\",\"m4\"],[\"m4\",\"b4\",\"f4\"],[\"s4\",null,\"d4\"],[\"s4\",null,\"m4\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s4, m4, f4, e14, e24, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4], [f4, null, e14], [e14, null, e24]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f4, m4]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([f4, m4], [f4], [f4], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f4, m4], [f4], [f4], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0056","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d924e6f690b121156bd2774adc15d9d44af4001aedfb5b52d795a01c95a9a15","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127056,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a5\",\"b5\"],\"final_states\":[\"f5\"],\"start_state\":\"s5\",\"states\":[\"d5\",\"e15\",\"e25\",\"f5\",\"m5\",\"s5\"],\"subset\":[\"d5\",\"m5\"],\"transitions\":[[\"e15\",null,\"e25\"],[\"f5\",null,\"e15\"],[\"m5\",\"a5\",\"m5\"],[\"m5\",\"b5\",\"f5\"],[\"s5\",null,\"d5\"],[\"s5\",null,\"m5\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s5, m5, f5, e15, e25, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5], [f5, null, e15], [e15, null, e25]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m5, d5]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([d5, m5], [f5], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d5, m5], [f5], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0057","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2230ba111a6b5474b2a114cd657d2a25e7ea2d076c87bc233e2c4da3236ccdd","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127057,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a6\",\"b6\"],\"final_states\":[\"f6\"],\"start_state\":\"s6\",\"states\":[\"d6\",\"e16\",\"e26\",\"f6\",\"m6\",\"s6\"],\"subset\":[\"f6\",\"m6\"],\"transitions\":[[\"e16\",null,\"e26\"],[\"f6\",null,\"e16\"],[\"m6\",\"a6\",\"m6\"],[\"m6\",\"b6\",\"f6\"],[\"s6\",null,\"d6\"],[\"s6\",null,\"m6\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s6, m6, f6, e16, e26, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6], [f6, null, e16], [e16, null, e26]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f6, m6]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([f6, m6], [f6], [f6], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f6, m6], [f6], [f6], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0058","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72d2f7d743dd89caedbcf8a47b64d054c10b16d9794bd86bf30c76d0fe97c4db","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127058,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a7\",\"b7\"],\"final_states\":[\"f7\"],\"start_state\":\"s7\",\"states\":[\"d7\",\"e17\",\"e27\",\"f7\",\"m7\",\"s7\"],\"subset\":[\"d7\",\"m7\"],\"transitions\":[[\"e17\",null,\"e27\"],[\"f7\",null,\"e17\"],[\"m7\",\"a7\",\"m7\"],[\"m7\",\"b7\",\"f7\"],[\"s7\",null,\"d7\"],[\"s7\",null,\"m7\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s7, m7, f7, e17, e27, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7], [f7, null, e17], [e17, null, e27]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m7, d7]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([d7, m7], [f7], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d7, m7], [f7], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0059","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"878039694e17d45c7dd1c361ea80bba7f6b79dc01ad5d820ff4a67f16d735c50","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127059,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a8\",\"b8\"],\"final_states\":[\"f8\"],\"start_state\":\"s8\",\"states\":[\"d8\",\"e18\",\"e28\",\"f8\",\"m8\",\"s8\"],\"subset\":[\"f8\",\"m8\"],\"transitions\":[[\"e18\",null,\"e28\"],[\"f8\",null,\"e18\"],[\"m8\",\"a8\",\"m8\"],[\"m8\",\"b8\",\"f8\"],[\"s8\",null,\"d8\"],[\"s8\",null,\"m8\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s8, m8, f8, e18, e28, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8], [f8, null, e18], [e18, null, e28]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f8, m8]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([f8, m8], [f8], [f8], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f8, m8], [f8], [f8], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0060","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"034379c68f3ad6600f970cd14dadbf5e09d2e28da7e79d33d9a85a0a21c32782","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127060,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a9\",\"b9\"],\"final_states\":[\"f9\"],\"start_state\":\"s9\",\"states\":[\"d9\",\"e19\",\"e29\",\"f9\",\"m9\",\"s9\"],\"subset\":[\"d9\",\"m9\"],\"transitions\":[[\"e19\",null,\"e29\"],[\"f9\",null,\"e19\"],[\"m9\",\"a9\",\"m9\"],[\"m9\",\"b9\",\"f9\"],[\"s9\",null,\"d9\"],[\"s9\",null,\"m9\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s9, m9, f9, e19, e29, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9], [f9, null, e19], [e19, null, e29]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m9, d9]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([d9, m9], [f9], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d9, m9], [f9], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0061","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23f566cf28a3f4af4ce490926987eaef8259c204bf768db7a5b7d876fb5800fd","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127061,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a10\",\"b10\"],\"final_states\":[\"f10\"],\"start_state\":\"s10\",\"states\":[\"d10\",\"e110\",\"e210\",\"f10\",\"m10\",\"s10\"],\"subset\":[\"f10\",\"m10\"],\"transitions\":[[\"e110\",null,\"e210\"],[\"f10\",null,\"e110\"],[\"m10\",\"a10\",\"m10\"],[\"m10\",\"b10\",\"f10\"],[\"s10\",null,\"d10\"],[\"s10\",null,\"m10\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s10, m10, f10, e110, e210, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10], [f10, null, e110], [e110, null, e210]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f10, m10]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([f10, m10], [f10], [f10], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f10, m10], [f10], [f10], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0062","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"341d06a2971c92405ff365758b57d5220ed1aae867a5eb5f69e0f285de4c8c6f","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127062,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a11\",\"b11\"],\"final_states\":[\"f11\"],\"start_state\":\"s11\",\"states\":[\"d11\",\"e111\",\"e211\",\"f11\",\"m11\",\"s11\"],\"subset\":[\"d11\",\"m11\"],\"transitions\":[[\"e111\",null,\"e211\"],[\"f11\",null,\"e111\"],[\"m11\",\"a11\",\"m11\"],[\"m11\",\"b11\",\"f11\"],[\"s11\",null,\"d11\"],[\"s11\",null,\"m11\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s11, m11, f11, e111, e211, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11], [f11, null, e111], [e111, null, e211]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m11, d11]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([d11, m11], [f11], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d11, m11], [f11], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0063","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be7c41418968fa86eb5f22e5de41299bf533fc0ce4404972a4088310ba6ce49c","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127063,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a12\",\"b12\"],\"final_states\":[\"f12\"],\"start_state\":\"s12\",\"states\":[\"d12\",\"e112\",\"e212\",\"f12\",\"m12\",\"s12\"],\"subset\":[\"f12\",\"m12\"],\"transitions\":[[\"e112\",null,\"e212\"],[\"f12\",null,\"e112\"],[\"m12\",\"a12\",\"m12\"],[\"m12\",\"b12\",\"f12\"],[\"s12\",null,\"d12\"],[\"s12\",null,\"m12\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s12, m12, f12, e112, e212, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12], [f12, null, e112], [e112, null, e212]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f12, m12]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([f12, m12], [f12], [f12], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f12, m12], [f12], [f12], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0064","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e5f79aaff5c46afe568e18fcdf977c85cd5a3116d8a9c66e512b1c7e918c550","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127064,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a13\",\"b13\"],\"final_states\":[\"f13\"],\"start_state\":\"s13\",\"states\":[\"d13\",\"e113\",\"e213\",\"f13\",\"m13\",\"s13\"],\"subset\":[\"d13\",\"m13\"],\"transitions\":[[\"e113\",null,\"e213\"],[\"f13\",null,\"e113\"],[\"m13\",\"a13\",\"m13\"],[\"m13\",\"b13\",\"f13\"],[\"s13\",null,\"d13\"],[\"s13\",null,\"m13\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s13, m13, f13, e113, e213, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13], [f13, null, e113], [e113, null, e213]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m13, d13]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([d13, m13], [f13], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d13, m13], [f13], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0065","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8d41732b0027fe61268d4c4fb764490af75d40ee0a72d2caf253e0dbd245e9a","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127065,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a14\",\"b14\"],\"final_states\":[\"f14\"],\"start_state\":\"s14\",\"states\":[\"d14\",\"e114\",\"e214\",\"f14\",\"m14\",\"s14\"],\"subset\":[\"f14\",\"m14\"],\"transitions\":[[\"e114\",null,\"e214\"],[\"f14\",null,\"e114\"],[\"m14\",\"a14\",\"m14\"],[\"m14\",\"b14\",\"f14\"],[\"s14\",null,\"d14\"],[\"s14\",null,\"m14\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s14, m14, f14, e114, e214, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14], [f14, null, e114], [e114, null, e214]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f14, m14]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([f14, m14], [f14], [f14], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f14, m14], [f14], [f14], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0066","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"276afb898662708d3a0ea7d30cc0fe26523eac2bdcdf5d2f8a11f1aa2311bb63","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127066,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a15\",\"b15\"],\"final_states\":[\"f15\"],\"start_state\":\"s15\",\"states\":[\"d15\",\"e115\",\"e215\",\"f15\",\"m15\",\"s15\"],\"subset\":[\"d15\",\"m15\"],\"transitions\":[[\"e115\",null,\"e215\"],[\"f15\",null,\"e115\"],[\"m15\",\"a15\",\"m15\"],[\"m15\",\"b15\",\"f15\"],[\"s15\",null,\"d15\"],[\"s15\",null,\"m15\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s15, m15, f15, e115, e215, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15], [f15, null, e115], [e115, null, e215]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m15, d15]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([d15, m15], [f15], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d15, m15], [f15], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0067","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a93fdd569190ef2da3b00f2f6ff21388e1d8277ebc71b2339ee2ec6ba2aa5784","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127067,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a16\",\"b16\"],\"final_states\":[\"f16\"],\"start_state\":\"s16\",\"states\":[\"d16\",\"e116\",\"e216\",\"f16\",\"m16\",\"s16\"],\"subset\":[\"f16\",\"m16\"],\"transitions\":[[\"e116\",null,\"e216\"],[\"f16\",null,\"e116\"],[\"m16\",\"a16\",\"m16\"],[\"m16\",\"b16\",\"f16\"],[\"s16\",null,\"d16\"],[\"s16\",null,\"m16\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s16, m16, f16, e116, e216, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16], [f16, null, e116], [e116, null, e216]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f16, m16]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([f16, m16], [f16], [f16], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f16, m16], [f16], [f16], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0068","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"057979f72f9e879e5bcff49573e9e39ed8d658ad0b6aef13f32f906d5ad6b78a","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127068,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a17\",\"b17\"],\"final_states\":[\"f17\"],\"start_state\":\"s17\",\"states\":[\"d17\",\"e117\",\"e217\",\"f17\",\"m17\",\"s17\"],\"subset\":[\"d17\",\"m17\"],\"transitions\":[[\"e117\",null,\"e217\"],[\"f17\",null,\"e117\"],[\"m17\",\"a17\",\"m17\"],[\"m17\",\"b17\",\"f17\"],[\"s17\",null,\"d17\"],[\"s17\",null,\"m17\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s17, m17, f17, e117, e217, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17], [f17, null, e117], [e117, null, e217]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m17, d17]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([d17, m17], [f17], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d17, m17], [f17], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0069","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bebc938f996152d085b16df40fdb5efb3a8c20286c64f5863b9fcae345ef6d92","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127069,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a18\",\"b18\"],\"final_states\":[\"f18\"],\"start_state\":\"s18\",\"states\":[\"d18\",\"e118\",\"e218\",\"f18\",\"m18\",\"s18\"],\"subset\":[\"f18\",\"m18\"],\"transitions\":[[\"e118\",null,\"e218\"],[\"f18\",null,\"e118\"],[\"m18\",\"a18\",\"m18\"],[\"m18\",\"b18\",\"f18\"],[\"s18\",null,\"d18\"],[\"s18\",null,\"m18\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s18, m18, f18, e118, e218, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18], [f18, null, e118], [e118, null, e218]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f18, m18]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([f18, m18], [f18], [f18], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f18, m18], [f18], [f18], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0070","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96387f6134dd26f7fe96ff5fda727f496e8c9b999c55b690bf318a6718316599","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127070,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a19\",\"b19\"],\"final_states\":[\"f19\"],\"start_state\":\"s19\",\"states\":[\"d19\",\"e119\",\"e219\",\"f19\",\"m19\",\"s19\"],\"subset\":[\"d19\",\"m19\"],\"transitions\":[[\"e119\",null,\"e219\"],[\"f19\",null,\"e119\"],[\"m19\",\"a19\",\"m19\"],[\"m19\",\"b19\",\"f19\"],[\"s19\",null,\"d19\"],[\"s19\",null,\"m19\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s19, m19, f19, e119, e219, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19], [f19, null, e119], [e119, null, e219]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m19, d19]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([d19, m19], [f19], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d19, m19], [f19], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0071","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"669cba570953f6729acf240b3de1fd7f3adfacd0dfc5937912848f571e0259e0","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127071,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a20\",\"b20\"],\"final_states\":[\"f20\"],\"start_state\":\"s20\",\"states\":[\"d20\",\"e120\",\"e220\",\"f20\",\"m20\",\"s20\"],\"subset\":[\"f20\",\"m20\"],\"transitions\":[[\"e120\",null,\"e220\"],[\"f20\",null,\"e120\"],[\"m20\",\"a20\",\"m20\"],[\"m20\",\"b20\",\"f20\"],[\"s20\",null,\"d20\"],[\"s20\",null,\"m20\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s20, m20, f20, e120, e220, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20], [f20, null, e120], [e120, null, e220]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f20, m20]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([f20, m20], [f20], [f20], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f20, m20], [f20], [f20], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0072","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83c311154c131cbb842d4c62056387499a1a6ab24fff2561321dbe1c2d0fc0d4","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127072,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a21\",\"b21\"],\"final_states\":[\"f21\"],\"start_state\":\"s21\",\"states\":[\"d21\",\"e121\",\"e221\",\"f21\",\"m21\",\"s21\"],\"subset\":[\"d21\",\"m21\"],\"transitions\":[[\"e121\",null,\"e221\"],[\"f21\",null,\"e121\"],[\"m21\",\"a21\",\"m21\"],[\"m21\",\"b21\",\"f21\"],[\"s21\",null,\"d21\"],[\"s21\",null,\"m21\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s21, m21, f21, e121, e221, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m21, d21]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([d21, m21], [f21], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d21, m21], [f21], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0073","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6048ec845e6dd54ca91337cbedcd621458084b6f08c428e94be3e05df8321b71","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127073,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a22\",\"b22\"],\"final_states\":[\"f22\"],\"start_state\":\"s22\",\"states\":[\"d22\",\"e122\",\"e222\",\"f22\",\"m22\",\"s22\"],\"subset\":[\"f22\",\"m22\"],\"transitions\":[[\"e122\",null,\"e222\"],[\"f22\",null,\"e122\"],[\"m22\",\"a22\",\"m22\"],[\"m22\",\"b22\",\"f22\"],[\"s22\",null,\"d22\"],[\"s22\",null,\"m22\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s22, m22, f22, e122, e222, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22], [f22, null, e122], [e122, null, e222]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f22, m22]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([f22, m22], [f22], [f22], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f22, m22], [f22], [f22], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0074","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03eb32ad2464f57f22e622f004625b77d5e859b7de1cce5735286ae0cef24775","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127074,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a23\",\"b23\"],\"final_states\":[\"f23\"],\"start_state\":\"s23\",\"states\":[\"d23\",\"e123\",\"e223\",\"f23\",\"m23\",\"s23\"],\"subset\":[\"d23\",\"m23\"],\"transitions\":[[\"e123\",null,\"e223\"],[\"f23\",null,\"e123\"],[\"m23\",\"a23\",\"m23\"],[\"m23\",\"b23\",\"f23\"],[\"s23\",null,\"d23\"],[\"s23\",null,\"m23\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s23, m23, f23, e123, e223, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23], [f23, null, e123], [e123, null, e223]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[m23, d23]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([d23, m23], [f23], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([d23, m23], [f23], [], No, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0075","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5331e759655b58ddc167f6aa72153ed34aaf88b95151fed2a73aaabcff10a10","family_id":"qf-mod43-subsetdfa-accepting-subset","seed":202611127075,"semantic_template":"qf-mod43-subsetdfa-accepting-subset/{\"alphabet\":[\"a24\",\"b24\"],\"final_states\":[\"f24\"],\"start_state\":\"s24\",\"states\":[\"d24\",\"e124\",\"e224\",\"f24\",\"m24\",\"s24\"],\"subset\":[\"f24\",\"m24\"],\"transitions\":[[\"e124\",null,\"e224\"],[\"f24\",null,\"e124\"],[\"m24\",\"a24\",\"m24\"],[\"m24\",\"b24\",\"f24\"],[\"s24\",null,\"d24\"],[\"s24\",null,\"m24\"]]}"},"fields":{"title":"A Subset Accepts Exactly When It Intersects the NFA Final Set","stem":"Use complete epsilon-NFA states=[s24, m24, f24, e124, e224, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24], [f24, null, e124], [e124, null, e224]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Given subset S=[f24, m24]. Give (subset, NFA_final_states, intersection, accepting_subset, certificate).","answer":{"kind":"expression","value":"([f24, m24], [f24], [f24], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)","explanation":"Intersect the exact supplied subset with final states and test nonemptiness. Empty subset is a legal rejecting dead state. Do not insert an additional unrequested epsilon closure into this acceptance rule. The exact result, retaining the requested order, is ([f24, m24], [f24], [f24], Yes, DFA_subset_accepts_exactly_when_it_intersects_the_NFA_final_set)."},"skills":["A Subset Accepts Exactly When It Intersects the NFA Final Set"],"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-20261112-r07-subset-construction-nfa-to-dfa-0076","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99f2e2c42c55f03b031ed09fcf097aa200dd89c9d1a65bcd79f077afb9ab063a","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127076,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a0\",\"b0\"],\"final_states\":[\"f0\"],\"input\":[\"b0\"],\"start_state\":\"s0\",\"states\":[\"d0\",\"e10\",\"e20\",\"f0\",\"m0\",\"s0\"],\"transitions\":[[\"e10\",null,\"e20\"],[\"f0\",null,\"e10\"],[\"m0\",\"a0\",\"m0\"],[\"m0\",\"b0\",\"f0\"],[\"s0\",null,\"d0\"],[\"s0\",null,\"m0\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s0, m0, f0, e10, e20, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0], [f0, null, e10], [e10, null, e20]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[b0]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([b0], [[d0, m0, s0], [e10, e20, f0]], [[d0, m0, s0], [e10, e20, f0]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([b0], [[d0, m0, s0], [e10, e20, f0]], [[d0, m0, s0], [e10, e20, f0]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0077","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec0ce6b3d08e60b082a8a82c3d37c624674b10fef4294e9c61229520f9099f7e","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127077,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a1\",\"b1\"],\"final_states\":[\"f1\"],\"input\":[\"a1\",\"b1\"],\"start_state\":\"s1\",\"states\":[\"d1\",\"e11\",\"e21\",\"f1\",\"m1\",\"s1\"],\"transitions\":[[\"e11\",null,\"e21\"],[\"f1\",null,\"e11\"],[\"m1\",\"a1\",\"m1\"],[\"m1\",\"b1\",\"f1\"],[\"s1\",null,\"d1\"],[\"s1\",null,\"m1\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s1, m1, f1, e11, e21, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1], [f1, null, e11], [e11, null, e21]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a1, b1]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a1, b1], [[d1, m1, s1], [m1], [e11, e21, f1]], [[d1, m1, s1], [m1], [e11, e21, f1]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a1, b1], [[d1, m1, s1], [m1], [e11, e21, f1]], [[d1, m1, s1], [m1], [e11, e21, f1]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0078","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0453781916f1700601cc04ea9f318b3097a9b570b494bc5f611a3a86157dcce","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127078,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a2\",\"b2\"],\"final_states\":[\"f2\"],\"input\":[\"a2\",\"a2\",\"b2\"],\"start_state\":\"s2\",\"states\":[\"d2\",\"e12\",\"e22\",\"f2\",\"m2\",\"s2\"],\"transitions\":[[\"e12\",null,\"e22\"],[\"f2\",null,\"e12\"],[\"m2\",\"a2\",\"m2\"],[\"m2\",\"b2\",\"f2\"],[\"s2\",null,\"d2\"],[\"s2\",null,\"m2\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s2, m2, f2, e12, e22, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2], [f2, null, e12], [e12, null, e22]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a2, a2, b2]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a2, a2, b2], [[d2, m2, s2], [m2], [m2], [e12, e22, f2]], [[d2, m2, s2], [m2], [m2], [e12, e22, f2]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a2, a2, b2], [[d2, m2, s2], [m2], [m2], [e12, e22, f2]], [[d2, m2, s2], [m2], [m2], [e12, e22, f2]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0079","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d98597603d857e395fdaa1c5add214f8f43a68ffbe16518714d65dca917a0905","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127079,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a3\",\"b3\"],\"final_states\":[\"f3\"],\"input\":[\"a3\",\"a3\",\"a3\",\"b3\"],\"start_state\":\"s3\",\"states\":[\"d3\",\"e13\",\"e23\",\"f3\",\"m3\",\"s3\"],\"transitions\":[[\"e13\",null,\"e23\"],[\"f3\",null,\"e13\"],[\"m3\",\"a3\",\"m3\"],[\"m3\",\"b3\",\"f3\"],[\"s3\",null,\"d3\"],[\"s3\",null,\"m3\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s3, m3, f3, e13, e23, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3], [f3, null, e13], [e13, null, e23]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a3, a3, a3, b3]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a3, a3, a3, b3], [[d3, m3, s3], [m3], [m3], [m3], [e13, e23, f3]], [[d3, m3, s3], [m3], [m3], [m3], [e13, e23, f3]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a3, a3, a3, b3], [[d3, m3, s3], [m3], [m3], [m3], [e13, e23, f3]], [[d3, m3, s3], [m3], [m3], [m3], [e13, e23, f3]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0080","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f52fd87d48644a79aa103cd1d0b898934298829d656aa637700fe3d8269f8c35","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127080,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a4\",\"b4\"],\"final_states\":[\"f4\"],\"input\":[\"b4\"],\"start_state\":\"s4\",\"states\":[\"d4\",\"e14\",\"e24\",\"f4\",\"m4\",\"s4\"],\"transitions\":[[\"e14\",null,\"e24\"],[\"f4\",null,\"e14\"],[\"m4\",\"a4\",\"m4\"],[\"m4\",\"b4\",\"f4\"],[\"s4\",null,\"d4\"],[\"s4\",null,\"m4\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s4, m4, f4, e14, e24, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4], [f4, null, e14], [e14, null, e24]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[b4]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([b4], [[d4, m4, s4], [e14, e24, f4]], [[d4, m4, s4], [e14, e24, f4]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([b4], [[d4, m4, s4], [e14, e24, f4]], [[d4, m4, s4], [e14, e24, f4]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0081","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a714d629539689cc93799124c0b1573b49557e04e96d659b8af5f98618044de","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127081,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a5\",\"b5\"],\"final_states\":[\"f5\"],\"input\":[\"a5\",\"b5\"],\"start_state\":\"s5\",\"states\":[\"d5\",\"e15\",\"e25\",\"f5\",\"m5\",\"s5\"],\"transitions\":[[\"e15\",null,\"e25\"],[\"f5\",null,\"e15\"],[\"m5\",\"a5\",\"m5\"],[\"m5\",\"b5\",\"f5\"],[\"s5\",null,\"d5\"],[\"s5\",null,\"m5\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s5, m5, f5, e15, e25, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5], [f5, null, e15], [e15, null, e25]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a5, b5]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a5, b5], [[d5, m5, s5], [m5], [e15, e25, f5]], [[d5, m5, s5], [m5], [e15, e25, f5]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a5, b5], [[d5, m5, s5], [m5], [e15, e25, f5]], [[d5, m5, s5], [m5], [e15, e25, f5]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0082","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ce4c67acf3df8d5d944a9715f9fcbc48529a4a37793a0904f97926e8093884f","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127082,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a6\",\"b6\"],\"final_states\":[\"f6\"],\"input\":[\"a6\",\"a6\",\"b6\"],\"start_state\":\"s6\",\"states\":[\"d6\",\"e16\",\"e26\",\"f6\",\"m6\",\"s6\"],\"transitions\":[[\"e16\",null,\"e26\"],[\"f6\",null,\"e16\"],[\"m6\",\"a6\",\"m6\"],[\"m6\",\"b6\",\"f6\"],[\"s6\",null,\"d6\"],[\"s6\",null,\"m6\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s6, m6, f6, e16, e26, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6], [f6, null, e16], [e16, null, e26]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a6, a6, b6]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a6, a6, b6], [[d6, m6, s6], [m6], [m6], [e16, e26, f6]], [[d6, m6, s6], [m6], [m6], [e16, e26, f6]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a6, a6, b6], [[d6, m6, s6], [m6], [m6], [e16, e26, f6]], [[d6, m6, s6], [m6], [m6], [e16, e26, f6]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0083","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5561aa3366b85d727ff5ff3d262d849c6c411417a76b0e0ef6aef8f157b18fa6","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127083,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a7\",\"b7\"],\"final_states\":[\"f7\"],\"input\":[\"a7\",\"a7\",\"a7\",\"b7\"],\"start_state\":\"s7\",\"states\":[\"d7\",\"e17\",\"e27\",\"f7\",\"m7\",\"s7\"],\"transitions\":[[\"e17\",null,\"e27\"],[\"f7\",null,\"e17\"],[\"m7\",\"a7\",\"m7\"],[\"m7\",\"b7\",\"f7\"],[\"s7\",null,\"d7\"],[\"s7\",null,\"m7\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s7, m7, f7, e17, e27, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7], [f7, null, e17], [e17, null, e27]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a7, a7, a7, b7]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a7, a7, a7, b7], [[d7, m7, s7], [m7], [m7], [m7], [e17, e27, f7]], [[d7, m7, s7], [m7], [m7], [m7], [e17, e27, f7]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a7, a7, a7, b7], [[d7, m7, s7], [m7], [m7], [m7], [e17, e27, f7]], [[d7, m7, s7], [m7], [m7], [m7], [e17, e27, f7]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0084","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ab14b11627d6281e529e0b1cbfb5bbfb9eba075be3bc9e45b29ac714436654a","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127084,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a8\",\"b8\"],\"final_states\":[\"f8\"],\"input\":[\"b8\"],\"start_state\":\"s8\",\"states\":[\"d8\",\"e18\",\"e28\",\"f8\",\"m8\",\"s8\"],\"transitions\":[[\"e18\",null,\"e28\"],[\"f8\",null,\"e18\"],[\"m8\",\"a8\",\"m8\"],[\"m8\",\"b8\",\"f8\"],[\"s8\",null,\"d8\"],[\"s8\",null,\"m8\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s8, m8, f8, e18, e28, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8], [f8, null, e18], [e18, null, e28]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[b8]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([b8], [[d8, m8, s8], [e18, e28, f8]], [[d8, m8, s8], [e18, e28, f8]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([b8], [[d8, m8, s8], [e18, e28, f8]], [[d8, m8, s8], [e18, e28, f8]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0085","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26a87ec662415eaa2428c09dce3590474f42a48aed9fccd157ddc389bccef42b","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127085,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a9\",\"b9\"],\"final_states\":[\"f9\"],\"input\":[\"a9\",\"b9\"],\"start_state\":\"s9\",\"states\":[\"d9\",\"e19\",\"e29\",\"f9\",\"m9\",\"s9\"],\"transitions\":[[\"e19\",null,\"e29\"],[\"f9\",null,\"e19\"],[\"m9\",\"a9\",\"m9\"],[\"m9\",\"b9\",\"f9\"],[\"s9\",null,\"d9\"],[\"s9\",null,\"m9\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s9, m9, f9, e19, e29, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9], [f9, null, e19], [e19, null, e29]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a9, b9]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a9, b9], [[d9, m9, s9], [m9], [e19, e29, f9]], [[d9, m9, s9], [m9], [e19, e29, f9]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a9, b9], [[d9, m9, s9], [m9], [e19, e29, f9]], [[d9, m9, s9], [m9], [e19, e29, f9]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0086","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43d285b3789fb4196d78f0a7e3745da69de76d271b70bf083c496e7e89f5d69b","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127086,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a10\",\"b10\"],\"final_states\":[\"f10\"],\"input\":[\"a10\",\"a10\",\"b10\"],\"start_state\":\"s10\",\"states\":[\"d10\",\"e110\",\"e210\",\"f10\",\"m10\",\"s10\"],\"transitions\":[[\"e110\",null,\"e210\"],[\"f10\",null,\"e110\"],[\"m10\",\"a10\",\"m10\"],[\"m10\",\"b10\",\"f10\"],[\"s10\",null,\"d10\"],[\"s10\",null,\"m10\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s10, m10, f10, e110, e210, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10], [f10, null, e110], [e110, null, e210]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a10, a10, b10]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a10, a10, b10], [[d10, m10, s10], [m10], [m10], [e110, e210, f10]], [[d10, m10, s10], [m10], [m10], [e110, e210, f10]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a10, a10, b10], [[d10, m10, s10], [m10], [m10], [e110, e210, f10]], [[d10, m10, s10], [m10], [m10], [e110, e210, f10]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0087","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9108c6370d43a9f80aee0a5454a07cc8ca41f1cd04ee86ef7521cd727ab137c0","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127087,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a11\",\"b11\"],\"final_states\":[\"f11\"],\"input\":[\"a11\",\"a11\",\"a11\",\"b11\"],\"start_state\":\"s11\",\"states\":[\"d11\",\"e111\",\"e211\",\"f11\",\"m11\",\"s11\"],\"transitions\":[[\"e111\",null,\"e211\"],[\"f11\",null,\"e111\"],[\"m11\",\"a11\",\"m11\"],[\"m11\",\"b11\",\"f11\"],[\"s11\",null,\"d11\"],[\"s11\",null,\"m11\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s11, m11, f11, e111, e211, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11], [f11, null, e111], [e111, null, e211]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a11, a11, a11, b11]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a11, a11, a11, b11], [[d11, m11, s11], [m11], [m11], [m11], [e111, e211, f11]], [[d11, m11, s11], [m11], [m11], [m11], [e111, e211, f11]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a11, a11, a11, b11], [[d11, m11, s11], [m11], [m11], [m11], [e111, e211, f11]], [[d11, m11, s11], [m11], [m11], [m11], [e111, e211, f11]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0088","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"078097f36a1e422d58816251facd7c21e3e9f7c60010dbd126309e79724966bd","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127088,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a12\",\"b12\"],\"final_states\":[\"f12\"],\"input\":[\"b12\"],\"start_state\":\"s12\",\"states\":[\"d12\",\"e112\",\"e212\",\"f12\",\"m12\",\"s12\"],\"transitions\":[[\"e112\",null,\"e212\"],[\"f12\",null,\"e112\"],[\"m12\",\"a12\",\"m12\"],[\"m12\",\"b12\",\"f12\"],[\"s12\",null,\"d12\"],[\"s12\",null,\"m12\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s12, m12, f12, e112, e212, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12], [f12, null, e112], [e112, null, e212]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[b12]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([b12], [[d12, m12, s12], [e112, e212, f12]], [[d12, m12, s12], [e112, e212, f12]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([b12], [[d12, m12, s12], [e112, e212, f12]], [[d12, m12, s12], [e112, e212, f12]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0089","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01a9f5f74d5f538723b2dea4029246f300de58fd1593d5d405c3a6e2766624e0","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127089,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a13\",\"b13\"],\"final_states\":[\"f13\"],\"input\":[\"a13\",\"b13\"],\"start_state\":\"s13\",\"states\":[\"d13\",\"e113\",\"e213\",\"f13\",\"m13\",\"s13\"],\"transitions\":[[\"e113\",null,\"e213\"],[\"f13\",null,\"e113\"],[\"m13\",\"a13\",\"m13\"],[\"m13\",\"b13\",\"f13\"],[\"s13\",null,\"d13\"],[\"s13\",null,\"m13\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s13, m13, f13, e113, e213, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13], [f13, null, e113], [e113, null, e213]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a13, b13]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a13, b13], [[d13, m13, s13], [m13], [e113, e213, f13]], [[d13, m13, s13], [m13], [e113, e213, f13]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a13, b13], [[d13, m13, s13], [m13], [e113, e213, f13]], [[d13, m13, s13], [m13], [e113, e213, f13]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0090","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b62789979fbe4927cbbf5577011a233837e8c1a7f178c6bc40b1e9264323542c","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127090,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a14\",\"b14\"],\"final_states\":[\"f14\"],\"input\":[\"a14\",\"a14\",\"b14\"],\"start_state\":\"s14\",\"states\":[\"d14\",\"e114\",\"e214\",\"f14\",\"m14\",\"s14\"],\"transitions\":[[\"e114\",null,\"e214\"],[\"f14\",null,\"e114\"],[\"m14\",\"a14\",\"m14\"],[\"m14\",\"b14\",\"f14\"],[\"s14\",null,\"d14\"],[\"s14\",null,\"m14\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s14, m14, f14, e114, e214, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14], [f14, null, e114], [e114, null, e214]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a14, a14, b14]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a14, a14, b14], [[d14, m14, s14], [m14], [m14], [e114, e214, f14]], [[d14, m14, s14], [m14], [m14], [e114, e214, f14]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a14, a14, b14], [[d14, m14, s14], [m14], [m14], [e114, e214, f14]], [[d14, m14, s14], [m14], [m14], [e114, e214, f14]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0091","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c17fe66952089e81c49b8925bdaf154af959adbb8fa09f21756d3d44054c7b6","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127091,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a15\",\"b15\"],\"final_states\":[\"f15\"],\"input\":[\"a15\",\"a15\",\"a15\",\"b15\"],\"start_state\":\"s15\",\"states\":[\"d15\",\"e115\",\"e215\",\"f15\",\"m15\",\"s15\"],\"transitions\":[[\"e115\",null,\"e215\"],[\"f15\",null,\"e115\"],[\"m15\",\"a15\",\"m15\"],[\"m15\",\"b15\",\"f15\"],[\"s15\",null,\"d15\"],[\"s15\",null,\"m15\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s15, m15, f15, e115, e215, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15], [f15, null, e115], [e115, null, e215]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a15, a15, a15, b15]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a15, a15, a15, b15], [[d15, m15, s15], [m15], [m15], [m15], [e115, e215, f15]], [[d15, m15, s15], [m15], [m15], [m15], [e115, e215, f15]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a15, a15, a15, b15], [[d15, m15, s15], [m15], [m15], [m15], [e115, e215, f15]], [[d15, m15, s15], [m15], [m15], [m15], [e115, e215, f15]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0092","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fa7a22bb6b1542fc2ddb01bef8680597d0b450666bad9fd8a335434a9571a04","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127092,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a16\",\"b16\"],\"final_states\":[\"f16\"],\"input\":[\"b16\"],\"start_state\":\"s16\",\"states\":[\"d16\",\"e116\",\"e216\",\"f16\",\"m16\",\"s16\"],\"transitions\":[[\"e116\",null,\"e216\"],[\"f16\",null,\"e116\"],[\"m16\",\"a16\",\"m16\"],[\"m16\",\"b16\",\"f16\"],[\"s16\",null,\"d16\"],[\"s16\",null,\"m16\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s16, m16, f16, e116, e216, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16], [f16, null, e116], [e116, null, e216]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[b16]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([b16], [[d16, m16, s16], [e116, e216, f16]], [[d16, m16, s16], [e116, e216, f16]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([b16], [[d16, m16, s16], [e116, e216, f16]], [[d16, m16, s16], [e116, e216, f16]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0093","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d21ffd4ef893a4ded5474c2ee1afd6bbb3e24c646069750a0103d7de61745bc9","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127093,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a17\",\"b17\"],\"final_states\":[\"f17\"],\"input\":[\"a17\",\"b17\"],\"start_state\":\"s17\",\"states\":[\"d17\",\"e117\",\"e217\",\"f17\",\"m17\",\"s17\"],\"transitions\":[[\"e117\",null,\"e217\"],[\"f17\",null,\"e117\"],[\"m17\",\"a17\",\"m17\"],[\"m17\",\"b17\",\"f17\"],[\"s17\",null,\"d17\"],[\"s17\",null,\"m17\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s17, m17, f17, e117, e217, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17], [f17, null, e117], [e117, null, e217]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a17, b17]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a17, b17], [[d17, m17, s17], [m17], [e117, e217, f17]], [[d17, m17, s17], [m17], [e117, e217, f17]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a17, b17], [[d17, m17, s17], [m17], [e117, e217, f17]], [[d17, m17, s17], [m17], [e117, e217, f17]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0094","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ed6a23cecf514a097ba2d2732e13e21ffe46ce56a00c25631c11d7ba72ff175","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127094,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a18\",\"b18\"],\"final_states\":[\"f18\"],\"input\":[\"a18\",\"a18\",\"b18\"],\"start_state\":\"s18\",\"states\":[\"d18\",\"e118\",\"e218\",\"f18\",\"m18\",\"s18\"],\"transitions\":[[\"e118\",null,\"e218\"],[\"f18\",null,\"e118\"],[\"m18\",\"a18\",\"m18\"],[\"m18\",\"b18\",\"f18\"],[\"s18\",null,\"d18\"],[\"s18\",null,\"m18\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s18, m18, f18, e118, e218, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18], [f18, null, e118], [e118, null, e218]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a18, a18, b18]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a18, a18, b18], [[d18, m18, s18], [m18], [m18], [e118, e218, f18]], [[d18, m18, s18], [m18], [m18], [e118, e218, f18]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a18, a18, b18], [[d18, m18, s18], [m18], [m18], [e118, e218, f18]], [[d18, m18, s18], [m18], [m18], [e118, e218, f18]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0095","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba9e5299152bf83c262f145057a618b4248be13206cb6ae205c474f15f86125a","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127095,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a19\",\"b19\"],\"final_states\":[\"f19\"],\"input\":[\"a19\",\"a19\",\"a19\",\"b19\"],\"start_state\":\"s19\",\"states\":[\"d19\",\"e119\",\"e219\",\"f19\",\"m19\",\"s19\"],\"transitions\":[[\"e119\",null,\"e219\"],[\"f19\",null,\"e119\"],[\"m19\",\"a19\",\"m19\"],[\"m19\",\"b19\",\"f19\"],[\"s19\",null,\"d19\"],[\"s19\",null,\"m19\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s19, m19, f19, e119, e219, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19], [f19, null, e119], [e119, null, e219]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a19, a19, a19, b19]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a19, a19, a19, b19], [[d19, m19, s19], [m19], [m19], [m19], [e119, e219, f19]], [[d19, m19, s19], [m19], [m19], [m19], [e119, e219, f19]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a19, a19, a19, b19], [[d19, m19, s19], [m19], [m19], [m19], [e119, e219, f19]], [[d19, m19, s19], [m19], [m19], [m19], [e119, e219, f19]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0096","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8dd003ebe2afc215002c628ec7db3e8def605cda105927eaedfddf7efa14d0f4","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127096,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a20\",\"b20\"],\"final_states\":[\"f20\"],\"input\":[\"b20\"],\"start_state\":\"s20\",\"states\":[\"d20\",\"e120\",\"e220\",\"f20\",\"m20\",\"s20\"],\"transitions\":[[\"e120\",null,\"e220\"],[\"f20\",null,\"e120\"],[\"m20\",\"a20\",\"m20\"],[\"m20\",\"b20\",\"f20\"],[\"s20\",null,\"d20\"],[\"s20\",null,\"m20\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s20, m20, f20, e120, e220, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20], [f20, null, e120], [e120, null, e220]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[b20]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([b20], [[d20, m20, s20], [e120, e220, f20]], [[d20, m20, s20], [e120, e220, f20]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([b20], [[d20, m20, s20], [e120, e220, f20]], [[d20, m20, s20], [e120, e220, f20]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0097","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2dce1f6ae419945d3fb54e870c36e059efd2034ed2f45132184df6a4ca9e0f4c","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127097,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a21\",\"b21\"],\"final_states\":[\"f21\"],\"input\":[\"a21\",\"b21\"],\"start_state\":\"s21\",\"states\":[\"d21\",\"e121\",\"e221\",\"f21\",\"m21\",\"s21\"],\"transitions\":[[\"e121\",null,\"e221\"],[\"f21\",null,\"e121\"],[\"m21\",\"a21\",\"m21\"],[\"m21\",\"b21\",\"f21\"],[\"s21\",null,\"d21\"],[\"s21\",null,\"m21\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s21, m21, f21, e121, e221, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21], [f21, null, e121], [e121, null, e221]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a21, b21]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a21, b21], [[d21, m21, s21], [m21], [e121, e221, f21]], [[d21, m21, s21], [m21], [e121, e221, f21]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a21, b21], [[d21, m21, s21], [m21], [e121, e221, f21]], [[d21, m21, s21], [m21], [e121, e221, f21]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0098","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0b973021534b4bfd240df8d63da5c5f72eea03b5b37ad147dc3e8b352831426","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127098,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a22\",\"b22\"],\"final_states\":[\"f22\"],\"input\":[\"a22\",\"a22\",\"b22\"],\"start_state\":\"s22\",\"states\":[\"d22\",\"e122\",\"e222\",\"f22\",\"m22\",\"s22\"],\"transitions\":[[\"e122\",null,\"e222\"],[\"f22\",null,\"e122\"],[\"m22\",\"a22\",\"m22\"],[\"m22\",\"b22\",\"f22\"],[\"s22\",null,\"d22\"],[\"s22\",null,\"m22\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s22, m22, f22, e122, e222, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22], [f22, null, e122], [e122, null, e222]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a22, a22, b22]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a22, a22, b22], [[d22, m22, s22], [m22], [m22], [e122, e222, f22]], [[d22, m22, s22], [m22], [m22], [e122, e222, f22]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a22, a22, b22], [[d22, m22, s22], [m22], [m22], [e122, e222, f22]], [[d22, m22, s22], [m22], [m22], [e122, e222, f22]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0099","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"537b93cdc0fbffbb8a97ba0c0ee8120ef8877b43092ac9e9cb614ac42759d1e8","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127099,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a23\",\"b23\"],\"final_states\":[\"f23\"],\"input\":[\"a23\",\"a23\",\"a23\",\"b23\"],\"start_state\":\"s23\",\"states\":[\"d23\",\"e123\",\"e223\",\"f23\",\"m23\",\"s23\"],\"transitions\":[[\"e123\",null,\"e223\"],[\"f23\",null,\"e123\"],[\"m23\",\"a23\",\"m23\"],[\"m23\",\"b23\",\"f23\"],[\"s23\",null,\"d23\"],[\"s23\",null,\"m23\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s23, m23, f23, e123, e223, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23], [f23, null, e123], [e123, null, e223]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[a23, a23, a23, b23]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([a23, a23, a23, b23], [[d23, m23, s23], [m23], [m23], [m23], [e123, e223, f23]], [[d23, m23, s23], [m23], [m23], [m23], [e123, e223, f23]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([a23, a23, a23, b23], [[d23, m23, s23], [m23], [m23], [m23], [e123, e223, f23]], [[d23, m23, s23], [m23], [m23], [m23], [e123, e223, f23]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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-20261112-r07-subset-construction-nfa-to-dfa-0100","batch_id":"batch-20261014-advance-20261112-r07-subset-construction-nfa-to-dfa-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef5f8e4ac6f1e62861bdd918280ce0a46d1db058e6fccbf6319e1761ff4d8a09","family_id":"qf-mod43-subsetdfa-full-word-trace-invariant","seed":202611127100,"semantic_template":"qf-mod43-subsetdfa-full-word-trace-invariant/{\"alphabet\":[\"a24\",\"b24\"],\"final_states\":[\"f24\"],\"input\":[\"b24\"],\"start_state\":\"s24\",\"states\":[\"d24\",\"e124\",\"e224\",\"f24\",\"m24\",\"s24\"],\"transitions\":[[\"e124\",null,\"e224\"],[\"f24\",null,\"e124\"],[\"m24\",\"a24\",\"m24\"],[\"m24\",\"b24\",\"f24\"],[\"s24\",null,\"d24\"],[\"s24\",null,\"m24\"]]}"},"fields":{"title":"Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States","stem":"Use complete epsilon-NFA states=[s24, m24, f24, e124, e224, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24], [f24, null, e124], [e124, null, e224]]. Each transition is [from,input symbol or null,to], with null consuming no input. States/labels and transitions must qualify, and subsets are unique carrier subsets sorted lexicographically. Empty alphabet/final sets and empty subset states are allowed. Supplied full input=[b24]. Give (input, DFA_prefix_subsets, direct_NFA_prefix_subsets, prefix_invariant, accepted, certificate, finite_tests_language_equivalence_claim, boundary).","answer":{"kind":"expression","value":"([b24], [[d24, m24, s24], [e124, e224, f24]], [[d24, m24, s24], [e124, e224, f24]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)","explanation":"Start epsilon-closed and apply symbol move then closure for every prefix. Compare DFA and direct NFA prefix sets and their final acceptance. The fixed No refuses a general language-equivalence proof from finitely many word tests; it does not negate the local invariant, acceptance or the subset-construction theorem. General equivalence requires the arbitrary-prefix induction invariant. The exact result, retaining the requested order, is ([b24], [[d24, m24, s24], [e124, e224, f24]], [[d24, m24, s24], [e124, e224, f24]], Yes, Yes, each_DFA_prefix_subset_equals_all_epsilon_closed_NFA_states_reachable_after_that_prefix, No, finite_word_tests_do_not_prove_language_equivalence_without_the_prefix_induction_invariant)."},"skills":["Each Recorded Prefix Subset Must Equal All Closed NFA Reachable States"],"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."}}