{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0001","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28c5932f657b87f932745d3f299df5fd718706a55d098f0e2105aea1d86854e8","family_id":"qf-mod42-nfa-qualification","seed":202611117001,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a0\",\"b0\"],\"final_states\":[\"f0\"],\"start_state\":\"s0\",\"states\":[\"d0\",\"f0\",\"m0\",\"s0\"],\"transitions\":[[\"m0\",\"a0\",\"m0\"],[\"m0\",\"b0\",\"f0\"],[\"s0\",null,\"d0\"],[\"s0\",null,\"m0\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s0, m0, f0, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s0, m0, f0, d0], [a0, b0], s0, [f0], [[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0]], [[s0, a0, []], [s0, b0, []], [f0, a0, []], [f0, b0, []], [d0, a0, []], [d0, b0, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s0, m0, f0, d0], [a0, b0], s0, [f0], [[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0]], [[s0, a0, []], [s0, b0, []], [f0, a0, []], [f0, b0, []], [d0, a0, []], [d0, b0, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0002","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de617c19673e6d1f64298d8f7109479d90068c40ce23df8d5498fbdc81804654","family_id":"qf-mod42-nfa-qualification","seed":202611117002,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a1\",\"b1\"],\"final_states\":[\"f1\"],\"start_state\":\"s1\",\"states\":[\"d1\",\"f1\",\"m1\",\"s1\"],\"transitions\":[[\"m1\",\"a1\",\"m1\"],[\"m1\",\"b1\",\"f1\"],[\"s1\",null,\"d1\"],[\"s1\",null,\"m1\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s1, m1, f1, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s1, m1, f1, d1], [a1, b1], s1, [f1], [[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1]], [[s1, a1, []], [s1, b1, []], [f1, a1, []], [f1, b1, []], [d1, a1, []], [d1, b1, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s1, m1, f1, d1], [a1, b1], s1, [f1], [[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1]], [[s1, a1, []], [s1, b1, []], [f1, a1, []], [f1, b1, []], [d1, a1, []], [d1, b1, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0003","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d77cdf05077df745419cf3da7c0fddecd26177f23919ec31b3ab1b76a19d647d","family_id":"qf-mod42-nfa-qualification","seed":202611117003,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a2\",\"b2\"],\"final_states\":[\"f2\"],\"start_state\":\"s2\",\"states\":[\"d2\",\"f2\",\"m2\",\"s2\"],\"transitions\":[[\"m2\",\"a2\",\"m2\"],[\"m2\",\"b2\",\"f2\"],[\"s2\",null,\"d2\"],[\"s2\",null,\"m2\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s2, m2, f2, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s2, m2, f2, d2], [a2, b2], s2, [f2], [[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2]], [[s2, a2, []], [s2, b2, []], [f2, a2, []], [f2, b2, []], [d2, a2, []], [d2, b2, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s2, m2, f2, d2], [a2, b2], s2, [f2], [[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2]], [[s2, a2, []], [s2, b2, []], [f2, a2, []], [f2, b2, []], [d2, a2, []], [d2, b2, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0004","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45b0f34c341053cf9262c9436cdd210fc1f2cfe0d9b647f4f9c2ffb66c90cf16","family_id":"qf-mod42-nfa-qualification","seed":202611117004,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a3\",\"b3\"],\"final_states\":[\"f3\"],\"start_state\":\"s3\",\"states\":[\"d3\",\"f3\",\"m3\",\"s3\"],\"transitions\":[[\"m3\",\"a3\",\"m3\"],[\"m3\",\"b3\",\"f3\"],[\"s3\",null,\"d3\"],[\"s3\",null,\"m3\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s3, m3, f3, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s3, m3, f3, d3], [a3, b3], s3, [f3], [[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]], [[s3, a3, []], [s3, b3, []], [f3, a3, []], [f3, b3, []], [d3, a3, []], [d3, b3, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s3, m3, f3, d3], [a3, b3], s3, [f3], [[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]], [[s3, a3, []], [s3, b3, []], [f3, a3, []], [f3, b3, []], [d3, a3, []], [d3, b3, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0005","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a5759a976f776a04e52d0dd5389eb0f010ec8c6bab694efc64551be6b1013fd","family_id":"qf-mod42-nfa-qualification","seed":202611117005,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a4\",\"b4\"],\"final_states\":[\"f4\"],\"start_state\":\"s4\",\"states\":[\"d4\",\"f4\",\"m4\",\"s4\"],\"transitions\":[[\"m4\",\"a4\",\"m4\"],[\"m4\",\"b4\",\"f4\"],[\"s4\",null,\"d4\"],[\"s4\",null,\"m4\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s4, m4, f4, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s4, m4, f4, d4], [a4, b4], s4, [f4], [[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]], [[s4, a4, []], [s4, b4, []], [f4, a4, []], [f4, b4, []], [d4, a4, []], [d4, b4, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s4, m4, f4, d4], [a4, b4], s4, [f4], [[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]], [[s4, a4, []], [s4, b4, []], [f4, a4, []], [f4, b4, []], [d4, a4, []], [d4, b4, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0006","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4d13187971b33fb5d8c598191e91dff436ccc306d62857a1bf8022c4cca9f3d","family_id":"qf-mod42-nfa-qualification","seed":202611117006,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a5\",\"b5\"],\"final_states\":[\"f5\"],\"start_state\":\"s5\",\"states\":[\"d5\",\"f5\",\"m5\",\"s5\"],\"transitions\":[[\"m5\",\"a5\",\"m5\"],[\"m5\",\"b5\",\"f5\"],[\"s5\",null,\"d5\"],[\"s5\",null,\"m5\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s5, m5, f5, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s5, m5, f5, d5], [a5, b5], s5, [f5], [[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]], [[s5, a5, []], [s5, b5, []], [f5, a5, []], [f5, b5, []], [d5, a5, []], [d5, b5, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s5, m5, f5, d5], [a5, b5], s5, [f5], [[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]], [[s5, a5, []], [s5, b5, []], [f5, a5, []], [f5, b5, []], [d5, a5, []], [d5, b5, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0007","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d863edab8b2c67bff49c781b1a46434b44c39a8cc3b13e6267c50eb0e026a723","family_id":"qf-mod42-nfa-qualification","seed":202611117007,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a6\",\"b6\"],\"final_states\":[\"f6\"],\"start_state\":\"s6\",\"states\":[\"d6\",\"f6\",\"m6\",\"s6\"],\"transitions\":[[\"m6\",\"a6\",\"m6\"],[\"m6\",\"b6\",\"f6\"],[\"s6\",null,\"d6\"],[\"s6\",null,\"m6\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s6, m6, f6, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s6, m6, f6, d6], [a6, b6], s6, [f6], [[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6]], [[s6, a6, []], [s6, b6, []], [f6, a6, []], [f6, b6, []], [d6, a6, []], [d6, b6, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s6, m6, f6, d6], [a6, b6], s6, [f6], [[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6]], [[s6, a6, []], [s6, b6, []], [f6, a6, []], [f6, b6, []], [d6, a6, []], [d6, b6, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0008","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c6d764febbf0323da236d320a7ee67fc976177ea9370eba3e2eada7b979fa46","family_id":"qf-mod42-nfa-qualification","seed":202611117008,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a7\",\"b7\"],\"final_states\":[\"f7\"],\"start_state\":\"s7\",\"states\":[\"d7\",\"f7\",\"m7\",\"s7\"],\"transitions\":[[\"m7\",\"a7\",\"m7\"],[\"m7\",\"b7\",\"f7\"],[\"s7\",null,\"d7\"],[\"s7\",null,\"m7\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s7, m7, f7, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s7, m7, f7, d7], [a7, b7], s7, [f7], [[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7]], [[s7, a7, []], [s7, b7, []], [f7, a7, []], [f7, b7, []], [d7, a7, []], [d7, b7, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s7, m7, f7, d7], [a7, b7], s7, [f7], [[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7]], [[s7, a7, []], [s7, b7, []], [f7, a7, []], [f7, b7, []], [d7, a7, []], [d7, b7, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0009","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a28556e25cf8dc9e5f95de64d69758bfd79cc80794ed7108b5c3e311b57ec19","family_id":"qf-mod42-nfa-qualification","seed":202611117009,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a8\",\"b8\"],\"final_states\":[\"f8\"],\"start_state\":\"s8\",\"states\":[\"d8\",\"f8\",\"m8\",\"s8\"],\"transitions\":[[\"m8\",\"a8\",\"m8\"],[\"m8\",\"b8\",\"f8\"],[\"s8\",null,\"d8\"],[\"s8\",null,\"m8\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s8, m8, f8, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s8, m8, f8, d8], [a8, b8], s8, [f8], [[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]], [[s8, a8, []], [s8, b8, []], [f8, a8, []], [f8, b8, []], [d8, a8, []], [d8, b8, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s8, m8, f8, d8], [a8, b8], s8, [f8], [[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]], [[s8, a8, []], [s8, b8, []], [f8, a8, []], [f8, b8, []], [d8, a8, []], [d8, b8, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0010","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1152cdfc72c2345ffc334b4f9c39ea23d4171b3a1776ad51850bb027b01d72ae","family_id":"qf-mod42-nfa-qualification","seed":202611117010,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a9\",\"b9\"],\"final_states\":[\"f9\"],\"start_state\":\"s9\",\"states\":[\"d9\",\"f9\",\"m9\",\"s9\"],\"transitions\":[[\"m9\",\"a9\",\"m9\"],[\"m9\",\"b9\",\"f9\"],[\"s9\",null,\"d9\"],[\"s9\",null,\"m9\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s9, m9, f9, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s9, m9, f9, d9], [a9, b9], s9, [f9], [[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]], [[s9, a9, []], [s9, b9, []], [f9, a9, []], [f9, b9, []], [d9, a9, []], [d9, b9, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s9, m9, f9, d9], [a9, b9], s9, [f9], [[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]], [[s9, a9, []], [s9, b9, []], [f9, a9, []], [f9, b9, []], [d9, a9, []], [d9, b9, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0011","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10fd31ca595d3f733a76f5b209e27cd71fe3305da45dec9dd8b31893c5bb688a","family_id":"qf-mod42-nfa-qualification","seed":202611117011,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a10\",\"b10\"],\"final_states\":[\"f10\"],\"start_state\":\"s10\",\"states\":[\"d10\",\"f10\",\"m10\",\"s10\"],\"transitions\":[[\"m10\",\"a10\",\"m10\"],[\"m10\",\"b10\",\"f10\"],[\"s10\",null,\"d10\"],[\"s10\",null,\"m10\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s10, m10, f10, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s10, m10, f10, d10], [a10, b10], s10, [f10], [[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]], [[s10, a10, []], [s10, b10, []], [f10, a10, []], [f10, b10, []], [d10, a10, []], [d10, b10, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s10, m10, f10, d10], [a10, b10], s10, [f10], [[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]], [[s10, a10, []], [s10, b10, []], [f10, a10, []], [f10, b10, []], [d10, a10, []], [d10, b10, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0012","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b0c9530f7b5f0c1b398e9c7b375c02e436c7306b4d257687b5bb93eca00d7d4","family_id":"qf-mod42-nfa-qualification","seed":202611117012,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a11\",\"b11\"],\"final_states\":[\"f11\"],\"start_state\":\"s11\",\"states\":[\"d11\",\"f11\",\"m11\",\"s11\"],\"transitions\":[[\"m11\",\"a11\",\"m11\"],[\"m11\",\"b11\",\"f11\"],[\"s11\",null,\"d11\"],[\"s11\",null,\"m11\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s11, m11, f11, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s11, m11, f11, d11], [a11, b11], s11, [f11], [[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11]], [[s11, a11, []], [s11, b11, []], [f11, a11, []], [f11, b11, []], [d11, a11, []], [d11, b11, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s11, m11, f11, d11], [a11, b11], s11, [f11], [[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11]], [[s11, a11, []], [s11, b11, []], [f11, a11, []], [f11, b11, []], [d11, a11, []], [d11, b11, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0013","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21019fecefb42345ee2d11ddf29660f170ba3a7cde18980bd2fb5443df957b94","family_id":"qf-mod42-nfa-qualification","seed":202611117013,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a12\",\"b12\"],\"final_states\":[\"f12\"],\"start_state\":\"s12\",\"states\":[\"d12\",\"f12\",\"m12\",\"s12\"],\"transitions\":[[\"m12\",\"a12\",\"m12\"],[\"m12\",\"b12\",\"f12\"],[\"s12\",null,\"d12\"],[\"s12\",null,\"m12\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s12, m12, f12, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s12, m12, f12, d12], [a12, b12], s12, [f12], [[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]], [[s12, a12, []], [s12, b12, []], [f12, a12, []], [f12, b12, []], [d12, a12, []], [d12, b12, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s12, m12, f12, d12], [a12, b12], s12, [f12], [[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]], [[s12, a12, []], [s12, b12, []], [f12, a12, []], [f12, b12, []], [d12, a12, []], [d12, b12, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0014","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6712237a27efa2369ce0b0c52832a0d2e38c184d2d5bc7db48f676c83244267","family_id":"qf-mod42-nfa-qualification","seed":202611117014,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a13\",\"b13\"],\"final_states\":[\"f13\"],\"start_state\":\"s13\",\"states\":[\"d13\",\"f13\",\"m13\",\"s13\"],\"transitions\":[[\"m13\",\"a13\",\"m13\"],[\"m13\",\"b13\",\"f13\"],[\"s13\",null,\"d13\"],[\"s13\",null,\"m13\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s13, m13, f13, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s13, m13, f13, d13], [a13, b13], s13, [f13], [[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]], [[s13, a13, []], [s13, b13, []], [f13, a13, []], [f13, b13, []], [d13, a13, []], [d13, b13, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s13, m13, f13, d13], [a13, b13], s13, [f13], [[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]], [[s13, a13, []], [s13, b13, []], [f13, a13, []], [f13, b13, []], [d13, a13, []], [d13, b13, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0015","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1f3ae58dec913be81e50955326f5ff0f105895e487382df99d7b3852e2e7947","family_id":"qf-mod42-nfa-qualification","seed":202611117015,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a14\",\"b14\"],\"final_states\":[\"f14\"],\"start_state\":\"s14\",\"states\":[\"d14\",\"f14\",\"m14\",\"s14\"],\"transitions\":[[\"m14\",\"a14\",\"m14\"],[\"m14\",\"b14\",\"f14\"],[\"s14\",null,\"d14\"],[\"s14\",null,\"m14\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s14, m14, f14, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s14, m14, f14, d14], [a14, b14], s14, [f14], [[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14]], [[s14, a14, []], [s14, b14, []], [f14, a14, []], [f14, b14, []], [d14, a14, []], [d14, b14, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s14, m14, f14, d14], [a14, b14], s14, [f14], [[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14]], [[s14, a14, []], [s14, b14, []], [f14, a14, []], [f14, b14, []], [d14, a14, []], [d14, b14, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0016","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77d66b5625713de9058c95a985bbbbfd6b1e0f4ab109f591cd91c64f6f2a6e30","family_id":"qf-mod42-nfa-qualification","seed":202611117016,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a15\",\"b15\"],\"final_states\":[\"f15\"],\"start_state\":\"s15\",\"states\":[\"d15\",\"f15\",\"m15\",\"s15\"],\"transitions\":[[\"m15\",\"a15\",\"m15\"],[\"m15\",\"b15\",\"f15\"],[\"s15\",null,\"d15\"],[\"s15\",null,\"m15\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s15, m15, f15, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s15, m15, f15, d15], [a15, b15], s15, [f15], [[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]], [[s15, a15, []], [s15, b15, []], [f15, a15, []], [f15, b15, []], [d15, a15, []], [d15, b15, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s15, m15, f15, d15], [a15, b15], s15, [f15], [[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]], [[s15, a15, []], [s15, b15, []], [f15, a15, []], [f15, b15, []], [d15, a15, []], [d15, b15, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0017","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ae839d4c2662d35c0530a69ac09d6db4c58328086d6c4483f3530dd84cd5b44","family_id":"qf-mod42-nfa-qualification","seed":202611117017,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a16\",\"b16\"],\"final_states\":[\"f16\"],\"start_state\":\"s16\",\"states\":[\"d16\",\"f16\",\"m16\",\"s16\"],\"transitions\":[[\"m16\",\"a16\",\"m16\"],[\"m16\",\"b16\",\"f16\"],[\"s16\",null,\"d16\"],[\"s16\",null,\"m16\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s16, m16, f16, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s16, m16, f16, d16], [a16, b16], s16, [f16], [[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]], [[s16, a16, []], [s16, b16, []], [f16, a16, []], [f16, b16, []], [d16, a16, []], [d16, b16, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s16, m16, f16, d16], [a16, b16], s16, [f16], [[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]], [[s16, a16, []], [s16, b16, []], [f16, a16, []], [f16, b16, []], [d16, a16, []], [d16, b16, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0018","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5601733042515773fc411b905c234a9fb38c3afd74d901365c5f1a379999a053","family_id":"qf-mod42-nfa-qualification","seed":202611117018,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a17\",\"b17\"],\"final_states\":[\"f17\"],\"start_state\":\"s17\",\"states\":[\"d17\",\"f17\",\"m17\",\"s17\"],\"transitions\":[[\"m17\",\"a17\",\"m17\"],[\"m17\",\"b17\",\"f17\"],[\"s17\",null,\"d17\"],[\"s17\",null,\"m17\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s17, m17, f17, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s17, m17, f17, d17], [a17, b17], s17, [f17], [[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17]], [[s17, a17, []], [s17, b17, []], [f17, a17, []], [f17, b17, []], [d17, a17, []], [d17, b17, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s17, m17, f17, d17], [a17, b17], s17, [f17], [[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17]], [[s17, a17, []], [s17, b17, []], [f17, a17, []], [f17, b17, []], [d17, a17, []], [d17, b17, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0019","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a976a2e7793300cbbf9264c68215fa2851fa573904c3dabdb6db1b6792e291f","family_id":"qf-mod42-nfa-qualification","seed":202611117019,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a18\",\"b18\"],\"final_states\":[\"f18\"],\"start_state\":\"s18\",\"states\":[\"d18\",\"f18\",\"m18\",\"s18\"],\"transitions\":[[\"m18\",\"a18\",\"m18\"],[\"m18\",\"b18\",\"f18\"],[\"s18\",null,\"d18\"],[\"s18\",null,\"m18\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s18, m18, f18, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s18, m18, f18, d18], [a18, b18], s18, [f18], [[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]], [[s18, a18, []], [s18, b18, []], [f18, a18, []], [f18, b18, []], [d18, a18, []], [d18, b18, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s18, m18, f18, d18], [a18, b18], s18, [f18], [[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]], [[s18, a18, []], [s18, b18, []], [f18, a18, []], [f18, b18, []], [d18, a18, []], [d18, b18, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0020","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"888247f607f38b8e9346576c34ff2c43949bd0041a0df3aa14793301eca4cc8d","family_id":"qf-mod42-nfa-qualification","seed":202611117020,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a19\",\"b19\"],\"final_states\":[\"f19\"],\"start_state\":\"s19\",\"states\":[\"d19\",\"f19\",\"m19\",\"s19\"],\"transitions\":[[\"m19\",\"a19\",\"m19\"],[\"m19\",\"b19\",\"f19\"],[\"s19\",null,\"d19\"],[\"s19\",null,\"m19\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s19, m19, f19, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s19, m19, f19, d19], [a19, b19], s19, [f19], [[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19]], [[s19, a19, []], [s19, b19, []], [f19, a19, []], [f19, b19, []], [d19, a19, []], [d19, b19, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s19, m19, f19, d19], [a19, b19], s19, [f19], [[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19]], [[s19, a19, []], [s19, b19, []], [f19, a19, []], [f19, b19, []], [d19, a19, []], [d19, b19, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0021","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"351edcba0639563eb260ddbd66de2b46bc4cafb9744ca6f2fc797b05d41b9536","family_id":"qf-mod42-nfa-qualification","seed":202611117021,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a20\",\"b20\"],\"final_states\":[\"f20\"],\"start_state\":\"s20\",\"states\":[\"d20\",\"f20\",\"m20\",\"s20\"],\"transitions\":[[\"m20\",\"a20\",\"m20\"],[\"m20\",\"b20\",\"f20\"],[\"s20\",null,\"d20\"],[\"s20\",null,\"m20\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s20, m20, f20, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s20, m20, f20, d20], [a20, b20], s20, [f20], [[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20]], [[s20, a20, []], [s20, b20, []], [f20, a20, []], [f20, b20, []], [d20, a20, []], [d20, b20, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s20, m20, f20, d20], [a20, b20], s20, [f20], [[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20]], [[s20, a20, []], [s20, b20, []], [f20, a20, []], [f20, b20, []], [d20, a20, []], [d20, b20, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0022","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa3f462fd1641d9d443789b0d9f53ce80d999cc17639c0dc4f7e5864be39fe79","family_id":"qf-mod42-nfa-qualification","seed":202611117022,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a21\",\"b21\"],\"final_states\":[\"f21\"],\"start_state\":\"s21\",\"states\":[\"d21\",\"f21\",\"m21\",\"s21\"],\"transitions\":[[\"m21\",\"a21\",\"m21\"],[\"m21\",\"b21\",\"f21\"],[\"s21\",null,\"d21\"],[\"s21\",null,\"m21\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s21, m21, f21, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s21, m21, f21, d21], [a21, b21], s21, [f21], [[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21]], [[s21, a21, []], [s21, b21, []], [f21, a21, []], [f21, b21, []], [d21, a21, []], [d21, b21, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s21, m21, f21, d21], [a21, b21], s21, [f21], [[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21]], [[s21, a21, []], [s21, b21, []], [f21, a21, []], [f21, b21, []], [d21, a21, []], [d21, b21, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0023","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97045c3ff2c3dcb7f979a63a99079e14f203796edf8448514710a4f9fc57dd22","family_id":"qf-mod42-nfa-qualification","seed":202611117023,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a22\",\"b22\"],\"final_states\":[\"f22\"],\"start_state\":\"s22\",\"states\":[\"d22\",\"f22\",\"m22\",\"s22\"],\"transitions\":[[\"m22\",\"a22\",\"m22\"],[\"m22\",\"b22\",\"f22\"],[\"s22\",null,\"d22\"],[\"s22\",null,\"m22\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s22, m22, f22, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s22, m22, f22, d22], [a22, b22], s22, [f22], [[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]], [[s22, a22, []], [s22, b22, []], [f22, a22, []], [f22, b22, []], [d22, a22, []], [d22, b22, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s22, m22, f22, d22], [a22, b22], s22, [f22], [[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]], [[s22, a22, []], [s22, b22, []], [f22, a22, []], [f22, b22, []], [d22, a22, []], [d22, b22, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0024","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"202fd8eff621552ce294d90474e162a071507d838f9e98ed29f493814ee76b4a","family_id":"qf-mod42-nfa-qualification","seed":202611117024,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a23\",\"b23\"],\"final_states\":[\"f23\"],\"start_state\":\"s23\",\"states\":[\"d23\",\"f23\",\"m23\",\"s23\"],\"transitions\":[[\"m23\",\"a23\",\"m23\"],[\"m23\",\"b23\",\"f23\"],[\"s23\",null,\"d23\"],[\"s23\",null,\"m23\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s23, m23, f23, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s23, m23, f23, d23], [a23, b23], s23, [f23], [[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]], [[s23, a23, []], [s23, b23, []], [f23, a23, []], [f23, b23, []], [d23, a23, []], [d23, b23, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s23, m23, f23, d23], [a23, b23], s23, [f23], [[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]], [[s23, a23, []], [s23, b23, []], [f23, a23, []], [f23, b23, []], [d23, a23, []], [d23, b23, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0025","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f6c341ef41a6a5c5befedc3f3919797a3707d4551b432917361add55dba90a3","family_id":"qf-mod42-nfa-qualification","seed":202611117025,"semantic_template":"qf-mod42-nfa-qualification/{\"alphabet\":[\"a24\",\"b24\"],\"final_states\":[\"f24\"],\"start_state\":\"s24\",\"states\":[\"d24\",\"f24\",\"m24\",\"s24\"],\"transitions\":[[\"m24\",\"a24\",\"m24\"],[\"m24\",\"b24\",\"f24\"],[\"s24\",null,\"d24\"],[\"s24\",null,\"m24\"]]}"},"fields":{"title":"An Epsilon-NFA Relation May Be Empty, Partial or Multivalued","stem":"Use the complete finite epsilon-NFA: states=[s24, m24, f24, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Give (states, alphabet, start, final_states, transitions, empty_target_rows, qualified, convention).","answer":{"kind":"expression","value":"([s24, m24, f24, d24], [a24, b24], s24, [f24], [[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24]], [[s24, a24, []], [s24, b24, []], [f24, a24, []], [f24, b24, []], [d24, a24, []], [d24, b24, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)","explanation":"Retain the original state/alphabet/transition order and every empty [state,input symbol,target list] row. Empty final sets are valid and recognize the empty language; an empty alphabet is valid. Qualification does not claim determinism, totality, DFA equivalence or the subset-construction theorem. The exact result, retaining the requested order, is ([s24, m24, f24, d24], [a24, b24], s24, [f24], [[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24]], [[s24, a24, []], [s24, b24, []], [f24, a24, []], [f24, b24, []], [d24, a24, []], [d24, b24, []]], Yes, transition_relation_may_be_empty_multi_valued_and_epsilon_enabled)."},"skills":["An Epsilon-NFA Relation May Be Empty, Partial or Multivalued"],"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-20261111-r07-nondeterministic-finite-automaton-0026","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bba14d719200e62fed7c4f02cbb3b26fa9fb050bbb431173280758ef6d43efab","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117026,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a0\",\"b0\"],\"final_states\":[\"f0\"],\"seed_states\":[\"s0\"],\"start_state\":\"s0\",\"states\":[\"d0\",\"f0\",\"m0\",\"s0\"],\"transitions\":[[\"m0\",\"a0\",\"m0\"],[\"m0\",\"b0\",\"f0\"],[\"s0\",null,\"d0\"],[\"s0\",null,\"m0\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s0, m0, f0, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s0]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s0], [d0, m0, s0], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s0], [d0, m0, s0], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0027","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e3128197385638b5b58f67391b396938030993e086416bfaebdce4e7a4b8ff6","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117027,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a1\",\"b1\"],\"final_states\":[\"f1\"],\"seed_states\":[\"s1\"],\"start_state\":\"s1\",\"states\":[\"d1\",\"f1\",\"m1\",\"s1\"],\"transitions\":[[\"m1\",\"a1\",\"m1\"],[\"m1\",\"b1\",\"f1\"],[\"s1\",null,\"d1\"],[\"s1\",null,\"m1\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s1, m1, f1, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s1]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s1], [d1, m1, s1], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s1], [d1, m1, s1], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0028","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00ec0f7c7ce27fdcdea062cda57fbff0701562a27e61c749db2e30ffba622c62","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117028,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a2\",\"b2\"],\"final_states\":[\"f2\"],\"seed_states\":[\"s2\"],\"start_state\":\"s2\",\"states\":[\"d2\",\"f2\",\"m2\",\"s2\"],\"transitions\":[[\"m2\",\"a2\",\"m2\"],[\"m2\",\"b2\",\"f2\"],[\"s2\",null,\"d2\"],[\"s2\",null,\"m2\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s2, m2, f2, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s2]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s2], [d2, m2, s2], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s2], [d2, m2, s2], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0029","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"381029202648f90af797df4a68a1de94555ffa27ccac025fabeba979d83b7d73","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117029,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a3\",\"b3\"],\"final_states\":[\"f3\"],\"seed_states\":[\"s3\"],\"start_state\":\"s3\",\"states\":[\"d3\",\"f3\",\"m3\",\"s3\"],\"transitions\":[[\"m3\",\"a3\",\"m3\"],[\"m3\",\"b3\",\"f3\"],[\"s3\",null,\"d3\"],[\"s3\",null,\"m3\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s3, m3, f3, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s3]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s3], [d3, m3, s3], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s3], [d3, m3, s3], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0030","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b853b0dbfaba14b14c1d309cf344f19272e6d2317b5c6ccbdc0d182b63a9c95c","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117030,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a4\",\"b4\"],\"final_states\":[\"f4\"],\"seed_states\":[\"s4\"],\"start_state\":\"s4\",\"states\":[\"d4\",\"f4\",\"m4\",\"s4\"],\"transitions\":[[\"m4\",\"a4\",\"m4\"],[\"m4\",\"b4\",\"f4\"],[\"s4\",null,\"d4\"],[\"s4\",null,\"m4\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s4, m4, f4, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s4]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s4], [d4, m4, s4], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s4], [d4, m4, s4], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0031","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc2cd02b23fe9aa06742452c03cc4aedc1bf366114cf9b1d41962acd14f4bc54","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117031,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a5\",\"b5\"],\"final_states\":[\"f5\"],\"seed_states\":[\"s5\"],\"start_state\":\"s5\",\"states\":[\"d5\",\"f5\",\"m5\",\"s5\"],\"transitions\":[[\"m5\",\"a5\",\"m5\"],[\"m5\",\"b5\",\"f5\"],[\"s5\",null,\"d5\"],[\"s5\",null,\"m5\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s5, m5, f5, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s5]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s5], [d5, m5, s5], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s5], [d5, m5, s5], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0032","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db2f1f1536ec080a6783b205b4186d20a3fe54ddc30c278e0375dca7952ecd45","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117032,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a6\",\"b6\"],\"final_states\":[\"f6\"],\"seed_states\":[\"s6\"],\"start_state\":\"s6\",\"states\":[\"d6\",\"f6\",\"m6\",\"s6\"],\"transitions\":[[\"m6\",\"a6\",\"m6\"],[\"m6\",\"b6\",\"f6\"],[\"s6\",null,\"d6\"],[\"s6\",null,\"m6\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s6, m6, f6, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s6]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s6], [d6, m6, s6], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s6], [d6, m6, s6], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0033","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61d67c77d588f5c812ae7ac8d7f0b65e703ac513f365827272487322c7c26a65","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117033,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a7\",\"b7\"],\"final_states\":[\"f7\"],\"seed_states\":[\"s7\"],\"start_state\":\"s7\",\"states\":[\"d7\",\"f7\",\"m7\",\"s7\"],\"transitions\":[[\"m7\",\"a7\",\"m7\"],[\"m7\",\"b7\",\"f7\"],[\"s7\",null,\"d7\"],[\"s7\",null,\"m7\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s7, m7, f7, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s7]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s7], [d7, m7, s7], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s7], [d7, m7, s7], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0034","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"288bbc5778ee68f1e3b61d693185803ff07aa47bfc1d29c0070fc2babc84cfde","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117034,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a8\",\"b8\"],\"final_states\":[\"f8\"],\"seed_states\":[\"s8\"],\"start_state\":\"s8\",\"states\":[\"d8\",\"f8\",\"m8\",\"s8\"],\"transitions\":[[\"m8\",\"a8\",\"m8\"],[\"m8\",\"b8\",\"f8\"],[\"s8\",null,\"d8\"],[\"s8\",null,\"m8\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s8, m8, f8, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s8]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s8], [d8, m8, s8], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s8], [d8, m8, s8], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0035","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be311da8c14e769fb7da282f3f07b448dab7d96bdc0b1ba7fd44d3776c756e58","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117035,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a9\",\"b9\"],\"final_states\":[\"f9\"],\"seed_states\":[\"s9\"],\"start_state\":\"s9\",\"states\":[\"d9\",\"f9\",\"m9\",\"s9\"],\"transitions\":[[\"m9\",\"a9\",\"m9\"],[\"m9\",\"b9\",\"f9\"],[\"s9\",null,\"d9\"],[\"s9\",null,\"m9\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s9, m9, f9, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s9]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s9], [d9, m9, s9], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s9], [d9, m9, s9], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0036","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06af3c923191c38d263563a34502680acf4ec6e9a626f9c7a1bb5451ee2c9745","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117036,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a10\",\"b10\"],\"final_states\":[\"f10\"],\"seed_states\":[\"s10\"],\"start_state\":\"s10\",\"states\":[\"d10\",\"f10\",\"m10\",\"s10\"],\"transitions\":[[\"m10\",\"a10\",\"m10\"],[\"m10\",\"b10\",\"f10\"],[\"s10\",null,\"d10\"],[\"s10\",null,\"m10\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s10, m10, f10, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s10]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s10], [d10, m10, s10], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s10], [d10, m10, s10], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0037","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5faf427555fddad82a742776cee097b68d0c0fb246f686530e1e238e988374d3","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117037,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a11\",\"b11\"],\"final_states\":[\"f11\"],\"seed_states\":[\"s11\"],\"start_state\":\"s11\",\"states\":[\"d11\",\"f11\",\"m11\",\"s11\"],\"transitions\":[[\"m11\",\"a11\",\"m11\"],[\"m11\",\"b11\",\"f11\"],[\"s11\",null,\"d11\"],[\"s11\",null,\"m11\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s11, m11, f11, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s11]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s11], [d11, m11, s11], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s11], [d11, m11, s11], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0038","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2a1f7be0cbee4ecfe4ca022e2d50c62c6dbb8c54b51472514cd422af257a684","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117038,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a12\",\"b12\"],\"final_states\":[\"f12\"],\"seed_states\":[\"s12\"],\"start_state\":\"s12\",\"states\":[\"d12\",\"f12\",\"m12\",\"s12\"],\"transitions\":[[\"m12\",\"a12\",\"m12\"],[\"m12\",\"b12\",\"f12\"],[\"s12\",null,\"d12\"],[\"s12\",null,\"m12\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s12, m12, f12, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s12]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s12], [d12, m12, s12], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s12], [d12, m12, s12], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0039","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f09ef27ef0440a20b287c7dedfa23af765ae35f2301fe6a3401cf36ba1fd679","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117039,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a13\",\"b13\"],\"final_states\":[\"f13\"],\"seed_states\":[\"s13\"],\"start_state\":\"s13\",\"states\":[\"d13\",\"f13\",\"m13\",\"s13\"],\"transitions\":[[\"m13\",\"a13\",\"m13\"],[\"m13\",\"b13\",\"f13\"],[\"s13\",null,\"d13\"],[\"s13\",null,\"m13\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s13, m13, f13, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s13]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s13], [d13, m13, s13], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s13], [d13, m13, s13], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0040","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7c8b9fb40d9988585bafe1765e0b984b1b5deb9ac4f555b30a8c5453d653976","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117040,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a14\",\"b14\"],\"final_states\":[\"f14\"],\"seed_states\":[\"s14\"],\"start_state\":\"s14\",\"states\":[\"d14\",\"f14\",\"m14\",\"s14\"],\"transitions\":[[\"m14\",\"a14\",\"m14\"],[\"m14\",\"b14\",\"f14\"],[\"s14\",null,\"d14\"],[\"s14\",null,\"m14\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s14, m14, f14, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s14]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s14], [d14, m14, s14], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s14], [d14, m14, s14], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0041","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a63015d7960b170073dbdf16c9fd9dcb987df3d70870e28fe15affc58df5ef6","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117041,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a15\",\"b15\"],\"final_states\":[\"f15\"],\"seed_states\":[\"s15\"],\"start_state\":\"s15\",\"states\":[\"d15\",\"f15\",\"m15\",\"s15\"],\"transitions\":[[\"m15\",\"a15\",\"m15\"],[\"m15\",\"b15\",\"f15\"],[\"s15\",null,\"d15\"],[\"s15\",null,\"m15\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s15, m15, f15, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s15]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s15], [d15, m15, s15], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s15], [d15, m15, s15], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0042","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a644c50980e96590f4656f8d1029f280edf9db1ca03728b0f0a51b58405eff07","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117042,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a16\",\"b16\"],\"final_states\":[\"f16\"],\"seed_states\":[\"s16\"],\"start_state\":\"s16\",\"states\":[\"d16\",\"f16\",\"m16\",\"s16\"],\"transitions\":[[\"m16\",\"a16\",\"m16\"],[\"m16\",\"b16\",\"f16\"],[\"s16\",null,\"d16\"],[\"s16\",null,\"m16\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s16, m16, f16, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s16]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s16], [d16, m16, s16], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s16], [d16, m16, s16], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0043","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cc75eaa526e02f3d4d3863c07c8e31f60a9609f10b4bc8aea0aab1e5bd4d1f3","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117043,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a17\",\"b17\"],\"final_states\":[\"f17\"],\"seed_states\":[\"s17\"],\"start_state\":\"s17\",\"states\":[\"d17\",\"f17\",\"m17\",\"s17\"],\"transitions\":[[\"m17\",\"a17\",\"m17\"],[\"m17\",\"b17\",\"f17\"],[\"s17\",null,\"d17\"],[\"s17\",null,\"m17\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s17, m17, f17, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s17]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s17], [d17, m17, s17], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s17], [d17, m17, s17], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0044","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b320a7ab2867dd3edb1042f135e2858a8de241b3a524b46b96891539009f5cba","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117044,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a18\",\"b18\"],\"final_states\":[\"f18\"],\"seed_states\":[\"s18\"],\"start_state\":\"s18\",\"states\":[\"d18\",\"f18\",\"m18\",\"s18\"],\"transitions\":[[\"m18\",\"a18\",\"m18\"],[\"m18\",\"b18\",\"f18\"],[\"s18\",null,\"d18\"],[\"s18\",null,\"m18\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s18, m18, f18, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s18]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s18], [d18, m18, s18], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s18], [d18, m18, s18], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0045","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d3d470a3afca1cfc600b91d769e5540d16ee3d489105f678c6cc318f80d2fa3","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117045,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a19\",\"b19\"],\"final_states\":[\"f19\"],\"seed_states\":[\"s19\"],\"start_state\":\"s19\",\"states\":[\"d19\",\"f19\",\"m19\",\"s19\"],\"transitions\":[[\"m19\",\"a19\",\"m19\"],[\"m19\",\"b19\",\"f19\"],[\"s19\",null,\"d19\"],[\"s19\",null,\"m19\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s19, m19, f19, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s19]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s19], [d19, m19, s19], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s19], [d19, m19, s19], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0046","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31d01b08c0f0a99ef6a6b9487a33c9a3ef06c57df3eff9840a46958d898230c4","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117046,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a20\",\"b20\"],\"final_states\":[\"f20\"],\"seed_states\":[\"s20\"],\"start_state\":\"s20\",\"states\":[\"d20\",\"f20\",\"m20\",\"s20\"],\"transitions\":[[\"m20\",\"a20\",\"m20\"],[\"m20\",\"b20\",\"f20\"],[\"s20\",null,\"d20\"],[\"s20\",null,\"m20\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s20, m20, f20, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s20]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s20], [d20, m20, s20], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s20], [d20, m20, s20], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0047","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9b50261897cc68b22b402eecf29fcd92d463ef51e68d543a50f13488d2ad7df","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117047,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a21\",\"b21\"],\"final_states\":[\"f21\"],\"seed_states\":[\"s21\"],\"start_state\":\"s21\",\"states\":[\"d21\",\"f21\",\"m21\",\"s21\"],\"transitions\":[[\"m21\",\"a21\",\"m21\"],[\"m21\",\"b21\",\"f21\"],[\"s21\",null,\"d21\"],[\"s21\",null,\"m21\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s21, m21, f21, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s21]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s21], [d21, m21, s21], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s21], [d21, m21, s21], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0048","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d92a6e86804a809232c3448114968170f113b274e2b934e89bbb9bf26c005f5","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117048,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a22\",\"b22\"],\"final_states\":[\"f22\"],\"seed_states\":[\"s22\"],\"start_state\":\"s22\",\"states\":[\"d22\",\"f22\",\"m22\",\"s22\"],\"transitions\":[[\"m22\",\"a22\",\"m22\"],[\"m22\",\"b22\",\"f22\"],[\"s22\",null,\"d22\"],[\"s22\",null,\"m22\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s22, m22, f22, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s22]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s22], [d22, m22, s22], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s22], [d22, m22, s22], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0049","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"347897c8e1bc342044a2c7ff8e6321ba939e70e3bd8d7bc6b47bfd777c2dd779","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117049,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a23\",\"b23\"],\"final_states\":[\"f23\"],\"seed_states\":[\"s23\"],\"start_state\":\"s23\",\"states\":[\"d23\",\"f23\",\"m23\",\"s23\"],\"transitions\":[[\"m23\",\"a23\",\"m23\"],[\"m23\",\"b23\",\"f23\"],[\"s23\",null,\"d23\"],[\"s23\",null,\"m23\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s23, m23, f23, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s23]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s23], [d23, m23, s23], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s23], [d23, m23, s23], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0050","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cce9f2b99e5702c40c71e9de06ea6c3ac7518bfe66507f4e78c59de14f856bc","family_id":"qf-mod42-nfa-epsilon-closure","seed":202611117050,"semantic_template":"qf-mod42-nfa-epsilon-closure/{\"alphabet\":[\"a24\",\"b24\"],\"final_states\":[\"f24\"],\"seed_states\":[\"s24\"],\"start_state\":\"s24\",\"states\":[\"d24\",\"f24\",\"m24\",\"s24\"],\"transitions\":[[\"m24\",\"a24\",\"m24\"],[\"m24\",\"b24\",\"f24\"],[\"s24\",null,\"d24\"],[\"s24\",null,\"m24\"]]}"},"fields":{"title":"Epsilon Closure Reaches States without Consuming Input","stem":"Use the complete finite epsilon-NFA: states=[s24, m24, f24, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Seed states=[s24]. Give (seed_states, epsilon_closure, closure_certificate, certificate).","answer":{"kind":"expression","value":"([s24], [d24, m24, s24], Yes, finite_epsilon_reachability_without_input_consumption)","explanation":"Use all finite epsilon-reachable states, including seeds, and sort the resulting state set lexicographically as in the source convention. Epsilon cycles add reachable states but do not by themselves accept a word. The exact result, retaining the requested order, is ([s24], [d24, m24, s24], Yes, finite_epsilon_reachability_without_input_consumption)."},"skills":["Epsilon Closure Reaches States without Consuming Input"],"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-20261111-r07-nondeterministic-finite-automaton-0051","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7eb7cbf4aba1fe2655b32be73b0c5aff4aec90c8867a9137bee8b0c5023e00c9","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117051,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a0\",\"b0\"],\"final_states\":[\"f0\"],\"input\":[\"b0\"],\"path\":[[\"s0\",0],[\"m0\",0],[\"f0\",1]],\"start_state\":\"s0\",\"states\":[\"d0\",\"f0\",\"m0\",\"s0\"],\"transitions\":[[\"m0\",\"a0\",\"m0\"],[\"m0\",\"b0\",\"f0\"],[\"s0\",null,\"d0\"],[\"s0\",null,\"m0\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s0, m0, f0, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b0], supplied path=[[s0, 0], [m0, 0], [f0, 1]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([b0], [[s0, 0], [m0, 0], [f0, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([b0], [[s0, 0], [m0, 0], [f0, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0052","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"200e7c087ea6fc75b614a7b1a67feb9ec0f7c09f5a479c0ad6654235c66ad626","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117052,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a1\",\"b1\"],\"final_states\":[\"f1\"],\"input\":[\"a1\",\"b1\"],\"path\":[[\"s1\",0],[\"m1\",0],[\"m1\",1],[\"f1\",2]],\"start_state\":\"s1\",\"states\":[\"d1\",\"f1\",\"m1\",\"s1\"],\"transitions\":[[\"m1\",\"a1\",\"m1\"],[\"m1\",\"b1\",\"f1\"],[\"s1\",null,\"d1\"],[\"s1\",null,\"m1\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s1, m1, f1, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a1, b1], supplied path=[[s1, 0], [m1, 0], [m1, 1], [f1, 2]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a1, b1], [[s1, 0], [m1, 0], [m1, 1], [f1, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a1, b1], [[s1, 0], [m1, 0], [m1, 1], [f1, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0053","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bf92cc9ff03bea19a12ee1cf20f02da9e5c0c0ce671a371aa44912f0031b894","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117053,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a2\",\"b2\"],\"final_states\":[\"f2\"],\"input\":[\"a2\",\"a2\",\"b2\"],\"path\":[[\"s2\",0],[\"m2\",0],[\"m2\",1],[\"m2\",2],[\"f2\",3]],\"start_state\":\"s2\",\"states\":[\"d2\",\"f2\",\"m2\",\"s2\"],\"transitions\":[[\"m2\",\"a2\",\"m2\"],[\"m2\",\"b2\",\"f2\"],[\"s2\",null,\"d2\"],[\"s2\",null,\"m2\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s2, m2, f2, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a2, a2, b2], supplied path=[[s2, 0], [m2, 0], [m2, 1], [m2, 2], [f2, 3]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a2, a2, b2], [[s2, 0], [m2, 0], [m2, 1], [m2, 2], [f2, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a2, a2, b2], [[s2, 0], [m2, 0], [m2, 1], [m2, 2], [f2, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0054","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73d073ed776723db165b2f6d29267123351b22542929f417d78524dfcf584715","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117054,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a3\",\"b3\"],\"final_states\":[\"f3\"],\"input\":[\"a3\",\"a3\",\"a3\",\"b3\"],\"path\":[[\"s3\",0],[\"m3\",0],[\"m3\",1],[\"m3\",2],[\"m3\",3],[\"f3\",4]],\"start_state\":\"s3\",\"states\":[\"d3\",\"f3\",\"m3\",\"s3\"],\"transitions\":[[\"m3\",\"a3\",\"m3\"],[\"m3\",\"b3\",\"f3\"],[\"s3\",null,\"d3\"],[\"s3\",null,\"m3\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s3, m3, f3, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a3, a3, a3, b3], supplied path=[[s3, 0], [m3, 0], [m3, 1], [m3, 2], [m3, 3], [f3, 4]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a3, a3, a3, b3], [[s3, 0], [m3, 0], [m3, 1], [m3, 2], [m3, 3], [f3, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a3, a3, a3, b3], [[s3, 0], [m3, 0], [m3, 1], [m3, 2], [m3, 3], [f3, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0055","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3209db0c32f8e0ff516a25b88377e2a3ee1da6bffe41b16e75325b1aa9fe192b","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117055,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a4\",\"b4\"],\"final_states\":[\"f4\"],\"input\":[\"b4\"],\"path\":[[\"s4\",0],[\"m4\",0],[\"f4\",1]],\"start_state\":\"s4\",\"states\":[\"d4\",\"f4\",\"m4\",\"s4\"],\"transitions\":[[\"m4\",\"a4\",\"m4\"],[\"m4\",\"b4\",\"f4\"],[\"s4\",null,\"d4\"],[\"s4\",null,\"m4\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s4, m4, f4, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b4], supplied path=[[s4, 0], [m4, 0], [f4, 1]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([b4], [[s4, 0], [m4, 0], [f4, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([b4], [[s4, 0], [m4, 0], [f4, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0056","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"185ed34cc02bff295b0e9a1de60f2d483b983429fd95bd85df32a3fad9588209","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117056,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a5\",\"b5\"],\"final_states\":[\"f5\"],\"input\":[\"a5\",\"b5\"],\"path\":[[\"s5\",0],[\"m5\",0],[\"m5\",1],[\"f5\",2]],\"start_state\":\"s5\",\"states\":[\"d5\",\"f5\",\"m5\",\"s5\"],\"transitions\":[[\"m5\",\"a5\",\"m5\"],[\"m5\",\"b5\",\"f5\"],[\"s5\",null,\"d5\"],[\"s5\",null,\"m5\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s5, m5, f5, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a5, b5], supplied path=[[s5, 0], [m5, 0], [m5, 1], [f5, 2]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a5, b5], [[s5, 0], [m5, 0], [m5, 1], [f5, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a5, b5], [[s5, 0], [m5, 0], [m5, 1], [f5, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0057","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41f9c03b58aec491cf6126095bbbb0d7a2d5552ad4e2ae47d3e60431450604fe","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117057,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a6\",\"b6\"],\"final_states\":[\"f6\"],\"input\":[\"a6\",\"a6\",\"b6\"],\"path\":[[\"s6\",0],[\"m6\",0],[\"m6\",1],[\"m6\",2],[\"f6\",3]],\"start_state\":\"s6\",\"states\":[\"d6\",\"f6\",\"m6\",\"s6\"],\"transitions\":[[\"m6\",\"a6\",\"m6\"],[\"m6\",\"b6\",\"f6\"],[\"s6\",null,\"d6\"],[\"s6\",null,\"m6\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s6, m6, f6, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a6, a6, b6], supplied path=[[s6, 0], [m6, 0], [m6, 1], [m6, 2], [f6, 3]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a6, a6, b6], [[s6, 0], [m6, 0], [m6, 1], [m6, 2], [f6, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a6, a6, b6], [[s6, 0], [m6, 0], [m6, 1], [m6, 2], [f6, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0058","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e691f20bab444471717a4e3b1afaa3ba5c702e6ee5ff2ced25e8491b3fb3654d","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117058,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a7\",\"b7\"],\"final_states\":[\"f7\"],\"input\":[\"a7\",\"a7\",\"a7\",\"b7\"],\"path\":[[\"s7\",0],[\"m7\",0],[\"m7\",1],[\"m7\",2],[\"m7\",3],[\"f7\",4]],\"start_state\":\"s7\",\"states\":[\"d7\",\"f7\",\"m7\",\"s7\"],\"transitions\":[[\"m7\",\"a7\",\"m7\"],[\"m7\",\"b7\",\"f7\"],[\"s7\",null,\"d7\"],[\"s7\",null,\"m7\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s7, m7, f7, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a7, a7, a7, b7], supplied path=[[s7, 0], [m7, 0], [m7, 1], [m7, 2], [m7, 3], [f7, 4]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a7, a7, a7, b7], [[s7, 0], [m7, 0], [m7, 1], [m7, 2], [m7, 3], [f7, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a7, a7, a7, b7], [[s7, 0], [m7, 0], [m7, 1], [m7, 2], [m7, 3], [f7, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0059","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68828b43c09a985b845031a8a10e5bab7306c5bdf05155b678560d673e1e173a","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117059,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a8\",\"b8\"],\"final_states\":[\"f8\"],\"input\":[\"b8\"],\"path\":[[\"s8\",0],[\"m8\",0],[\"f8\",1]],\"start_state\":\"s8\",\"states\":[\"d8\",\"f8\",\"m8\",\"s8\"],\"transitions\":[[\"m8\",\"a8\",\"m8\"],[\"m8\",\"b8\",\"f8\"],[\"s8\",null,\"d8\"],[\"s8\",null,\"m8\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s8, m8, f8, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b8], supplied path=[[s8, 0], [m8, 0], [f8, 1]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([b8], [[s8, 0], [m8, 0], [f8, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([b8], [[s8, 0], [m8, 0], [f8, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0060","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2d53fabfa9f15215ef1cbeed9e71f9347b41ac86abe107150faa542602b228d","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117060,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a9\",\"b9\"],\"final_states\":[\"f9\"],\"input\":[\"a9\",\"b9\"],\"path\":[[\"s9\",0],[\"m9\",0],[\"m9\",1],[\"f9\",2]],\"start_state\":\"s9\",\"states\":[\"d9\",\"f9\",\"m9\",\"s9\"],\"transitions\":[[\"m9\",\"a9\",\"m9\"],[\"m9\",\"b9\",\"f9\"],[\"s9\",null,\"d9\"],[\"s9\",null,\"m9\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s9, m9, f9, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a9, b9], supplied path=[[s9, 0], [m9, 0], [m9, 1], [f9, 2]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a9, b9], [[s9, 0], [m9, 0], [m9, 1], [f9, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a9, b9], [[s9, 0], [m9, 0], [m9, 1], [f9, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0061","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"803d2c88ec52cc6ec8877fb759f047307b03d04dfe40671d03391b35f9660c88","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117061,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a10\",\"b10\"],\"final_states\":[\"f10\"],\"input\":[\"a10\",\"a10\",\"b10\"],\"path\":[[\"s10\",0],[\"m10\",0],[\"m10\",1],[\"m10\",2],[\"f10\",3]],\"start_state\":\"s10\",\"states\":[\"d10\",\"f10\",\"m10\",\"s10\"],\"transitions\":[[\"m10\",\"a10\",\"m10\"],[\"m10\",\"b10\",\"f10\"],[\"s10\",null,\"d10\"],[\"s10\",null,\"m10\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s10, m10, f10, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a10, a10, b10], supplied path=[[s10, 0], [m10, 0], [m10, 1], [m10, 2], [f10, 3]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a10, a10, b10], [[s10, 0], [m10, 0], [m10, 1], [m10, 2], [f10, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a10, a10, b10], [[s10, 0], [m10, 0], [m10, 1], [m10, 2], [f10, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0062","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7cb061cefdf12e2caf2748ea605152f53e69dd8b98c7b69dddbfd922c0eaf3d","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117062,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a11\",\"b11\"],\"final_states\":[\"f11\"],\"input\":[\"a11\",\"a11\",\"a11\",\"b11\"],\"path\":[[\"s11\",0],[\"m11\",0],[\"m11\",1],[\"m11\",2],[\"m11\",3],[\"f11\",4]],\"start_state\":\"s11\",\"states\":[\"d11\",\"f11\",\"m11\",\"s11\"],\"transitions\":[[\"m11\",\"a11\",\"m11\"],[\"m11\",\"b11\",\"f11\"],[\"s11\",null,\"d11\"],[\"s11\",null,\"m11\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s11, m11, f11, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a11, a11, a11, b11], supplied path=[[s11, 0], [m11, 0], [m11, 1], [m11, 2], [m11, 3], [f11, 4]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a11, a11, a11, b11], [[s11, 0], [m11, 0], [m11, 1], [m11, 2], [m11, 3], [f11, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a11, a11, a11, b11], [[s11, 0], [m11, 0], [m11, 1], [m11, 2], [m11, 3], [f11, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0063","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7760301effda5002b8e5a717ebcbc1f30827879e5962bed52b8dc038ca035402","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117063,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a12\",\"b12\"],\"final_states\":[\"f12\"],\"input\":[\"b12\"],\"path\":[[\"s12\",0],[\"m12\",0],[\"f12\",1]],\"start_state\":\"s12\",\"states\":[\"d12\",\"f12\",\"m12\",\"s12\"],\"transitions\":[[\"m12\",\"a12\",\"m12\"],[\"m12\",\"b12\",\"f12\"],[\"s12\",null,\"d12\"],[\"s12\",null,\"m12\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s12, m12, f12, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b12], supplied path=[[s12, 0], [m12, 0], [f12, 1]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([b12], [[s12, 0], [m12, 0], [f12, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([b12], [[s12, 0], [m12, 0], [f12, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0064","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a97733700d5db2e05d137f4f6f9aac4b48397c45bf1f72dae68d1fdfb55e35f5","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117064,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a13\",\"b13\"],\"final_states\":[\"f13\"],\"input\":[\"a13\",\"b13\"],\"path\":[[\"s13\",0],[\"m13\",0],[\"m13\",1],[\"f13\",2]],\"start_state\":\"s13\",\"states\":[\"d13\",\"f13\",\"m13\",\"s13\"],\"transitions\":[[\"m13\",\"a13\",\"m13\"],[\"m13\",\"b13\",\"f13\"],[\"s13\",null,\"d13\"],[\"s13\",null,\"m13\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s13, m13, f13, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a13, b13], supplied path=[[s13, 0], [m13, 0], [m13, 1], [f13, 2]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a13, b13], [[s13, 0], [m13, 0], [m13, 1], [f13, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a13, b13], [[s13, 0], [m13, 0], [m13, 1], [f13, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0065","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86679ceb0d98bb99053977ee389e6e664f12bff1aee023a385279f1752fabfe0","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117065,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a14\",\"b14\"],\"final_states\":[\"f14\"],\"input\":[\"a14\",\"a14\",\"b14\"],\"path\":[[\"s14\",0],[\"m14\",0],[\"m14\",1],[\"m14\",2],[\"f14\",3]],\"start_state\":\"s14\",\"states\":[\"d14\",\"f14\",\"m14\",\"s14\"],\"transitions\":[[\"m14\",\"a14\",\"m14\"],[\"m14\",\"b14\",\"f14\"],[\"s14\",null,\"d14\"],[\"s14\",null,\"m14\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s14, m14, f14, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a14, a14, b14], supplied path=[[s14, 0], [m14, 0], [m14, 1], [m14, 2], [f14, 3]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a14, a14, b14], [[s14, 0], [m14, 0], [m14, 1], [m14, 2], [f14, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a14, a14, b14], [[s14, 0], [m14, 0], [m14, 1], [m14, 2], [f14, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0066","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b020e01d21d3e47d01950d69a150977fe09f9470bf4c25229ff23a5c133823bd","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117066,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a15\",\"b15\"],\"final_states\":[\"f15\"],\"input\":[\"a15\",\"a15\",\"a15\",\"b15\"],\"path\":[[\"s15\",0],[\"m15\",0],[\"m15\",1],[\"m15\",2],[\"m15\",3],[\"f15\",4]],\"start_state\":\"s15\",\"states\":[\"d15\",\"f15\",\"m15\",\"s15\"],\"transitions\":[[\"m15\",\"a15\",\"m15\"],[\"m15\",\"b15\",\"f15\"],[\"s15\",null,\"d15\"],[\"s15\",null,\"m15\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s15, m15, f15, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a15, a15, a15, b15], supplied path=[[s15, 0], [m15, 0], [m15, 1], [m15, 2], [m15, 3], [f15, 4]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a15, a15, a15, b15], [[s15, 0], [m15, 0], [m15, 1], [m15, 2], [m15, 3], [f15, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a15, a15, a15, b15], [[s15, 0], [m15, 0], [m15, 1], [m15, 2], [m15, 3], [f15, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0067","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afae7839b709cb6ae27e2b7fdd7a5ef7dacebdd673dcff63a042d166207f8558","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117067,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a16\",\"b16\"],\"final_states\":[\"f16\"],\"input\":[\"b16\"],\"path\":[[\"s16\",0],[\"m16\",0],[\"f16\",1]],\"start_state\":\"s16\",\"states\":[\"d16\",\"f16\",\"m16\",\"s16\"],\"transitions\":[[\"m16\",\"a16\",\"m16\"],[\"m16\",\"b16\",\"f16\"],[\"s16\",null,\"d16\"],[\"s16\",null,\"m16\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s16, m16, f16, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b16], supplied path=[[s16, 0], [m16, 0], [f16, 1]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([b16], [[s16, 0], [m16, 0], [f16, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([b16], [[s16, 0], [m16, 0], [f16, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0068","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"256b85311fbd334ed89c0e3e866a0b3cdd6612a6d57f20c6ce599ff705fb379f","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117068,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a17\",\"b17\"],\"final_states\":[\"f17\"],\"input\":[\"a17\",\"b17\"],\"path\":[[\"s17\",0],[\"m17\",0],[\"m17\",1],[\"f17\",2]],\"start_state\":\"s17\",\"states\":[\"d17\",\"f17\",\"m17\",\"s17\"],\"transitions\":[[\"m17\",\"a17\",\"m17\"],[\"m17\",\"b17\",\"f17\"],[\"s17\",null,\"d17\"],[\"s17\",null,\"m17\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s17, m17, f17, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a17, b17], supplied path=[[s17, 0], [m17, 0], [m17, 1], [f17, 2]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a17, b17], [[s17, 0], [m17, 0], [m17, 1], [f17, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a17, b17], [[s17, 0], [m17, 0], [m17, 1], [f17, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0069","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"734ce28146170be851992003638c3fd4331fe889b8a39248313247d117eccbb8","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117069,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a18\",\"b18\"],\"final_states\":[\"f18\"],\"input\":[\"a18\",\"a18\",\"b18\"],\"path\":[[\"s18\",0],[\"m18\",0],[\"m18\",1],[\"m18\",2],[\"f18\",3]],\"start_state\":\"s18\",\"states\":[\"d18\",\"f18\",\"m18\",\"s18\"],\"transitions\":[[\"m18\",\"a18\",\"m18\"],[\"m18\",\"b18\",\"f18\"],[\"s18\",null,\"d18\"],[\"s18\",null,\"m18\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s18, m18, f18, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a18, a18, b18], supplied path=[[s18, 0], [m18, 0], [m18, 1], [m18, 2], [f18, 3]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a18, a18, b18], [[s18, 0], [m18, 0], [m18, 1], [m18, 2], [f18, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a18, a18, b18], [[s18, 0], [m18, 0], [m18, 1], [m18, 2], [f18, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0070","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78d8f7bf9585940940adcf50ba569819c08fdc2c3417e3f195d9e34a20da3d9e","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117070,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a19\",\"b19\"],\"final_states\":[\"f19\"],\"input\":[\"a19\",\"a19\",\"a19\",\"b19\"],\"path\":[[\"s19\",0],[\"m19\",0],[\"m19\",1],[\"m19\",2],[\"m19\",3],[\"f19\",4]],\"start_state\":\"s19\",\"states\":[\"d19\",\"f19\",\"m19\",\"s19\"],\"transitions\":[[\"m19\",\"a19\",\"m19\"],[\"m19\",\"b19\",\"f19\"],[\"s19\",null,\"d19\"],[\"s19\",null,\"m19\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s19, m19, f19, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a19, a19, a19, b19], supplied path=[[s19, 0], [m19, 0], [m19, 1], [m19, 2], [m19, 3], [f19, 4]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a19, a19, a19, b19], [[s19, 0], [m19, 0], [m19, 1], [m19, 2], [m19, 3], [f19, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a19, a19, a19, b19], [[s19, 0], [m19, 0], [m19, 1], [m19, 2], [m19, 3], [f19, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0071","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cda457f42d5a0c9c6055792b5931f472dfa31a859ad13e8b509d77eb3f978767","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117071,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a20\",\"b20\"],\"final_states\":[\"f20\"],\"input\":[\"b20\"],\"path\":[[\"s20\",0],[\"m20\",0],[\"f20\",1]],\"start_state\":\"s20\",\"states\":[\"d20\",\"f20\",\"m20\",\"s20\"],\"transitions\":[[\"m20\",\"a20\",\"m20\"],[\"m20\",\"b20\",\"f20\"],[\"s20\",null,\"d20\"],[\"s20\",null,\"m20\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s20, m20, f20, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b20], supplied path=[[s20, 0], [m20, 0], [f20, 1]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([b20], [[s20, 0], [m20, 0], [f20, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([b20], [[s20, 0], [m20, 0], [f20, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0072","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"541c781db5c47979e97dc993e8636701ad24f2b892d0cb4aa9efc7096dd44bc1","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117072,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a21\",\"b21\"],\"final_states\":[\"f21\"],\"input\":[\"a21\",\"b21\"],\"path\":[[\"s21\",0],[\"m21\",0],[\"m21\",1],[\"f21\",2]],\"start_state\":\"s21\",\"states\":[\"d21\",\"f21\",\"m21\",\"s21\"],\"transitions\":[[\"m21\",\"a21\",\"m21\"],[\"m21\",\"b21\",\"f21\"],[\"s21\",null,\"d21\"],[\"s21\",null,\"m21\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s21, m21, f21, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a21, b21], supplied path=[[s21, 0], [m21, 0], [m21, 1], [f21, 2]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a21, b21], [[s21, 0], [m21, 0], [m21, 1], [f21, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a21, b21], [[s21, 0], [m21, 0], [m21, 1], [f21, 2]], [0, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0073","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0791f60f020dc72c1fb441091545dd5070f167286837d4f5a34fc390cf372a3","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117073,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a22\",\"b22\"],\"final_states\":[\"f22\"],\"input\":[\"a22\",\"a22\",\"b22\"],\"path\":[[\"s22\",0],[\"m22\",0],[\"m22\",1],[\"m22\",2],[\"f22\",3]],\"start_state\":\"s22\",\"states\":[\"d22\",\"f22\",\"m22\",\"s22\"],\"transitions\":[[\"m22\",\"a22\",\"m22\"],[\"m22\",\"b22\",\"f22\"],[\"s22\",null,\"d22\"],[\"s22\",null,\"m22\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s22, m22, f22, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a22, a22, b22], supplied path=[[s22, 0], [m22, 0], [m22, 1], [m22, 2], [f22, 3]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a22, a22, b22], [[s22, 0], [m22, 0], [m22, 1], [m22, 2], [f22, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a22, a22, b22], [[s22, 0], [m22, 0], [m22, 1], [m22, 2], [f22, 3]], [0, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0074","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f51169ef10e2c22766b9095b29279c2ad4348838abff6b8927401f463c826a8","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117074,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a23\",\"b23\"],\"final_states\":[\"f23\"],\"input\":[\"a23\",\"a23\",\"a23\",\"b23\"],\"path\":[[\"s23\",0],[\"m23\",0],[\"m23\",1],[\"m23\",2],[\"m23\",3],[\"f23\",4]],\"start_state\":\"s23\",\"states\":[\"d23\",\"f23\",\"m23\",\"s23\"],\"transitions\":[[\"m23\",\"a23\",\"m23\"],[\"m23\",\"b23\",\"f23\"],[\"s23\",null,\"d23\"],[\"s23\",null,\"m23\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s23, m23, f23, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a23, a23, a23, b23], supplied path=[[s23, 0], [m23, 0], [m23, 1], [m23, 2], [m23, 3], [f23, 4]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([a23, a23, a23, b23], [[s23, 0], [m23, 0], [m23, 1], [m23, 2], [m23, 3], [f23, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([a23, a23, a23, b23], [[s23, 0], [m23, 0], [m23, 1], [m23, 2], [m23, 3], [f23, 4]], [0, 2, 2, 2, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0075","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0f31b540b93b39022a02f7a17b1dd9922d1cb79f4b2aab7297864f60a5557a8","family_id":"qf-mod42-nfa-accepting-path-witness","seed":202611117075,"semantic_template":"qf-mod42-nfa-accepting-path-witness/{\"alphabet\":[\"a24\",\"b24\"],\"final_states\":[\"f24\"],\"input\":[\"b24\"],\"path\":[[\"s24\",0],[\"m24\",0],[\"f24\",1]],\"start_state\":\"s24\",\"states\":[\"d24\",\"f24\",\"m24\",\"s24\"],\"transitions\":[[\"m24\",\"a24\",\"m24\"],[\"m24\",\"b24\",\"f24\"],[\"s24\",null,\"d24\"],[\"s24\",null,\"m24\"]]}"},"fields":{"title":"One Full-Input Final Path Suffices Despite Dead Branches","stem":"Use the complete finite epsilon-NFA: states=[s24, m24, f24, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b24], supplied path=[[s24, 0], [m24, 0], [f24, 1]]. Give (input, path, transition_indices, valid_path, accepted, certificate).","answer":{"kind":"expression","value":"([b24], [[s24, 0], [m24, 0], [f24, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)","explanation":"Require the initial configuration [start,0] and each legal transition/input update, retaining the first matching original transition index. Keep valid_path separate from accepted: premature final states do not accept. A single finite accepting path proves membership; dead alternatives do not veto it. The exact result, retaining the requested order, is ([b24], [[s24, 0], [m24, 0], [f24, 1]], [0, 3], Yes, Yes, one_finite_path_consuming_full_input_and_ending_final_proves_membership)."},"skills":["One Full-Input Final Path Suffices Despite Dead Branches"],"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-20261111-r07-nondeterministic-finite-automaton-0076","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d62461ffa6cb99eebf8de53ed1993a861fa88692260f2cd7c5f3b538bce237c8","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117076,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a0\",\"b0\"],\"final_states\":[\"f0\"],\"input\":[\"b0\"],\"max_steps\":1,\"start_state\":\"s0\",\"states\":[\"d0\",\"f0\",\"m0\",\"s0\"],\"transitions\":[[\"m0\",\"a0\",\"m0\"],[\"m0\",\"b0\",\"f0\"],[\"s0\",null,\"d0\"],[\"s0\",null,\"m0\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s0, m0, f0, d0], alphabet=[a0, b0], start=s0, final_states=[f0], transitions=[[s0, null, m0], [s0, null, d0], [m0, a0, m0], [m0, b0, f0]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b0], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([b0], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([b0], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0077","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"164de753d1087452e4d1d133aee16e6ef620ccf9f17e34eec4988761b1abeb6a","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117077,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a1\",\"b1\"],\"final_states\":[\"f1\"],\"input\":[\"a1\",\"b1\"],\"max_steps\":1,\"start_state\":\"s1\",\"states\":[\"d1\",\"f1\",\"m1\",\"s1\"],\"transitions\":[[\"m1\",\"a1\",\"m1\"],[\"m1\",\"b1\",\"f1\"],[\"s1\",null,\"d1\"],[\"s1\",null,\"m1\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s1, m1, f1, d1], alphabet=[a1, b1], start=s1, final_states=[f1], transitions=[[s1, null, m1], [s1, null, d1], [m1, a1, m1], [m1, b1, f1]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a1, b1], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a1, b1], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a1, b1], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0078","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36cfe3b916b520683b5eb2263afe5a2e0884bc02bba22afebab1f64ee6b773db","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117078,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a2\",\"b2\"],\"final_states\":[\"f2\"],\"input\":[\"a2\",\"a2\",\"b2\"],\"max_steps\":1,\"start_state\":\"s2\",\"states\":[\"d2\",\"f2\",\"m2\",\"s2\"],\"transitions\":[[\"m2\",\"a2\",\"m2\"],[\"m2\",\"b2\",\"f2\"],[\"s2\",null,\"d2\"],[\"s2\",null,\"m2\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s2, m2, f2, d2], alphabet=[a2, b2], start=s2, final_states=[f2], transitions=[[s2, null, m2], [s2, null, d2], [m2, a2, m2], [m2, b2, f2]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a2, a2, b2], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a2, a2, b2], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a2, a2, b2], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0079","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c8efa3ad849b04017510112c1a705bfae2306bfc65e9d2e1de6241b2aaca3ca","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117079,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a3\",\"b3\"],\"final_states\":[\"f3\"],\"input\":[\"a3\",\"a3\",\"a3\",\"b3\"],\"max_steps\":1,\"start_state\":\"s3\",\"states\":[\"d3\",\"f3\",\"m3\",\"s3\"],\"transitions\":[[\"m3\",\"a3\",\"m3\"],[\"m3\",\"b3\",\"f3\"],[\"s3\",null,\"d3\"],[\"s3\",null,\"m3\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s3, m3, f3, d3], alphabet=[a3, b3], start=s3, final_states=[f3], transitions=[[s3, null, m3], [s3, null, d3], [m3, a3, m3], [m3, b3, f3]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a3, a3, a3, b3], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a3, a3, a3, b3], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a3, a3, a3, b3], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0080","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"204c8398b51a1aa9a148cff89f4e8b259c6197601fc0d081b942d48e218f1b5a","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117080,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a4\",\"b4\"],\"final_states\":[\"f4\"],\"input\":[\"b4\"],\"max_steps\":1,\"start_state\":\"s4\",\"states\":[\"d4\",\"f4\",\"m4\",\"s4\"],\"transitions\":[[\"m4\",\"a4\",\"m4\"],[\"m4\",\"b4\",\"f4\"],[\"s4\",null,\"d4\"],[\"s4\",null,\"m4\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s4, m4, f4, d4], alphabet=[a4, b4], start=s4, final_states=[f4], transitions=[[s4, null, m4], [s4, null, d4], [m4, a4, m4], [m4, b4, f4]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b4], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([b4], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([b4], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0081","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65512003d9103398c870426159d7520ef100d73ad601c1a1069564319d74cd4b","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117081,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a5\",\"b5\"],\"final_states\":[\"f5\"],\"input\":[\"a5\",\"b5\"],\"max_steps\":1,\"start_state\":\"s5\",\"states\":[\"d5\",\"f5\",\"m5\",\"s5\"],\"transitions\":[[\"m5\",\"a5\",\"m5\"],[\"m5\",\"b5\",\"f5\"],[\"s5\",null,\"d5\"],[\"s5\",null,\"m5\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s5, m5, f5, d5], alphabet=[a5, b5], start=s5, final_states=[f5], transitions=[[s5, null, m5], [s5, null, d5], [m5, a5, m5], [m5, b5, f5]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a5, b5], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a5, b5], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a5, b5], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0082","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1da0c9f6a677820fc7e1cc67033d2496188d742fd768f4e59ac362bff865bf5","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117082,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a6\",\"b6\"],\"final_states\":[\"f6\"],\"input\":[\"a6\",\"a6\",\"b6\"],\"max_steps\":1,\"start_state\":\"s6\",\"states\":[\"d6\",\"f6\",\"m6\",\"s6\"],\"transitions\":[[\"m6\",\"a6\",\"m6\"],[\"m6\",\"b6\",\"f6\"],[\"s6\",null,\"d6\"],[\"s6\",null,\"m6\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s6, m6, f6, d6], alphabet=[a6, b6], start=s6, final_states=[f6], transitions=[[s6, null, m6], [s6, null, d6], [m6, a6, m6], [m6, b6, f6]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a6, a6, b6], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a6, a6, b6], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a6, a6, b6], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0083","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3f8e10003e7a534d2fbb4e976d66a2ebdc3d7261f12ee62dd4e69e39b7582ad","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117083,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a7\",\"b7\"],\"final_states\":[\"f7\"],\"input\":[\"a7\",\"a7\",\"a7\",\"b7\"],\"max_steps\":1,\"start_state\":\"s7\",\"states\":[\"d7\",\"f7\",\"m7\",\"s7\"],\"transitions\":[[\"m7\",\"a7\",\"m7\"],[\"m7\",\"b7\",\"f7\"],[\"s7\",null,\"d7\"],[\"s7\",null,\"m7\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s7, m7, f7, d7], alphabet=[a7, b7], start=s7, final_states=[f7], transitions=[[s7, null, m7], [s7, null, d7], [m7, a7, m7], [m7, b7, f7]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a7, a7, a7, b7], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a7, a7, a7, b7], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a7, a7, a7, b7], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0084","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03f346c7a853435eea94234c6b6daef6e623e2f4a3cc009a68895b994e218809","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117084,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a8\",\"b8\"],\"final_states\":[\"f8\"],\"input\":[\"b8\"],\"max_steps\":1,\"start_state\":\"s8\",\"states\":[\"d8\",\"f8\",\"m8\",\"s8\"],\"transitions\":[[\"m8\",\"a8\",\"m8\"],[\"m8\",\"b8\",\"f8\"],[\"s8\",null,\"d8\"],[\"s8\",null,\"m8\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s8, m8, f8, d8], alphabet=[a8, b8], start=s8, final_states=[f8], transitions=[[s8, null, m8], [s8, null, d8], [m8, a8, m8], [m8, b8, f8]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b8], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([b8], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([b8], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0085","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9a9a47d616f6ad235590e85934b100ddd4c3c918af2d9608c041ceb1e68a5c2","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117085,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a9\",\"b9\"],\"final_states\":[\"f9\"],\"input\":[\"a9\",\"b9\"],\"max_steps\":1,\"start_state\":\"s9\",\"states\":[\"d9\",\"f9\",\"m9\",\"s9\"],\"transitions\":[[\"m9\",\"a9\",\"m9\"],[\"m9\",\"b9\",\"f9\"],[\"s9\",null,\"d9\"],[\"s9\",null,\"m9\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s9, m9, f9, d9], alphabet=[a9, b9], start=s9, final_states=[f9], transitions=[[s9, null, m9], [s9, null, d9], [m9, a9, m9], [m9, b9, f9]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a9, b9], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a9, b9], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a9, b9], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0086","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da3714c1129642ac9412bdb8466e607caeccfabf022c376d1e70b042d1dcfbc9","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117086,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a10\",\"b10\"],\"final_states\":[\"f10\"],\"input\":[\"a10\",\"a10\",\"b10\"],\"max_steps\":1,\"start_state\":\"s10\",\"states\":[\"d10\",\"f10\",\"m10\",\"s10\"],\"transitions\":[[\"m10\",\"a10\",\"m10\"],[\"m10\",\"b10\",\"f10\"],[\"s10\",null,\"d10\"],[\"s10\",null,\"m10\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s10, m10, f10, d10], alphabet=[a10, b10], start=s10, final_states=[f10], transitions=[[s10, null, m10], [s10, null, d10], [m10, a10, m10], [m10, b10, f10]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a10, a10, b10], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a10, a10, b10], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a10, a10, b10], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0087","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a288c8ca5bc624dbe141a519b62e0d8aa1a95c08e4daa00a22b74240b0972d0","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117087,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a11\",\"b11\"],\"final_states\":[\"f11\"],\"input\":[\"a11\",\"a11\",\"a11\",\"b11\"],\"max_steps\":1,\"start_state\":\"s11\",\"states\":[\"d11\",\"f11\",\"m11\",\"s11\"],\"transitions\":[[\"m11\",\"a11\",\"m11\"],[\"m11\",\"b11\",\"f11\"],[\"s11\",null,\"d11\"],[\"s11\",null,\"m11\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s11, m11, f11, d11], alphabet=[a11, b11], start=s11, final_states=[f11], transitions=[[s11, null, m11], [s11, null, d11], [m11, a11, m11], [m11, b11, f11]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a11, a11, a11, b11], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a11, a11, a11, b11], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a11, a11, a11, b11], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0088","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59968f9b09568215f917ed7c0edb72da3bfac0b81e896506ee1dc74149fdaaf8","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117088,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a12\",\"b12\"],\"final_states\":[\"f12\"],\"input\":[\"b12\"],\"max_steps\":1,\"start_state\":\"s12\",\"states\":[\"d12\",\"f12\",\"m12\",\"s12\"],\"transitions\":[[\"m12\",\"a12\",\"m12\"],[\"m12\",\"b12\",\"f12\"],[\"s12\",null,\"d12\"],[\"s12\",null,\"m12\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s12, m12, f12, d12], alphabet=[a12, b12], start=s12, final_states=[f12], transitions=[[s12, null, m12], [s12, null, d12], [m12, a12, m12], [m12, b12, f12]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b12], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([b12], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([b12], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0089","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b33408cf83a554d7187be7123ad12a9de6e101700eb7213bedf26dfd05449ff6","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117089,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a13\",\"b13\"],\"final_states\":[\"f13\"],\"input\":[\"a13\",\"b13\"],\"max_steps\":1,\"start_state\":\"s13\",\"states\":[\"d13\",\"f13\",\"m13\",\"s13\"],\"transitions\":[[\"m13\",\"a13\",\"m13\"],[\"m13\",\"b13\",\"f13\"],[\"s13\",null,\"d13\"],[\"s13\",null,\"m13\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s13, m13, f13, d13], alphabet=[a13, b13], start=s13, final_states=[f13], transitions=[[s13, null, m13], [s13, null, d13], [m13, a13, m13], [m13, b13, f13]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a13, b13], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a13, b13], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a13, b13], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0090","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e0f4471abb762968b435f2a7161958e81840bdc24bb180930ad924a63f22537","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117090,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a14\",\"b14\"],\"final_states\":[\"f14\"],\"input\":[\"a14\",\"a14\",\"b14\"],\"max_steps\":1,\"start_state\":\"s14\",\"states\":[\"d14\",\"f14\",\"m14\",\"s14\"],\"transitions\":[[\"m14\",\"a14\",\"m14\"],[\"m14\",\"b14\",\"f14\"],[\"s14\",null,\"d14\"],[\"s14\",null,\"m14\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s14, m14, f14, d14], alphabet=[a14, b14], start=s14, final_states=[f14], transitions=[[s14, null, m14], [s14, null, d14], [m14, a14, m14], [m14, b14, f14]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a14, a14, b14], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a14, a14, b14], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a14, a14, b14], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0091","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"389b7eb5e20154353cc60fb23f64e25e429581236e9f2ee6ac3bea4883d604e2","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117091,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a15\",\"b15\"],\"final_states\":[\"f15\"],\"input\":[\"a15\",\"a15\",\"a15\",\"b15\"],\"max_steps\":1,\"start_state\":\"s15\",\"states\":[\"d15\",\"f15\",\"m15\",\"s15\"],\"transitions\":[[\"m15\",\"a15\",\"m15\"],[\"m15\",\"b15\",\"f15\"],[\"s15\",null,\"d15\"],[\"s15\",null,\"m15\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s15, m15, f15, d15], alphabet=[a15, b15], start=s15, final_states=[f15], transitions=[[s15, null, m15], [s15, null, d15], [m15, a15, m15], [m15, b15, f15]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a15, a15, a15, b15], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a15, a15, a15, b15], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a15, a15, a15, b15], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0092","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7428188f1b24c5b920f0c2c340910f723ce7c74528790f52e4e645cb3b8e3eff","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117092,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a16\",\"b16\"],\"final_states\":[\"f16\"],\"input\":[\"b16\"],\"max_steps\":1,\"start_state\":\"s16\",\"states\":[\"d16\",\"f16\",\"m16\",\"s16\"],\"transitions\":[[\"m16\",\"a16\",\"m16\"],[\"m16\",\"b16\",\"f16\"],[\"s16\",null,\"d16\"],[\"s16\",null,\"m16\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s16, m16, f16, d16], alphabet=[a16, b16], start=s16, final_states=[f16], transitions=[[s16, null, m16], [s16, null, d16], [m16, a16, m16], [m16, b16, f16]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b16], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([b16], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([b16], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0093","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a62b223707b90b9406eb90d23d8aea2008da4312c6bca32961c221cd91a6a8af","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117093,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a17\",\"b17\"],\"final_states\":[\"f17\"],\"input\":[\"a17\",\"b17\"],\"max_steps\":1,\"start_state\":\"s17\",\"states\":[\"d17\",\"f17\",\"m17\",\"s17\"],\"transitions\":[[\"m17\",\"a17\",\"m17\"],[\"m17\",\"b17\",\"f17\"],[\"s17\",null,\"d17\"],[\"s17\",null,\"m17\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s17, m17, f17, d17], alphabet=[a17, b17], start=s17, final_states=[f17], transitions=[[s17, null, m17], [s17, null, d17], [m17, a17, m17], [m17, b17, f17]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a17, b17], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a17, b17], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a17, b17], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0094","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07b7cd6f591c506a5e72cb74dcc2e07e03a75b839223177a2096a2adc39d758f","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117094,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a18\",\"b18\"],\"final_states\":[\"f18\"],\"input\":[\"a18\",\"a18\",\"b18\"],\"max_steps\":1,\"start_state\":\"s18\",\"states\":[\"d18\",\"f18\",\"m18\",\"s18\"],\"transitions\":[[\"m18\",\"a18\",\"m18\"],[\"m18\",\"b18\",\"f18\"],[\"s18\",null,\"d18\"],[\"s18\",null,\"m18\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s18, m18, f18, d18], alphabet=[a18, b18], start=s18, final_states=[f18], transitions=[[s18, null, m18], [s18, null, d18], [m18, a18, m18], [m18, b18, f18]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a18, a18, b18], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a18, a18, b18], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a18, a18, b18], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0095","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d03b1dc6b5d205f2a12c9b3dd94e5b6948893aa6278aad0a4fe11d55937b1d82","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117095,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a19\",\"b19\"],\"final_states\":[\"f19\"],\"input\":[\"a19\",\"a19\",\"a19\",\"b19\"],\"max_steps\":1,\"start_state\":\"s19\",\"states\":[\"d19\",\"f19\",\"m19\",\"s19\"],\"transitions\":[[\"m19\",\"a19\",\"m19\"],[\"m19\",\"b19\",\"f19\"],[\"s19\",null,\"d19\"],[\"s19\",null,\"m19\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s19, m19, f19, d19], alphabet=[a19, b19], start=s19, final_states=[f19], transitions=[[s19, null, m19], [s19, null, d19], [m19, a19, m19], [m19, b19, f19]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a19, a19, a19, b19], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a19, a19, a19, b19], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a19, a19, a19, b19], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0096","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5dd7d3cf6f2d421eb260a82f7608e76dfc053bbd3a610598760c05ca8fa9de1","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117096,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a20\",\"b20\"],\"final_states\":[\"f20\"],\"input\":[\"b20\"],\"max_steps\":1,\"start_state\":\"s20\",\"states\":[\"d20\",\"f20\",\"m20\",\"s20\"],\"transitions\":[[\"m20\",\"a20\",\"m20\"],[\"m20\",\"b20\",\"f20\"],[\"s20\",null,\"d20\"],[\"s20\",null,\"m20\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s20, m20, f20, d20], alphabet=[a20, b20], start=s20, final_states=[f20], transitions=[[s20, null, m20], [s20, null, d20], [m20, a20, m20], [m20, b20, f20]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b20], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([b20], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([b20], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0097","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ad30d6343c1b7fab4f78bf87096918e53ec332c82ec5616ce647dfbee8e87ac","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117097,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a21\",\"b21\"],\"final_states\":[\"f21\"],\"input\":[\"a21\",\"b21\"],\"max_steps\":1,\"start_state\":\"s21\",\"states\":[\"d21\",\"f21\",\"m21\",\"s21\"],\"transitions\":[[\"m21\",\"a21\",\"m21\"],[\"m21\",\"b21\",\"f21\"],[\"s21\",null,\"d21\"],[\"s21\",null,\"m21\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s21, m21, f21, d21], alphabet=[a21, b21], start=s21, final_states=[f21], transitions=[[s21, null, m21], [s21, null, d21], [m21, a21, m21], [m21, b21, f21]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a21, b21], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a21, b21], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a21, b21], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0098","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c6f4dee72da71dc4b2ad411c8990c7250d34f611f623b84d7e61aeeb6facbd0","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117098,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a22\",\"b22\"],\"final_states\":[\"f22\"],\"input\":[\"a22\",\"a22\",\"b22\"],\"max_steps\":1,\"start_state\":\"s22\",\"states\":[\"d22\",\"f22\",\"m22\",\"s22\"],\"transitions\":[[\"m22\",\"a22\",\"m22\"],[\"m22\",\"b22\",\"f22\"],[\"s22\",null,\"d22\"],[\"s22\",null,\"m22\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s22, m22, f22, d22], alphabet=[a22, b22], start=s22, final_states=[f22], transitions=[[s22, null, m22], [s22, null, d22], [m22, a22, m22], [m22, b22, f22]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a22, a22, b22], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a22, a22, b22], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a22, a22, b22], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0099","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ce54305ec756385ab943138e59a942323accac7729ea5395eebd549a838ac8a","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117099,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a23\",\"b23\"],\"final_states\":[\"f23\"],\"input\":[\"a23\",\"a23\",\"a23\",\"b23\"],\"max_steps\":1,\"start_state\":\"s23\",\"states\":[\"d23\",\"f23\",\"m23\",\"s23\"],\"transitions\":[[\"m23\",\"a23\",\"m23\"],[\"m23\",\"b23\",\"f23\"],[\"s23\",null,\"d23\"],[\"s23\",null,\"m23\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s23, m23, f23, d23], alphabet=[a23, b23], start=s23, final_states=[f23], transitions=[[s23, null, m23], [s23, null, d23], [m23, a23, m23], [m23, b23, f23]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[a23, a23, a23, b23], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([a23, a23, a23, b23], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([a23, a23, a23, b23], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r07-nondeterministic-finite-automaton-0100","batch_id":"batch-20261014-advance-20261111-r07-nondeterministic-finite-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59677f7358a7d1b2ad15aaa142601bef948f7530dd4fd82f5523f9f861f5eb26","family_id":"qf-mod42-nfa-bounded-search-boundary","seed":202611117100,"semantic_template":"qf-mod42-nfa-bounded-search-boundary/{\"alphabet\":[\"a24\",\"b24\"],\"final_states\":[\"f24\"],\"input\":[\"b24\"],\"max_steps\":1,\"start_state\":\"s24\",\"states\":[\"d24\",\"f24\",\"m24\",\"s24\"],\"transitions\":[[\"m24\",\"a24\",\"m24\"],[\"m24\",\"b24\",\"f24\"],[\"s24\",null,\"d24\"],[\"s24\",null,\"m24\"]]}"},"fields":{"title":"A Depth-Limited No-Hit Is Not an NFA Rejection Proof","stem":"Use the complete finite epsilon-NFA: states=[s24, m24, f24, d24], alphabet=[a24, b24], start=s24, final_states=[f24], transitions=[[s24, null, m24], [s24, null, d24], [m24, a24, m24], [m24, b24, f24]]. Transitions are [from,input symbol or null,to]; null is epsilon and consumes no input. States are nonempty because the start belongs to them, but alphabet/final sets and the transition relation may be empty. The relation may be multivalued, not deterministic or total. A configuration is [state,input position]; acceptance requires one finite path consuming the entire input and ending final. Input=[b24], max_steps=1. Give (input, max_steps, found_accepting_path, rejection_claim, boundary).","answer":{"kind":"expression","value":"([b24], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)","explanation":"Search only the declared number of transitions. Report the found flag separately from the fixed No rejection nonclaim. A longer finite accepting path may exist. Complete finite-configuration reachability is a separate decision procedure; an epsilon cycle alone is not an infinite accepting run. The exact result, retaining the requested order, is ([b24], 1, No, No, bounded_search_failure_cannot_prove_rejection_and_epsilon_cycles_are_not_infinite_accepting_runs)."},"skills":["A Depth-Limited No-Hit Is Not an NFA Rejection Proof"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}