{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0001","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"432e40be791ad05fc6fd266f4b8b111c3574af73296e0781b804c900b28bb39f","family_id":"qf-mod41-pda-qualification","seed":202611107001,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z0\",\"final_states\":[\"accept0\"],\"input_alphabet\":[\"a0\",\"b0\"],\"stack_alphabet\":[\"A0\",\"Z0\"],\"start_state\":\"push0\",\"states\":[\"accept0\",\"pop0\",\"push0\"],\"transitions\":[[\"pop0\",\"b0\",\"A0\",\"pop0\",[]],[\"pop0\",null,\"Z0\",\"accept0\",[\"Z0\"]],[\"push0\",\"a0\",\"A0\",\"push0\",[\"A0\",\"A0\"]],[\"push0\",\"a0\",\"Z0\",\"push0\",[\"Z0\",\"A0\"]],[\"push0\",\"b0\",\"A0\",\"pop0\",[]],[\"push0\",null,\"Z0\",\"accept0\",[\"Z0\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push0, pop0, accept0], input_alphabet=[a0, b0], stack_alphabet=[Z0, A0], start_state=push0, bottom_symbol=Z0, final_states=[accept0], transitions=[[push0, a0, Z0, push0, [Z0, A0]], [push0, a0, A0, push0, [A0, A0]], [push0, b0, A0, pop0, []], [pop0, b0, A0, pop0, []], [push0, null, Z0, accept0, [Z0]], [pop0, null, Z0, accept0, [Z0]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push0, pop0, accept0], [a0, b0], [Z0, A0], push0, Z0, [accept0], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push0, pop0, accept0], [a0, b0], [Z0, A0], push0, Z0, [accept0], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0002","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea23e38a748e7d4fc2207a7315379342c38266a8e44b0e7b1769187d2617b035","family_id":"qf-mod41-pda-qualification","seed":202611107002,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z1\",\"final_states\":[\"accept1\"],\"input_alphabet\":[\"a1\",\"b1\"],\"stack_alphabet\":[\"A1\",\"Z1\"],\"start_state\":\"push1\",\"states\":[\"accept1\",\"pop1\",\"push1\"],\"transitions\":[[\"pop1\",\"b1\",\"A1\",\"pop1\",[]],[\"pop1\",null,\"Z1\",\"accept1\",[\"Z1\"]],[\"push1\",\"a1\",\"A1\",\"push1\",[\"A1\",\"A1\"]],[\"push1\",\"a1\",\"Z1\",\"push1\",[\"Z1\",\"A1\"]],[\"push1\",\"b1\",\"A1\",\"pop1\",[]],[\"push1\",null,\"Z1\",\"accept1\",[\"Z1\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push1, pop1, accept1], input_alphabet=[a1, b1], stack_alphabet=[Z1, A1], start_state=push1, bottom_symbol=Z1, final_states=[accept1], transitions=[[push1, a1, Z1, push1, [Z1, A1]], [push1, a1, A1, push1, [A1, A1]], [push1, b1, A1, pop1, []], [pop1, b1, A1, pop1, []], [push1, null, Z1, accept1, [Z1]], [pop1, null, Z1, accept1, [Z1]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push1, pop1, accept1], [a1, b1], [Z1, A1], push1, Z1, [accept1], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push1, pop1, accept1], [a1, b1], [Z1, A1], push1, Z1, [accept1], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0003","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa6e6d07ac8b2c7336efb011e84a22751f62f7ef96b0f6e4fd5331fc3aec23c6","family_id":"qf-mod41-pda-qualification","seed":202611107003,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z2\",\"final_states\":[\"accept2\"],\"input_alphabet\":[\"a2\",\"b2\"],\"stack_alphabet\":[\"A2\",\"Z2\"],\"start_state\":\"push2\",\"states\":[\"accept2\",\"pop2\",\"push2\"],\"transitions\":[[\"pop2\",\"b2\",\"A2\",\"pop2\",[]],[\"pop2\",null,\"Z2\",\"accept2\",[\"Z2\"]],[\"push2\",\"a2\",\"A2\",\"push2\",[\"A2\",\"A2\"]],[\"push2\",\"a2\",\"Z2\",\"push2\",[\"Z2\",\"A2\"]],[\"push2\",\"b2\",\"A2\",\"pop2\",[]],[\"push2\",null,\"Z2\",\"accept2\",[\"Z2\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push2, pop2, accept2], input_alphabet=[a2, b2], stack_alphabet=[Z2, A2], start_state=push2, bottom_symbol=Z2, final_states=[accept2], transitions=[[push2, a2, Z2, push2, [Z2, A2]], [push2, a2, A2, push2, [A2, A2]], [push2, b2, A2, pop2, []], [pop2, b2, A2, pop2, []], [push2, null, Z2, accept2, [Z2]], [pop2, null, Z2, accept2, [Z2]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push2, pop2, accept2], [a2, b2], [Z2, A2], push2, Z2, [accept2], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push2, pop2, accept2], [a2, b2], [Z2, A2], push2, Z2, [accept2], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0004","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cadeff7df209b4d91f783d224a764fff7c5a5a5782635ceea01214954e188222","family_id":"qf-mod41-pda-qualification","seed":202611107004,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z3\",\"final_states\":[\"accept3\"],\"input_alphabet\":[\"a3\",\"b3\"],\"stack_alphabet\":[\"A3\",\"Z3\"],\"start_state\":\"push3\",\"states\":[\"accept3\",\"pop3\",\"push3\"],\"transitions\":[[\"pop3\",\"b3\",\"A3\",\"pop3\",[]],[\"pop3\",null,\"Z3\",\"accept3\",[\"Z3\"]],[\"push3\",\"a3\",\"A3\",\"push3\",[\"A3\",\"A3\"]],[\"push3\",\"a3\",\"Z3\",\"push3\",[\"Z3\",\"A3\"]],[\"push3\",\"b3\",\"A3\",\"pop3\",[]],[\"push3\",null,\"Z3\",\"accept3\",[\"Z3\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push3, pop3, accept3], input_alphabet=[a3, b3], stack_alphabet=[Z3, A3], start_state=push3, bottom_symbol=Z3, final_states=[accept3], transitions=[[push3, a3, Z3, push3, [Z3, A3]], [push3, a3, A3, push3, [A3, A3]], [push3, b3, A3, pop3, []], [pop3, b3, A3, pop3, []], [push3, null, Z3, accept3, [Z3]], [pop3, null, Z3, accept3, [Z3]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push3, pop3, accept3], [a3, b3], [Z3, A3], push3, Z3, [accept3], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push3, pop3, accept3], [a3, b3], [Z3, A3], push3, Z3, [accept3], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0005","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e80eb9a88f99b0cb84bc2124c5552c807ee57c89bab1d163936626754e98df7","family_id":"qf-mod41-pda-qualification","seed":202611107005,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z4\",\"final_states\":[\"accept4\"],\"input_alphabet\":[\"a4\",\"b4\"],\"stack_alphabet\":[\"A4\",\"Z4\"],\"start_state\":\"push4\",\"states\":[\"accept4\",\"pop4\",\"push4\"],\"transitions\":[[\"pop4\",\"b4\",\"A4\",\"pop4\",[]],[\"pop4\",null,\"Z4\",\"accept4\",[\"Z4\"]],[\"push4\",\"a4\",\"A4\",\"push4\",[\"A4\",\"A4\"]],[\"push4\",\"a4\",\"Z4\",\"push4\",[\"Z4\",\"A4\"]],[\"push4\",\"b4\",\"A4\",\"pop4\",[]],[\"push4\",null,\"Z4\",\"accept4\",[\"Z4\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push4, pop4, accept4], input_alphabet=[a4, b4], stack_alphabet=[Z4, A4], start_state=push4, bottom_symbol=Z4, final_states=[accept4], transitions=[[push4, a4, Z4, push4, [Z4, A4]], [push4, a4, A4, push4, [A4, A4]], [push4, b4, A4, pop4, []], [pop4, b4, A4, pop4, []], [push4, null, Z4, accept4, [Z4]], [pop4, null, Z4, accept4, [Z4]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push4, pop4, accept4], [a4, b4], [Z4, A4], push4, Z4, [accept4], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push4, pop4, accept4], [a4, b4], [Z4, A4], push4, Z4, [accept4], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0006","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fe347cdd2cd46b8a26b7a73ba7c92f1f41d0adf2608c25315592bc3250db3f6","family_id":"qf-mod41-pda-qualification","seed":202611107006,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z5\",\"final_states\":[\"accept5\"],\"input_alphabet\":[\"a5\",\"b5\"],\"stack_alphabet\":[\"A5\",\"Z5\"],\"start_state\":\"push5\",\"states\":[\"accept5\",\"pop5\",\"push5\"],\"transitions\":[[\"pop5\",\"b5\",\"A5\",\"pop5\",[]],[\"pop5\",null,\"Z5\",\"accept5\",[\"Z5\"]],[\"push5\",\"a5\",\"A5\",\"push5\",[\"A5\",\"A5\"]],[\"push5\",\"a5\",\"Z5\",\"push5\",[\"Z5\",\"A5\"]],[\"push5\",\"b5\",\"A5\",\"pop5\",[]],[\"push5\",null,\"Z5\",\"accept5\",[\"Z5\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push5, pop5, accept5], input_alphabet=[a5, b5], stack_alphabet=[Z5, A5], start_state=push5, bottom_symbol=Z5, final_states=[accept5], transitions=[[push5, a5, Z5, push5, [Z5, A5]], [push5, a5, A5, push5, [A5, A5]], [push5, b5, A5, pop5, []], [pop5, b5, A5, pop5, []], [push5, null, Z5, accept5, [Z5]], [pop5, null, Z5, accept5, [Z5]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push5, pop5, accept5], [a5, b5], [Z5, A5], push5, Z5, [accept5], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push5, pop5, accept5], [a5, b5], [Z5, A5], push5, Z5, [accept5], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0007","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5b948152735ac15fc6a16870df09ad10595f1ca63d0917c5d1f70dc48230f17","family_id":"qf-mod41-pda-qualification","seed":202611107007,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z6\",\"final_states\":[\"accept6\"],\"input_alphabet\":[\"a6\",\"b6\"],\"stack_alphabet\":[\"A6\",\"Z6\"],\"start_state\":\"push6\",\"states\":[\"accept6\",\"pop6\",\"push6\"],\"transitions\":[[\"pop6\",\"b6\",\"A6\",\"pop6\",[]],[\"pop6\",null,\"Z6\",\"accept6\",[\"Z6\"]],[\"push6\",\"a6\",\"A6\",\"push6\",[\"A6\",\"A6\"]],[\"push6\",\"a6\",\"Z6\",\"push6\",[\"Z6\",\"A6\"]],[\"push6\",\"b6\",\"A6\",\"pop6\",[]],[\"push6\",null,\"Z6\",\"accept6\",[\"Z6\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push6, pop6, accept6], input_alphabet=[a6, b6], stack_alphabet=[Z6, A6], start_state=push6, bottom_symbol=Z6, final_states=[accept6], transitions=[[push6, a6, Z6, push6, [Z6, A6]], [push6, a6, A6, push6, [A6, A6]], [push6, b6, A6, pop6, []], [pop6, b6, A6, pop6, []], [push6, null, Z6, accept6, [Z6]], [pop6, null, Z6, accept6, [Z6]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push6, pop6, accept6], [a6, b6], [Z6, A6], push6, Z6, [accept6], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push6, pop6, accept6], [a6, b6], [Z6, A6], push6, Z6, [accept6], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0008","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d763675ff8b706be22cb1dda807e6d3118777b5dfe5d609a07d41fed573fdedb","family_id":"qf-mod41-pda-qualification","seed":202611107008,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z7\",\"final_states\":[\"accept7\"],\"input_alphabet\":[\"a7\",\"b7\"],\"stack_alphabet\":[\"A7\",\"Z7\"],\"start_state\":\"push7\",\"states\":[\"accept7\",\"pop7\",\"push7\"],\"transitions\":[[\"pop7\",\"b7\",\"A7\",\"pop7\",[]],[\"pop7\",null,\"Z7\",\"accept7\",[\"Z7\"]],[\"push7\",\"a7\",\"A7\",\"push7\",[\"A7\",\"A7\"]],[\"push7\",\"a7\",\"Z7\",\"push7\",[\"Z7\",\"A7\"]],[\"push7\",\"b7\",\"A7\",\"pop7\",[]],[\"push7\",null,\"Z7\",\"accept7\",[\"Z7\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push7, pop7, accept7], input_alphabet=[a7, b7], stack_alphabet=[Z7, A7], start_state=push7, bottom_symbol=Z7, final_states=[accept7], transitions=[[push7, a7, Z7, push7, [Z7, A7]], [push7, a7, A7, push7, [A7, A7]], [push7, b7, A7, pop7, []], [pop7, b7, A7, pop7, []], [push7, null, Z7, accept7, [Z7]], [pop7, null, Z7, accept7, [Z7]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push7, pop7, accept7], [a7, b7], [Z7, A7], push7, Z7, [accept7], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push7, pop7, accept7], [a7, b7], [Z7, A7], push7, Z7, [accept7], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0009","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"225367f6442925de0abf448aa71e381d89a30abf4125f693f427808b9207da6a","family_id":"qf-mod41-pda-qualification","seed":202611107009,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z8\",\"final_states\":[\"accept8\"],\"input_alphabet\":[\"a8\",\"b8\"],\"stack_alphabet\":[\"A8\",\"Z8\"],\"start_state\":\"push8\",\"states\":[\"accept8\",\"pop8\",\"push8\"],\"transitions\":[[\"pop8\",\"b8\",\"A8\",\"pop8\",[]],[\"pop8\",null,\"Z8\",\"accept8\",[\"Z8\"]],[\"push8\",\"a8\",\"A8\",\"push8\",[\"A8\",\"A8\"]],[\"push8\",\"a8\",\"Z8\",\"push8\",[\"Z8\",\"A8\"]],[\"push8\",\"b8\",\"A8\",\"pop8\",[]],[\"push8\",null,\"Z8\",\"accept8\",[\"Z8\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push8, pop8, accept8], input_alphabet=[a8, b8], stack_alphabet=[Z8, A8], start_state=push8, bottom_symbol=Z8, final_states=[accept8], transitions=[[push8, a8, Z8, push8, [Z8, A8]], [push8, a8, A8, push8, [A8, A8]], [push8, b8, A8, pop8, []], [pop8, b8, A8, pop8, []], [push8, null, Z8, accept8, [Z8]], [pop8, null, Z8, accept8, [Z8]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push8, pop8, accept8], [a8, b8], [Z8, A8], push8, Z8, [accept8], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push8, pop8, accept8], [a8, b8], [Z8, A8], push8, Z8, [accept8], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0010","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e2d2c5f5ba4be86c333296fc3c4fdcdd6edeaf8e5d35dc567e601c28164a3d1","family_id":"qf-mod41-pda-qualification","seed":202611107010,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z9\",\"final_states\":[\"accept9\"],\"input_alphabet\":[\"a9\",\"b9\"],\"stack_alphabet\":[\"A9\",\"Z9\"],\"start_state\":\"push9\",\"states\":[\"accept9\",\"pop9\",\"push9\"],\"transitions\":[[\"pop9\",\"b9\",\"A9\",\"pop9\",[]],[\"pop9\",null,\"Z9\",\"accept9\",[\"Z9\"]],[\"push9\",\"a9\",\"A9\",\"push9\",[\"A9\",\"A9\"]],[\"push9\",\"a9\",\"Z9\",\"push9\",[\"Z9\",\"A9\"]],[\"push9\",\"b9\",\"A9\",\"pop9\",[]],[\"push9\",null,\"Z9\",\"accept9\",[\"Z9\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push9, pop9, accept9], input_alphabet=[a9, b9], stack_alphabet=[Z9, A9], start_state=push9, bottom_symbol=Z9, final_states=[accept9], transitions=[[push9, a9, Z9, push9, [Z9, A9]], [push9, a9, A9, push9, [A9, A9]], [push9, b9, A9, pop9, []], [pop9, b9, A9, pop9, []], [push9, null, Z9, accept9, [Z9]], [pop9, null, Z9, accept9, [Z9]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push9, pop9, accept9], [a9, b9], [Z9, A9], push9, Z9, [accept9], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push9, pop9, accept9], [a9, b9], [Z9, A9], push9, Z9, [accept9], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0011","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9c329c45c435a81fad5443553456872451f78ed8c874577a59803dbd6cbb71e","family_id":"qf-mod41-pda-qualification","seed":202611107011,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z10\",\"final_states\":[\"accept10\"],\"input_alphabet\":[\"a10\",\"b10\"],\"stack_alphabet\":[\"A10\",\"Z10\"],\"start_state\":\"push10\",\"states\":[\"accept10\",\"pop10\",\"push10\"],\"transitions\":[[\"pop10\",\"b10\",\"A10\",\"pop10\",[]],[\"pop10\",null,\"Z10\",\"accept10\",[\"Z10\"]],[\"push10\",\"a10\",\"A10\",\"push10\",[\"A10\",\"A10\"]],[\"push10\",\"a10\",\"Z10\",\"push10\",[\"Z10\",\"A10\"]],[\"push10\",\"b10\",\"A10\",\"pop10\",[]],[\"push10\",null,\"Z10\",\"accept10\",[\"Z10\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push10, pop10, accept10], input_alphabet=[a10, b10], stack_alphabet=[Z10, A10], start_state=push10, bottom_symbol=Z10, final_states=[accept10], transitions=[[push10, a10, Z10, push10, [Z10, A10]], [push10, a10, A10, push10, [A10, A10]], [push10, b10, A10, pop10, []], [pop10, b10, A10, pop10, []], [push10, null, Z10, accept10, [Z10]], [pop10, null, Z10, accept10, [Z10]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push10, pop10, accept10], [a10, b10], [Z10, A10], push10, Z10, [accept10], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push10, pop10, accept10], [a10, b10], [Z10, A10], push10, Z10, [accept10], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0012","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f9099903408981f2d4d8736911d1ba903dcb0c37308f4190eaddb6862a69e30","family_id":"qf-mod41-pda-qualification","seed":202611107012,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z11\",\"final_states\":[\"accept11\"],\"input_alphabet\":[\"a11\",\"b11\"],\"stack_alphabet\":[\"A11\",\"Z11\"],\"start_state\":\"push11\",\"states\":[\"accept11\",\"pop11\",\"push11\"],\"transitions\":[[\"pop11\",\"b11\",\"A11\",\"pop11\",[]],[\"pop11\",null,\"Z11\",\"accept11\",[\"Z11\"]],[\"push11\",\"a11\",\"A11\",\"push11\",[\"A11\",\"A11\"]],[\"push11\",\"a11\",\"Z11\",\"push11\",[\"Z11\",\"A11\"]],[\"push11\",\"b11\",\"A11\",\"pop11\",[]],[\"push11\",null,\"Z11\",\"accept11\",[\"Z11\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push11, pop11, accept11], input_alphabet=[a11, b11], stack_alphabet=[Z11, A11], start_state=push11, bottom_symbol=Z11, final_states=[accept11], transitions=[[push11, a11, Z11, push11, [Z11, A11]], [push11, a11, A11, push11, [A11, A11]], [push11, b11, A11, pop11, []], [pop11, b11, A11, pop11, []], [push11, null, Z11, accept11, [Z11]], [pop11, null, Z11, accept11, [Z11]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push11, pop11, accept11], [a11, b11], [Z11, A11], push11, Z11, [accept11], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push11, pop11, accept11], [a11, b11], [Z11, A11], push11, Z11, [accept11], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0013","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7de5cd716f8ea05e991899cfa9e2f8590355993e37de257d65f6bd9f6697a4a0","family_id":"qf-mod41-pda-qualification","seed":202611107013,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z12\",\"final_states\":[\"accept12\"],\"input_alphabet\":[\"a12\",\"b12\"],\"stack_alphabet\":[\"A12\",\"Z12\"],\"start_state\":\"push12\",\"states\":[\"accept12\",\"pop12\",\"push12\"],\"transitions\":[[\"pop12\",\"b12\",\"A12\",\"pop12\",[]],[\"pop12\",null,\"Z12\",\"accept12\",[\"Z12\"]],[\"push12\",\"a12\",\"A12\",\"push12\",[\"A12\",\"A12\"]],[\"push12\",\"a12\",\"Z12\",\"push12\",[\"Z12\",\"A12\"]],[\"push12\",\"b12\",\"A12\",\"pop12\",[]],[\"push12\",null,\"Z12\",\"accept12\",[\"Z12\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push12, pop12, accept12], input_alphabet=[a12, b12], stack_alphabet=[Z12, A12], start_state=push12, bottom_symbol=Z12, final_states=[accept12], transitions=[[push12, a12, Z12, push12, [Z12, A12]], [push12, a12, A12, push12, [A12, A12]], [push12, b12, A12, pop12, []], [pop12, b12, A12, pop12, []], [push12, null, Z12, accept12, [Z12]], [pop12, null, Z12, accept12, [Z12]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push12, pop12, accept12], [a12, b12], [Z12, A12], push12, Z12, [accept12], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push12, pop12, accept12], [a12, b12], [Z12, A12], push12, Z12, [accept12], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0014","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a593f03c5033829bff43db974d628ff0808029b3c3f772af4de89489cccb76cc","family_id":"qf-mod41-pda-qualification","seed":202611107014,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z13\",\"final_states\":[\"accept13\"],\"input_alphabet\":[\"a13\",\"b13\"],\"stack_alphabet\":[\"A13\",\"Z13\"],\"start_state\":\"push13\",\"states\":[\"accept13\",\"pop13\",\"push13\"],\"transitions\":[[\"pop13\",\"b13\",\"A13\",\"pop13\",[]],[\"pop13\",null,\"Z13\",\"accept13\",[\"Z13\"]],[\"push13\",\"a13\",\"A13\",\"push13\",[\"A13\",\"A13\"]],[\"push13\",\"a13\",\"Z13\",\"push13\",[\"Z13\",\"A13\"]],[\"push13\",\"b13\",\"A13\",\"pop13\",[]],[\"push13\",null,\"Z13\",\"accept13\",[\"Z13\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push13, pop13, accept13], input_alphabet=[a13, b13], stack_alphabet=[Z13, A13], start_state=push13, bottom_symbol=Z13, final_states=[accept13], transitions=[[push13, a13, Z13, push13, [Z13, A13]], [push13, a13, A13, push13, [A13, A13]], [push13, b13, A13, pop13, []], [pop13, b13, A13, pop13, []], [push13, null, Z13, accept13, [Z13]], [pop13, null, Z13, accept13, [Z13]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push13, pop13, accept13], [a13, b13], [Z13, A13], push13, Z13, [accept13], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push13, pop13, accept13], [a13, b13], [Z13, A13], push13, Z13, [accept13], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0015","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bba5c58ce8033f1487c0faf8339d3fdf9a556fcb7f8316f8281673f87ff586a4","family_id":"qf-mod41-pda-qualification","seed":202611107015,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z14\",\"final_states\":[\"accept14\"],\"input_alphabet\":[\"a14\",\"b14\"],\"stack_alphabet\":[\"A14\",\"Z14\"],\"start_state\":\"push14\",\"states\":[\"accept14\",\"pop14\",\"push14\"],\"transitions\":[[\"pop14\",\"b14\",\"A14\",\"pop14\",[]],[\"pop14\",null,\"Z14\",\"accept14\",[\"Z14\"]],[\"push14\",\"a14\",\"A14\",\"push14\",[\"A14\",\"A14\"]],[\"push14\",\"a14\",\"Z14\",\"push14\",[\"Z14\",\"A14\"]],[\"push14\",\"b14\",\"A14\",\"pop14\",[]],[\"push14\",null,\"Z14\",\"accept14\",[\"Z14\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push14, pop14, accept14], input_alphabet=[a14, b14], stack_alphabet=[Z14, A14], start_state=push14, bottom_symbol=Z14, final_states=[accept14], transitions=[[push14, a14, Z14, push14, [Z14, A14]], [push14, a14, A14, push14, [A14, A14]], [push14, b14, A14, pop14, []], [pop14, b14, A14, pop14, []], [push14, null, Z14, accept14, [Z14]], [pop14, null, Z14, accept14, [Z14]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push14, pop14, accept14], [a14, b14], [Z14, A14], push14, Z14, [accept14], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push14, pop14, accept14], [a14, b14], [Z14, A14], push14, Z14, [accept14], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0016","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d8c05b5b84c280dc5445c90faf55ac8129497d7ad29cf6baaf36b2867d5077a","family_id":"qf-mod41-pda-qualification","seed":202611107016,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z15\",\"final_states\":[\"accept15\"],\"input_alphabet\":[\"a15\",\"b15\"],\"stack_alphabet\":[\"A15\",\"Z15\"],\"start_state\":\"push15\",\"states\":[\"accept15\",\"pop15\",\"push15\"],\"transitions\":[[\"pop15\",\"b15\",\"A15\",\"pop15\",[]],[\"pop15\",null,\"Z15\",\"accept15\",[\"Z15\"]],[\"push15\",\"a15\",\"A15\",\"push15\",[\"A15\",\"A15\"]],[\"push15\",\"a15\",\"Z15\",\"push15\",[\"Z15\",\"A15\"]],[\"push15\",\"b15\",\"A15\",\"pop15\",[]],[\"push15\",null,\"Z15\",\"accept15\",[\"Z15\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push15, pop15, accept15], input_alphabet=[a15, b15], stack_alphabet=[Z15, A15], start_state=push15, bottom_symbol=Z15, final_states=[accept15], transitions=[[push15, a15, Z15, push15, [Z15, A15]], [push15, a15, A15, push15, [A15, A15]], [push15, b15, A15, pop15, []], [pop15, b15, A15, pop15, []], [push15, null, Z15, accept15, [Z15]], [pop15, null, Z15, accept15, [Z15]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push15, pop15, accept15], [a15, b15], [Z15, A15], push15, Z15, [accept15], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push15, pop15, accept15], [a15, b15], [Z15, A15], push15, Z15, [accept15], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0017","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92d23816c19ac1969c7c7a764103a5f40ec8ff3ff8a5ede6f9f3162cb39fe6c3","family_id":"qf-mod41-pda-qualification","seed":202611107017,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z16\",\"final_states\":[\"accept16\"],\"input_alphabet\":[\"a16\",\"b16\"],\"stack_alphabet\":[\"A16\",\"Z16\"],\"start_state\":\"push16\",\"states\":[\"accept16\",\"pop16\",\"push16\"],\"transitions\":[[\"pop16\",\"b16\",\"A16\",\"pop16\",[]],[\"pop16\",null,\"Z16\",\"accept16\",[\"Z16\"]],[\"push16\",\"a16\",\"A16\",\"push16\",[\"A16\",\"A16\"]],[\"push16\",\"a16\",\"Z16\",\"push16\",[\"Z16\",\"A16\"]],[\"push16\",\"b16\",\"A16\",\"pop16\",[]],[\"push16\",null,\"Z16\",\"accept16\",[\"Z16\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push16, pop16, accept16], input_alphabet=[a16, b16], stack_alphabet=[Z16, A16], start_state=push16, bottom_symbol=Z16, final_states=[accept16], transitions=[[push16, a16, Z16, push16, [Z16, A16]], [push16, a16, A16, push16, [A16, A16]], [push16, b16, A16, pop16, []], [pop16, b16, A16, pop16, []], [push16, null, Z16, accept16, [Z16]], [pop16, null, Z16, accept16, [Z16]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push16, pop16, accept16], [a16, b16], [Z16, A16], push16, Z16, [accept16], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push16, pop16, accept16], [a16, b16], [Z16, A16], push16, Z16, [accept16], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0018","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d51b2b7405ddc74b60e8edc98267988a12e947356cbbbd3c2d88ab25d6fc6316","family_id":"qf-mod41-pda-qualification","seed":202611107018,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z17\",\"final_states\":[\"accept17\"],\"input_alphabet\":[\"a17\",\"b17\"],\"stack_alphabet\":[\"A17\",\"Z17\"],\"start_state\":\"push17\",\"states\":[\"accept17\",\"pop17\",\"push17\"],\"transitions\":[[\"pop17\",\"b17\",\"A17\",\"pop17\",[]],[\"pop17\",null,\"Z17\",\"accept17\",[\"Z17\"]],[\"push17\",\"a17\",\"A17\",\"push17\",[\"A17\",\"A17\"]],[\"push17\",\"a17\",\"Z17\",\"push17\",[\"Z17\",\"A17\"]],[\"push17\",\"b17\",\"A17\",\"pop17\",[]],[\"push17\",null,\"Z17\",\"accept17\",[\"Z17\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push17, pop17, accept17], input_alphabet=[a17, b17], stack_alphabet=[Z17, A17], start_state=push17, bottom_symbol=Z17, final_states=[accept17], transitions=[[push17, a17, Z17, push17, [Z17, A17]], [push17, a17, A17, push17, [A17, A17]], [push17, b17, A17, pop17, []], [pop17, b17, A17, pop17, []], [push17, null, Z17, accept17, [Z17]], [pop17, null, Z17, accept17, [Z17]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push17, pop17, accept17], [a17, b17], [Z17, A17], push17, Z17, [accept17], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push17, pop17, accept17], [a17, b17], [Z17, A17], push17, Z17, [accept17], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0019","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e50b305d556193631cd3e94dbae194004afad3e64e43d83267ac41eaa24bf56","family_id":"qf-mod41-pda-qualification","seed":202611107019,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z18\",\"final_states\":[\"accept18\"],\"input_alphabet\":[\"a18\",\"b18\"],\"stack_alphabet\":[\"A18\",\"Z18\"],\"start_state\":\"push18\",\"states\":[\"accept18\",\"pop18\",\"push18\"],\"transitions\":[[\"pop18\",\"b18\",\"A18\",\"pop18\",[]],[\"pop18\",null,\"Z18\",\"accept18\",[\"Z18\"]],[\"push18\",\"a18\",\"A18\",\"push18\",[\"A18\",\"A18\"]],[\"push18\",\"a18\",\"Z18\",\"push18\",[\"Z18\",\"A18\"]],[\"push18\",\"b18\",\"A18\",\"pop18\",[]],[\"push18\",null,\"Z18\",\"accept18\",[\"Z18\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push18, pop18, accept18], input_alphabet=[a18, b18], stack_alphabet=[Z18, A18], start_state=push18, bottom_symbol=Z18, final_states=[accept18], transitions=[[push18, a18, Z18, push18, [Z18, A18]], [push18, a18, A18, push18, [A18, A18]], [push18, b18, A18, pop18, []], [pop18, b18, A18, pop18, []], [push18, null, Z18, accept18, [Z18]], [pop18, null, Z18, accept18, [Z18]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push18, pop18, accept18], [a18, b18], [Z18, A18], push18, Z18, [accept18], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push18, pop18, accept18], [a18, b18], [Z18, A18], push18, Z18, [accept18], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0020","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a63d1b4485448b04ffcb65b504d0c4b4ec2e324e6b32038a683810b7e4443af1","family_id":"qf-mod41-pda-qualification","seed":202611107020,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z19\",\"final_states\":[\"accept19\"],\"input_alphabet\":[\"a19\",\"b19\"],\"stack_alphabet\":[\"A19\",\"Z19\"],\"start_state\":\"push19\",\"states\":[\"accept19\",\"pop19\",\"push19\"],\"transitions\":[[\"pop19\",\"b19\",\"A19\",\"pop19\",[]],[\"pop19\",null,\"Z19\",\"accept19\",[\"Z19\"]],[\"push19\",\"a19\",\"A19\",\"push19\",[\"A19\",\"A19\"]],[\"push19\",\"a19\",\"Z19\",\"push19\",[\"Z19\",\"A19\"]],[\"push19\",\"b19\",\"A19\",\"pop19\",[]],[\"push19\",null,\"Z19\",\"accept19\",[\"Z19\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push19, pop19, accept19], input_alphabet=[a19, b19], stack_alphabet=[Z19, A19], start_state=push19, bottom_symbol=Z19, final_states=[accept19], transitions=[[push19, a19, Z19, push19, [Z19, A19]], [push19, a19, A19, push19, [A19, A19]], [push19, b19, A19, pop19, []], [pop19, b19, A19, pop19, []], [push19, null, Z19, accept19, [Z19]], [pop19, null, Z19, accept19, [Z19]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push19, pop19, accept19], [a19, b19], [Z19, A19], push19, Z19, [accept19], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push19, pop19, accept19], [a19, b19], [Z19, A19], push19, Z19, [accept19], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0021","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68be3fc5554e2c1a1c40152fa250072970c7e6c491ebc5b445c894fa16b5581d","family_id":"qf-mod41-pda-qualification","seed":202611107021,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z20\",\"final_states\":[\"accept20\"],\"input_alphabet\":[\"a20\",\"b20\"],\"stack_alphabet\":[\"A20\",\"Z20\"],\"start_state\":\"push20\",\"states\":[\"accept20\",\"pop20\",\"push20\"],\"transitions\":[[\"pop20\",\"b20\",\"A20\",\"pop20\",[]],[\"pop20\",null,\"Z20\",\"accept20\",[\"Z20\"]],[\"push20\",\"a20\",\"A20\",\"push20\",[\"A20\",\"A20\"]],[\"push20\",\"a20\",\"Z20\",\"push20\",[\"Z20\",\"A20\"]],[\"push20\",\"b20\",\"A20\",\"pop20\",[]],[\"push20\",null,\"Z20\",\"accept20\",[\"Z20\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push20, pop20, accept20], input_alphabet=[a20, b20], stack_alphabet=[Z20, A20], start_state=push20, bottom_symbol=Z20, final_states=[accept20], transitions=[[push20, a20, Z20, push20, [Z20, A20]], [push20, a20, A20, push20, [A20, A20]], [push20, b20, A20, pop20, []], [pop20, b20, A20, pop20, []], [push20, null, Z20, accept20, [Z20]], [pop20, null, Z20, accept20, [Z20]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push20, pop20, accept20], [a20, b20], [Z20, A20], push20, Z20, [accept20], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push20, pop20, accept20], [a20, b20], [Z20, A20], push20, Z20, [accept20], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0022","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8039d2d1803bc7f3cf295237c41ca3f2929afbae250fbbed7dd78c23d2ec7e20","family_id":"qf-mod41-pda-qualification","seed":202611107022,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z21\",\"final_states\":[\"accept21\"],\"input_alphabet\":[\"a21\",\"b21\"],\"stack_alphabet\":[\"A21\",\"Z21\"],\"start_state\":\"push21\",\"states\":[\"accept21\",\"pop21\",\"push21\"],\"transitions\":[[\"pop21\",\"b21\",\"A21\",\"pop21\",[]],[\"pop21\",null,\"Z21\",\"accept21\",[\"Z21\"]],[\"push21\",\"a21\",\"A21\",\"push21\",[\"A21\",\"A21\"]],[\"push21\",\"a21\",\"Z21\",\"push21\",[\"Z21\",\"A21\"]],[\"push21\",\"b21\",\"A21\",\"pop21\",[]],[\"push21\",null,\"Z21\",\"accept21\",[\"Z21\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push21, pop21, accept21], input_alphabet=[a21, b21], stack_alphabet=[Z21, A21], start_state=push21, bottom_symbol=Z21, final_states=[accept21], transitions=[[push21, a21, Z21, push21, [Z21, A21]], [push21, a21, A21, push21, [A21, A21]], [push21, b21, A21, pop21, []], [pop21, b21, A21, pop21, []], [push21, null, Z21, accept21, [Z21]], [pop21, null, Z21, accept21, [Z21]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push21, pop21, accept21], [a21, b21], [Z21, A21], push21, Z21, [accept21], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push21, pop21, accept21], [a21, b21], [Z21, A21], push21, Z21, [accept21], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0023","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f371c7b00a4ecfcffb485daa85ed003bba0991bf01e4466ce3002546a3d2cf4","family_id":"qf-mod41-pda-qualification","seed":202611107023,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z22\",\"final_states\":[\"accept22\"],\"input_alphabet\":[\"a22\",\"b22\"],\"stack_alphabet\":[\"A22\",\"Z22\"],\"start_state\":\"push22\",\"states\":[\"accept22\",\"pop22\",\"push22\"],\"transitions\":[[\"pop22\",\"b22\",\"A22\",\"pop22\",[]],[\"pop22\",null,\"Z22\",\"accept22\",[\"Z22\"]],[\"push22\",\"a22\",\"A22\",\"push22\",[\"A22\",\"A22\"]],[\"push22\",\"a22\",\"Z22\",\"push22\",[\"Z22\",\"A22\"]],[\"push22\",\"b22\",\"A22\",\"pop22\",[]],[\"push22\",null,\"Z22\",\"accept22\",[\"Z22\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push22, pop22, accept22], input_alphabet=[a22, b22], stack_alphabet=[Z22, A22], start_state=push22, bottom_symbol=Z22, final_states=[accept22], transitions=[[push22, a22, Z22, push22, [Z22, A22]], [push22, a22, A22, push22, [A22, A22]], [push22, b22, A22, pop22, []], [pop22, b22, A22, pop22, []], [push22, null, Z22, accept22, [Z22]], [pop22, null, Z22, accept22, [Z22]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push22, pop22, accept22], [a22, b22], [Z22, A22], push22, Z22, [accept22], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push22, pop22, accept22], [a22, b22], [Z22, A22], push22, Z22, [accept22], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0024","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f451ea55d340c5b506933235c2a2b67c6ba44a256ea33accb82cc5c049f7828b","family_id":"qf-mod41-pda-qualification","seed":202611107024,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z23\",\"final_states\":[\"accept23\"],\"input_alphabet\":[\"a23\",\"b23\"],\"stack_alphabet\":[\"A23\",\"Z23\"],\"start_state\":\"push23\",\"states\":[\"accept23\",\"pop23\",\"push23\"],\"transitions\":[[\"pop23\",\"b23\",\"A23\",\"pop23\",[]],[\"pop23\",null,\"Z23\",\"accept23\",[\"Z23\"]],[\"push23\",\"a23\",\"A23\",\"push23\",[\"A23\",\"A23\"]],[\"push23\",\"a23\",\"Z23\",\"push23\",[\"Z23\",\"A23\"]],[\"push23\",\"b23\",\"A23\",\"pop23\",[]],[\"push23\",null,\"Z23\",\"accept23\",[\"Z23\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push23, pop23, accept23], input_alphabet=[a23, b23], stack_alphabet=[Z23, A23], start_state=push23, bottom_symbol=Z23, final_states=[accept23], transitions=[[push23, a23, Z23, push23, [Z23, A23]], [push23, a23, A23, push23, [A23, A23]], [push23, b23, A23, pop23, []], [pop23, b23, A23, pop23, []], [push23, null, Z23, accept23, [Z23]], [pop23, null, Z23, accept23, [Z23]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push23, pop23, accept23], [a23, b23], [Z23, A23], push23, Z23, [accept23], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push23, pop23, accept23], [a23, b23], [Z23, A23], push23, Z23, [accept23], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0025","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf3d4895015aa0da1e0996a7d10ccc8440d3d1e06c8e9168a90adf38c2b5b682","family_id":"qf-mod41-pda-qualification","seed":202611107025,"semantic_template":"qf-mod41-pda-qualification/{\"bottom_symbol\":\"Z24\",\"final_states\":[\"accept24\"],\"input_alphabet\":[\"a24\",\"b24\"],\"stack_alphabet\":[\"A24\",\"Z24\"],\"start_state\":\"push24\",\"states\":[\"accept24\",\"pop24\",\"push24\"],\"transitions\":[[\"pop24\",\"b24\",\"A24\",\"pop24\",[]],[\"pop24\",null,\"Z24\",\"accept24\",[\"Z24\"]],[\"push24\",\"a24\",\"A24\",\"push24\",[\"A24\",\"A24\"]],[\"push24\",\"a24\",\"Z24\",\"push24\",[\"Z24\",\"A24\"]],[\"push24\",\"b24\",\"A24\",\"pop24\",[]],[\"push24\",null,\"Z24\",\"accept24\",[\"Z24\"]]]}"},"fields":{"title":"A Complete PDA Definition Retains Its Final-State Acceptance Convention","stem":"Use the complete finite nondeterministic PDA: states=[push24, pop24, accept24], input_alphabet=[a24, b24], stack_alphabet=[Z24, A24], start_state=push24, bottom_symbol=Z24, final_states=[accept24], transitions=[[push24, a24, Z24, push24, [Z24, A24]], [push24, a24, A24, push24, [A24, A24]], [push24, b24, A24, pop24, []], [pop24, b24, A24, pop24, []], [push24, null, Z24, accept24, [Z24]], [pop24, null, Z24, accept24, [Z24]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Give (states, input_alphabet, stack_alphabet, start_state, bottom_symbol, final_states, transition_count, qualified, convention).","answer":{"kind":"expression","value":"([push24, pop24, accept24], [a24, b24], [Z24, A24], push24, Z24, [accept24], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)","explanation":"Validate all finite states, alphabets, transitions, start, bottom and final states. Qualification alone is not an accepting run, determinism proof or CFG-equivalence claim. The exact result, retaining the requested order, is ([push24, pop24, accept24], [a24, b24], [Z24, A24], push24, Z24, [accept24], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)."},"skills":["A Complete PDA Definition Retains Its Final-State Acceptance Convention"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0026","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e53df793c585b918a6dbf7cfe46378412bffe318849be8ba6ac69e4b8f5e6ece","family_id":"qf-mod41-pda-trace-audit","seed":202611107026,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z0\",\"final_states\":[\"accept0\"],\"input\":[],\"input_alphabet\":[\"a0\",\"b0\"],\"stack_alphabet\":[\"A0\",\"Z0\"],\"start_state\":\"push0\",\"states\":[\"accept0\",\"pop0\",\"push0\"],\"trace\":[[\"push0\",0,[\"Z0\"]],[\"accept0\",0,[\"Z0\"]]],\"transitions\":[[\"pop0\",\"b0\",\"A0\",\"pop0\",[]],[\"pop0\",null,\"Z0\",\"accept0\",[\"Z0\"]],[\"push0\",\"a0\",\"A0\",\"push0\",[\"A0\",\"A0\"]],[\"push0\",\"a0\",\"Z0\",\"push0\",[\"Z0\",\"A0\"]],[\"push0\",\"b0\",\"A0\",\"pop0\",[]],[\"push0\",null,\"Z0\",\"accept0\",[\"Z0\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push0, pop0, accept0], input_alphabet=[a0, b0], stack_alphabet=[Z0, A0], start_state=push0, bottom_symbol=Z0, final_states=[accept0], transitions=[[push0, a0, Z0, push0, [Z0, A0]], [push0, a0, A0, push0, [A0, A0]], [push0, b0, A0, pop0, []], [pop0, b0, A0, pop0, []], [push0, null, Z0, accept0, [Z0]], [pop0, null, Z0, accept0, [Z0]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[], supplied trace=[[push0, 0, [Z0]], [accept0, 0, [Z0]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([], [[push0, 0, [Z0]], [accept0, 0, [Z0]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([], [[push0, 0, [Z0]], [accept0, 0, [Z0]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0027","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"899ba7c3f6a60090592e2c2964db130883ac18361066600d14d34fead9919f6f","family_id":"qf-mod41-pda-trace-audit","seed":202611107027,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z1\",\"final_states\":[\"accept1\"],\"input\":[\"a1\",\"b1\"],\"input_alphabet\":[\"a1\",\"b1\"],\"stack_alphabet\":[\"A1\",\"Z1\"],\"start_state\":\"push1\",\"states\":[\"accept1\",\"pop1\",\"push1\"],\"trace\":[[\"push1\",0,[\"Z1\"]],[\"push1\",1,[\"Z1\",\"A1\"]],[\"pop1\",2,[\"Z1\"]],[\"accept1\",2,[\"Z1\"]]],\"transitions\":[[\"pop1\",\"b1\",\"A1\",\"pop1\",[]],[\"pop1\",null,\"Z1\",\"accept1\",[\"Z1\"]],[\"push1\",\"a1\",\"A1\",\"push1\",[\"A1\",\"A1\"]],[\"push1\",\"a1\",\"Z1\",\"push1\",[\"Z1\",\"A1\"]],[\"push1\",\"b1\",\"A1\",\"pop1\",[]],[\"push1\",null,\"Z1\",\"accept1\",[\"Z1\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push1, pop1, accept1], input_alphabet=[a1, b1], stack_alphabet=[Z1, A1], start_state=push1, bottom_symbol=Z1, final_states=[accept1], transitions=[[push1, a1, Z1, push1, [Z1, A1]], [push1, a1, A1, push1, [A1, A1]], [push1, b1, A1, pop1, []], [pop1, b1, A1, pop1, []], [push1, null, Z1, accept1, [Z1]], [pop1, null, Z1, accept1, [Z1]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a1, b1], supplied trace=[[push1, 0, [Z1]], [push1, 1, [Z1, A1]], [pop1, 2, [Z1]], [accept1, 2, [Z1]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a1, b1], [[push1, 0, [Z1]], [push1, 1, [Z1, A1]], [pop1, 2, [Z1]], [accept1, 2, [Z1]]], [0, 2, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a1, b1], [[push1, 0, [Z1]], [push1, 1, [Z1, A1]], [pop1, 2, [Z1]], [accept1, 2, [Z1]]], [0, 2, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0028","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfade6b4ee076821cdebbeae6f792dc8e04fc2211c01e3dedcd6b4bd58aabfcb","family_id":"qf-mod41-pda-trace-audit","seed":202611107028,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z2\",\"final_states\":[\"accept2\"],\"input\":[\"a2\",\"a2\",\"b2\",\"b2\"],\"input_alphabet\":[\"a2\",\"b2\"],\"stack_alphabet\":[\"A2\",\"Z2\"],\"start_state\":\"push2\",\"states\":[\"accept2\",\"pop2\",\"push2\"],\"trace\":[[\"push2\",0,[\"Z2\"]],[\"push2\",1,[\"Z2\",\"A2\"]],[\"push2\",2,[\"Z2\",\"A2\",\"A2\"]],[\"pop2\",3,[\"Z2\",\"A2\"]],[\"pop2\",4,[\"Z2\"]],[\"accept2\",4,[\"Z2\"]]],\"transitions\":[[\"pop2\",\"b2\",\"A2\",\"pop2\",[]],[\"pop2\",null,\"Z2\",\"accept2\",[\"Z2\"]],[\"push2\",\"a2\",\"A2\",\"push2\",[\"A2\",\"A2\"]],[\"push2\",\"a2\",\"Z2\",\"push2\",[\"Z2\",\"A2\"]],[\"push2\",\"b2\",\"A2\",\"pop2\",[]],[\"push2\",null,\"Z2\",\"accept2\",[\"Z2\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push2, pop2, accept2], input_alphabet=[a2, b2], stack_alphabet=[Z2, A2], start_state=push2, bottom_symbol=Z2, final_states=[accept2], transitions=[[push2, a2, Z2, push2, [Z2, A2]], [push2, a2, A2, push2, [A2, A2]], [push2, b2, A2, pop2, []], [pop2, b2, A2, pop2, []], [push2, null, Z2, accept2, [Z2]], [pop2, null, Z2, accept2, [Z2]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a2, a2, b2, b2], supplied trace=[[push2, 0, [Z2]], [push2, 1, [Z2, A2]], [push2, 2, [Z2, A2, A2]], [pop2, 3, [Z2, A2]], [pop2, 4, [Z2]], [accept2, 4, [Z2]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a2, a2, b2, b2], [[push2, 0, [Z2]], [push2, 1, [Z2, A2]], [push2, 2, [Z2, A2, A2]], [pop2, 3, [Z2, A2]], [pop2, 4, [Z2]], [accept2, 4, [Z2]]], [0, 1, 2, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a2, a2, b2, b2], [[push2, 0, [Z2]], [push2, 1, [Z2, A2]], [push2, 2, [Z2, A2, A2]], [pop2, 3, [Z2, A2]], [pop2, 4, [Z2]], [accept2, 4, [Z2]]], [0, 1, 2, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0029","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aead4396dbc250521cb250e187ce8036ce1806e26c006290af6586aaabebaeec","family_id":"qf-mod41-pda-trace-audit","seed":202611107029,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z3\",\"final_states\":[\"accept3\"],\"input\":[\"a3\",\"a3\",\"a3\",\"b3\",\"b3\",\"b3\"],\"input_alphabet\":[\"a3\",\"b3\"],\"stack_alphabet\":[\"A3\",\"Z3\"],\"start_state\":\"push3\",\"states\":[\"accept3\",\"pop3\",\"push3\"],\"trace\":[[\"push3\",0,[\"Z3\"]],[\"push3\",1,[\"Z3\",\"A3\"]],[\"push3\",2,[\"Z3\",\"A3\",\"A3\"]],[\"push3\",3,[\"Z3\",\"A3\",\"A3\",\"A3\"]],[\"pop3\",4,[\"Z3\",\"A3\",\"A3\"]],[\"pop3\",5,[\"Z3\",\"A3\"]],[\"pop3\",6,[\"Z3\"]],[\"accept3\",6,[\"Z3\"]]],\"transitions\":[[\"pop3\",\"b3\",\"A3\",\"pop3\",[]],[\"pop3\",null,\"Z3\",\"accept3\",[\"Z3\"]],[\"push3\",\"a3\",\"A3\",\"push3\",[\"A3\",\"A3\"]],[\"push3\",\"a3\",\"Z3\",\"push3\",[\"Z3\",\"A3\"]],[\"push3\",\"b3\",\"A3\",\"pop3\",[]],[\"push3\",null,\"Z3\",\"accept3\",[\"Z3\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push3, pop3, accept3], input_alphabet=[a3, b3], stack_alphabet=[Z3, A3], start_state=push3, bottom_symbol=Z3, final_states=[accept3], transitions=[[push3, a3, Z3, push3, [Z3, A3]], [push3, a3, A3, push3, [A3, A3]], [push3, b3, A3, pop3, []], [pop3, b3, A3, pop3, []], [push3, null, Z3, accept3, [Z3]], [pop3, null, Z3, accept3, [Z3]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a3, a3, a3, b3, b3, b3], supplied trace=[[push3, 0, [Z3]], [push3, 1, [Z3, A3]], [push3, 2, [Z3, A3, A3]], [push3, 3, [Z3, A3, A3, A3]], [pop3, 4, [Z3, A3, A3]], [pop3, 5, [Z3, A3]], [pop3, 6, [Z3]], [accept3, 6, [Z3]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a3, a3, a3, b3, b3, b3], [[push3, 0, [Z3]], [push3, 1, [Z3, A3]], [push3, 2, [Z3, A3, A3]], [push3, 3, [Z3, A3, A3, A3]], [pop3, 4, [Z3, A3, A3]], [pop3, 5, [Z3, A3]], [pop3, 6, [Z3]], [accept3, 6, [Z3]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a3, a3, a3, b3, b3, b3], [[push3, 0, [Z3]], [push3, 1, [Z3, A3]], [push3, 2, [Z3, A3, A3]], [push3, 3, [Z3, A3, A3, A3]], [pop3, 4, [Z3, A3, A3]], [pop3, 5, [Z3, A3]], [pop3, 6, [Z3]], [accept3, 6, [Z3]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0030","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9135f38a3aebff56b483a66cb4e01397bb4383b9b1a3f33103f9ea4df9fdc7cf","family_id":"qf-mod41-pda-trace-audit","seed":202611107030,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z4\",\"final_states\":[\"accept4\"],\"input\":[],\"input_alphabet\":[\"a4\",\"b4\"],\"stack_alphabet\":[\"A4\",\"Z4\"],\"start_state\":\"push4\",\"states\":[\"accept4\",\"pop4\",\"push4\"],\"trace\":[[\"push4\",0,[\"Z4\"]],[\"accept4\",0,[\"Z4\"]]],\"transitions\":[[\"pop4\",\"b4\",\"A4\",\"pop4\",[]],[\"pop4\",null,\"Z4\",\"accept4\",[\"Z4\"]],[\"push4\",\"a4\",\"A4\",\"push4\",[\"A4\",\"A4\"]],[\"push4\",\"a4\",\"Z4\",\"push4\",[\"Z4\",\"A4\"]],[\"push4\",\"b4\",\"A4\",\"pop4\",[]],[\"push4\",null,\"Z4\",\"accept4\",[\"Z4\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push4, pop4, accept4], input_alphabet=[a4, b4], stack_alphabet=[Z4, A4], start_state=push4, bottom_symbol=Z4, final_states=[accept4], transitions=[[push4, a4, Z4, push4, [Z4, A4]], [push4, a4, A4, push4, [A4, A4]], [push4, b4, A4, pop4, []], [pop4, b4, A4, pop4, []], [push4, null, Z4, accept4, [Z4]], [pop4, null, Z4, accept4, [Z4]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[], supplied trace=[[push4, 0, [Z4]], [accept4, 0, [Z4]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([], [[push4, 0, [Z4]], [accept4, 0, [Z4]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([], [[push4, 0, [Z4]], [accept4, 0, [Z4]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0031","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fef66c352b554b057f7a2ab92e64da997f446387c6ce40da75f9ba506cb32aef","family_id":"qf-mod41-pda-trace-audit","seed":202611107031,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z5\",\"final_states\":[\"accept5\"],\"input\":[\"a5\",\"b5\"],\"input_alphabet\":[\"a5\",\"b5\"],\"stack_alphabet\":[\"A5\",\"Z5\"],\"start_state\":\"push5\",\"states\":[\"accept5\",\"pop5\",\"push5\"],\"trace\":[[\"push5\",0,[\"Z5\"]],[\"push5\",1,[\"Z5\",\"A5\"]],[\"pop5\",2,[\"Z5\"]],[\"accept5\",2,[\"Z5\"]]],\"transitions\":[[\"pop5\",\"b5\",\"A5\",\"pop5\",[]],[\"pop5\",null,\"Z5\",\"accept5\",[\"Z5\"]],[\"push5\",\"a5\",\"A5\",\"push5\",[\"A5\",\"A5\"]],[\"push5\",\"a5\",\"Z5\",\"push5\",[\"Z5\",\"A5\"]],[\"push5\",\"b5\",\"A5\",\"pop5\",[]],[\"push5\",null,\"Z5\",\"accept5\",[\"Z5\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push5, pop5, accept5], input_alphabet=[a5, b5], stack_alphabet=[Z5, A5], start_state=push5, bottom_symbol=Z5, final_states=[accept5], transitions=[[push5, a5, Z5, push5, [Z5, A5]], [push5, a5, A5, push5, [A5, A5]], [push5, b5, A5, pop5, []], [pop5, b5, A5, pop5, []], [push5, null, Z5, accept5, [Z5]], [pop5, null, Z5, accept5, [Z5]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a5, b5], supplied trace=[[push5, 0, [Z5]], [push5, 1, [Z5, A5]], [pop5, 2, [Z5]], [accept5, 2, [Z5]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a5, b5], [[push5, 0, [Z5]], [push5, 1, [Z5, A5]], [pop5, 2, [Z5]], [accept5, 2, [Z5]]], [0, 2, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a5, b5], [[push5, 0, [Z5]], [push5, 1, [Z5, A5]], [pop5, 2, [Z5]], [accept5, 2, [Z5]]], [0, 2, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0032","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b41c823ee0125080a8c5569eaea6bc17fb2bccddfcf8690a3582db0f4579114f","family_id":"qf-mod41-pda-trace-audit","seed":202611107032,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z6\",\"final_states\":[\"accept6\"],\"input\":[\"a6\",\"a6\",\"b6\",\"b6\"],\"input_alphabet\":[\"a6\",\"b6\"],\"stack_alphabet\":[\"A6\",\"Z6\"],\"start_state\":\"push6\",\"states\":[\"accept6\",\"pop6\",\"push6\"],\"trace\":[[\"push6\",0,[\"Z6\"]],[\"push6\",1,[\"Z6\",\"A6\"]],[\"push6\",2,[\"Z6\",\"A6\",\"A6\"]],[\"pop6\",3,[\"Z6\",\"A6\"]],[\"pop6\",4,[\"Z6\"]],[\"accept6\",4,[\"Z6\"]]],\"transitions\":[[\"pop6\",\"b6\",\"A6\",\"pop6\",[]],[\"pop6\",null,\"Z6\",\"accept6\",[\"Z6\"]],[\"push6\",\"a6\",\"A6\",\"push6\",[\"A6\",\"A6\"]],[\"push6\",\"a6\",\"Z6\",\"push6\",[\"Z6\",\"A6\"]],[\"push6\",\"b6\",\"A6\",\"pop6\",[]],[\"push6\",null,\"Z6\",\"accept6\",[\"Z6\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push6, pop6, accept6], input_alphabet=[a6, b6], stack_alphabet=[Z6, A6], start_state=push6, bottom_symbol=Z6, final_states=[accept6], transitions=[[push6, a6, Z6, push6, [Z6, A6]], [push6, a6, A6, push6, [A6, A6]], [push6, b6, A6, pop6, []], [pop6, b6, A6, pop6, []], [push6, null, Z6, accept6, [Z6]], [pop6, null, Z6, accept6, [Z6]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a6, a6, b6, b6], supplied trace=[[push6, 0, [Z6]], [push6, 1, [Z6, A6]], [push6, 2, [Z6, A6, A6]], [pop6, 3, [Z6, A6]], [pop6, 4, [Z6]], [accept6, 4, [Z6]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a6, a6, b6, b6], [[push6, 0, [Z6]], [push6, 1, [Z6, A6]], [push6, 2, [Z6, A6, A6]], [pop6, 3, [Z6, A6]], [pop6, 4, [Z6]], [accept6, 4, [Z6]]], [0, 1, 2, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a6, a6, b6, b6], [[push6, 0, [Z6]], [push6, 1, [Z6, A6]], [push6, 2, [Z6, A6, A6]], [pop6, 3, [Z6, A6]], [pop6, 4, [Z6]], [accept6, 4, [Z6]]], [0, 1, 2, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0033","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46921cb97b27ec65a186245e8934b08405d7918cbfb8326215c9ee223b0a0ff2","family_id":"qf-mod41-pda-trace-audit","seed":202611107033,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z7\",\"final_states\":[\"accept7\"],\"input\":[\"a7\",\"a7\",\"a7\",\"b7\",\"b7\",\"b7\"],\"input_alphabet\":[\"a7\",\"b7\"],\"stack_alphabet\":[\"A7\",\"Z7\"],\"start_state\":\"push7\",\"states\":[\"accept7\",\"pop7\",\"push7\"],\"trace\":[[\"push7\",0,[\"Z7\"]],[\"push7\",1,[\"Z7\",\"A7\"]],[\"push7\",2,[\"Z7\",\"A7\",\"A7\"]],[\"push7\",3,[\"Z7\",\"A7\",\"A7\",\"A7\"]],[\"pop7\",4,[\"Z7\",\"A7\",\"A7\"]],[\"pop7\",5,[\"Z7\",\"A7\"]],[\"pop7\",6,[\"Z7\"]],[\"accept7\",6,[\"Z7\"]]],\"transitions\":[[\"pop7\",\"b7\",\"A7\",\"pop7\",[]],[\"pop7\",null,\"Z7\",\"accept7\",[\"Z7\"]],[\"push7\",\"a7\",\"A7\",\"push7\",[\"A7\",\"A7\"]],[\"push7\",\"a7\",\"Z7\",\"push7\",[\"Z7\",\"A7\"]],[\"push7\",\"b7\",\"A7\",\"pop7\",[]],[\"push7\",null,\"Z7\",\"accept7\",[\"Z7\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push7, pop7, accept7], input_alphabet=[a7, b7], stack_alphabet=[Z7, A7], start_state=push7, bottom_symbol=Z7, final_states=[accept7], transitions=[[push7, a7, Z7, push7, [Z7, A7]], [push7, a7, A7, push7, [A7, A7]], [push7, b7, A7, pop7, []], [pop7, b7, A7, pop7, []], [push7, null, Z7, accept7, [Z7]], [pop7, null, Z7, accept7, [Z7]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a7, a7, a7, b7, b7, b7], supplied trace=[[push7, 0, [Z7]], [push7, 1, [Z7, A7]], [push7, 2, [Z7, A7, A7]], [push7, 3, [Z7, A7, A7, A7]], [pop7, 4, [Z7, A7, A7]], [pop7, 5, [Z7, A7]], [pop7, 6, [Z7]], [accept7, 6, [Z7]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a7, a7, a7, b7, b7, b7], [[push7, 0, [Z7]], [push7, 1, [Z7, A7]], [push7, 2, [Z7, A7, A7]], [push7, 3, [Z7, A7, A7, A7]], [pop7, 4, [Z7, A7, A7]], [pop7, 5, [Z7, A7]], [pop7, 6, [Z7]], [accept7, 6, [Z7]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a7, a7, a7, b7, b7, b7], [[push7, 0, [Z7]], [push7, 1, [Z7, A7]], [push7, 2, [Z7, A7, A7]], [push7, 3, [Z7, A7, A7, A7]], [pop7, 4, [Z7, A7, A7]], [pop7, 5, [Z7, A7]], [pop7, 6, [Z7]], [accept7, 6, [Z7]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0034","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"804f1668fc07fad2679353a5e22c100f85a56d0d60edeb318cab3f8de126233c","family_id":"qf-mod41-pda-trace-audit","seed":202611107034,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z8\",\"final_states\":[\"accept8\"],\"input\":[],\"input_alphabet\":[\"a8\",\"b8\"],\"stack_alphabet\":[\"A8\",\"Z8\"],\"start_state\":\"push8\",\"states\":[\"accept8\",\"pop8\",\"push8\"],\"trace\":[[\"push8\",0,[\"Z8\"]],[\"accept8\",0,[\"Z8\"]]],\"transitions\":[[\"pop8\",\"b8\",\"A8\",\"pop8\",[]],[\"pop8\",null,\"Z8\",\"accept8\",[\"Z8\"]],[\"push8\",\"a8\",\"A8\",\"push8\",[\"A8\",\"A8\"]],[\"push8\",\"a8\",\"Z8\",\"push8\",[\"Z8\",\"A8\"]],[\"push8\",\"b8\",\"A8\",\"pop8\",[]],[\"push8\",null,\"Z8\",\"accept8\",[\"Z8\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push8, pop8, accept8], input_alphabet=[a8, b8], stack_alphabet=[Z8, A8], start_state=push8, bottom_symbol=Z8, final_states=[accept8], transitions=[[push8, a8, Z8, push8, [Z8, A8]], [push8, a8, A8, push8, [A8, A8]], [push8, b8, A8, pop8, []], [pop8, b8, A8, pop8, []], [push8, null, Z8, accept8, [Z8]], [pop8, null, Z8, accept8, [Z8]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[], supplied trace=[[push8, 0, [Z8]], [accept8, 0, [Z8]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([], [[push8, 0, [Z8]], [accept8, 0, [Z8]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([], [[push8, 0, [Z8]], [accept8, 0, [Z8]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0035","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99425542bb5f11b42b451afb6ebbc941c9d96ac0b8aef9686e466beab6361172","family_id":"qf-mod41-pda-trace-audit","seed":202611107035,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z9\",\"final_states\":[\"accept9\"],\"input\":[\"a9\",\"b9\"],\"input_alphabet\":[\"a9\",\"b9\"],\"stack_alphabet\":[\"A9\",\"Z9\"],\"start_state\":\"push9\",\"states\":[\"accept9\",\"pop9\",\"push9\"],\"trace\":[[\"push9\",0,[\"Z9\"]],[\"push9\",1,[\"Z9\",\"A9\"]],[\"pop9\",2,[\"Z9\"]],[\"accept9\",2,[\"Z9\"]]],\"transitions\":[[\"pop9\",\"b9\",\"A9\",\"pop9\",[]],[\"pop9\",null,\"Z9\",\"accept9\",[\"Z9\"]],[\"push9\",\"a9\",\"A9\",\"push9\",[\"A9\",\"A9\"]],[\"push9\",\"a9\",\"Z9\",\"push9\",[\"Z9\",\"A9\"]],[\"push9\",\"b9\",\"A9\",\"pop9\",[]],[\"push9\",null,\"Z9\",\"accept9\",[\"Z9\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push9, pop9, accept9], input_alphabet=[a9, b9], stack_alphabet=[Z9, A9], start_state=push9, bottom_symbol=Z9, final_states=[accept9], transitions=[[push9, a9, Z9, push9, [Z9, A9]], [push9, a9, A9, push9, [A9, A9]], [push9, b9, A9, pop9, []], [pop9, b9, A9, pop9, []], [push9, null, Z9, accept9, [Z9]], [pop9, null, Z9, accept9, [Z9]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a9, b9], supplied trace=[[push9, 0, [Z9]], [push9, 1, [Z9, A9]], [pop9, 2, [Z9]], [accept9, 2, [Z9]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a9, b9], [[push9, 0, [Z9]], [push9, 1, [Z9, A9]], [pop9, 2, [Z9]], [accept9, 2, [Z9]]], [0, 2, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a9, b9], [[push9, 0, [Z9]], [push9, 1, [Z9, A9]], [pop9, 2, [Z9]], [accept9, 2, [Z9]]], [0, 2, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0036","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9f231e0994dccfb7fa4a257f15adeb08a66cdb5b1b1983821ec4c134945824e","family_id":"qf-mod41-pda-trace-audit","seed":202611107036,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z10\",\"final_states\":[\"accept10\"],\"input\":[\"a10\",\"a10\",\"b10\",\"b10\"],\"input_alphabet\":[\"a10\",\"b10\"],\"stack_alphabet\":[\"A10\",\"Z10\"],\"start_state\":\"push10\",\"states\":[\"accept10\",\"pop10\",\"push10\"],\"trace\":[[\"push10\",0,[\"Z10\"]],[\"push10\",1,[\"Z10\",\"A10\"]],[\"push10\",2,[\"Z10\",\"A10\",\"A10\"]],[\"pop10\",3,[\"Z10\",\"A10\"]],[\"pop10\",4,[\"Z10\"]],[\"accept10\",4,[\"Z10\"]]],\"transitions\":[[\"pop10\",\"b10\",\"A10\",\"pop10\",[]],[\"pop10\",null,\"Z10\",\"accept10\",[\"Z10\"]],[\"push10\",\"a10\",\"A10\",\"push10\",[\"A10\",\"A10\"]],[\"push10\",\"a10\",\"Z10\",\"push10\",[\"Z10\",\"A10\"]],[\"push10\",\"b10\",\"A10\",\"pop10\",[]],[\"push10\",null,\"Z10\",\"accept10\",[\"Z10\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push10, pop10, accept10], input_alphabet=[a10, b10], stack_alphabet=[Z10, A10], start_state=push10, bottom_symbol=Z10, final_states=[accept10], transitions=[[push10, a10, Z10, push10, [Z10, A10]], [push10, a10, A10, push10, [A10, A10]], [push10, b10, A10, pop10, []], [pop10, b10, A10, pop10, []], [push10, null, Z10, accept10, [Z10]], [pop10, null, Z10, accept10, [Z10]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a10, a10, b10, b10], supplied trace=[[push10, 0, [Z10]], [push10, 1, [Z10, A10]], [push10, 2, [Z10, A10, A10]], [pop10, 3, [Z10, A10]], [pop10, 4, [Z10]], [accept10, 4, [Z10]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a10, a10, b10, b10], [[push10, 0, [Z10]], [push10, 1, [Z10, A10]], [push10, 2, [Z10, A10, A10]], [pop10, 3, [Z10, A10]], [pop10, 4, [Z10]], [accept10, 4, [Z10]]], [0, 1, 2, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a10, a10, b10, b10], [[push10, 0, [Z10]], [push10, 1, [Z10, A10]], [push10, 2, [Z10, A10, A10]], [pop10, 3, [Z10, A10]], [pop10, 4, [Z10]], [accept10, 4, [Z10]]], [0, 1, 2, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0037","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96edefea6dad189094ea510e53baea84e09088789ddd26fbf2eb647f41655dda","family_id":"qf-mod41-pda-trace-audit","seed":202611107037,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z11\",\"final_states\":[\"accept11\"],\"input\":[\"a11\",\"a11\",\"a11\",\"b11\",\"b11\",\"b11\"],\"input_alphabet\":[\"a11\",\"b11\"],\"stack_alphabet\":[\"A11\",\"Z11\"],\"start_state\":\"push11\",\"states\":[\"accept11\",\"pop11\",\"push11\"],\"trace\":[[\"push11\",0,[\"Z11\"]],[\"push11\",1,[\"Z11\",\"A11\"]],[\"push11\",2,[\"Z11\",\"A11\",\"A11\"]],[\"push11\",3,[\"Z11\",\"A11\",\"A11\",\"A11\"]],[\"pop11\",4,[\"Z11\",\"A11\",\"A11\"]],[\"pop11\",5,[\"Z11\",\"A11\"]],[\"pop11\",6,[\"Z11\"]],[\"accept11\",6,[\"Z11\"]]],\"transitions\":[[\"pop11\",\"b11\",\"A11\",\"pop11\",[]],[\"pop11\",null,\"Z11\",\"accept11\",[\"Z11\"]],[\"push11\",\"a11\",\"A11\",\"push11\",[\"A11\",\"A11\"]],[\"push11\",\"a11\",\"Z11\",\"push11\",[\"Z11\",\"A11\"]],[\"push11\",\"b11\",\"A11\",\"pop11\",[]],[\"push11\",null,\"Z11\",\"accept11\",[\"Z11\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push11, pop11, accept11], input_alphabet=[a11, b11], stack_alphabet=[Z11, A11], start_state=push11, bottom_symbol=Z11, final_states=[accept11], transitions=[[push11, a11, Z11, push11, [Z11, A11]], [push11, a11, A11, push11, [A11, A11]], [push11, b11, A11, pop11, []], [pop11, b11, A11, pop11, []], [push11, null, Z11, accept11, [Z11]], [pop11, null, Z11, accept11, [Z11]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a11, a11, a11, b11, b11, b11], supplied trace=[[push11, 0, [Z11]], [push11, 1, [Z11, A11]], [push11, 2, [Z11, A11, A11]], [push11, 3, [Z11, A11, A11, A11]], [pop11, 4, [Z11, A11, A11]], [pop11, 5, [Z11, A11]], [pop11, 6, [Z11]], [accept11, 6, [Z11]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a11, a11, a11, b11, b11, b11], [[push11, 0, [Z11]], [push11, 1, [Z11, A11]], [push11, 2, [Z11, A11, A11]], [push11, 3, [Z11, A11, A11, A11]], [pop11, 4, [Z11, A11, A11]], [pop11, 5, [Z11, A11]], [pop11, 6, [Z11]], [accept11, 6, [Z11]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a11, a11, a11, b11, b11, b11], [[push11, 0, [Z11]], [push11, 1, [Z11, A11]], [push11, 2, [Z11, A11, A11]], [push11, 3, [Z11, A11, A11, A11]], [pop11, 4, [Z11, A11, A11]], [pop11, 5, [Z11, A11]], [pop11, 6, [Z11]], [accept11, 6, [Z11]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0038","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a0ea3c2686729fadb6f422bac46293c468f643adaa4bc7e7bc0adf5981c6f7f","family_id":"qf-mod41-pda-trace-audit","seed":202611107038,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z12\",\"final_states\":[\"accept12\"],\"input\":[],\"input_alphabet\":[\"a12\",\"b12\"],\"stack_alphabet\":[\"A12\",\"Z12\"],\"start_state\":\"push12\",\"states\":[\"accept12\",\"pop12\",\"push12\"],\"trace\":[[\"push12\",0,[\"Z12\"]],[\"accept12\",0,[\"Z12\"]]],\"transitions\":[[\"pop12\",\"b12\",\"A12\",\"pop12\",[]],[\"pop12\",null,\"Z12\",\"accept12\",[\"Z12\"]],[\"push12\",\"a12\",\"A12\",\"push12\",[\"A12\",\"A12\"]],[\"push12\",\"a12\",\"Z12\",\"push12\",[\"Z12\",\"A12\"]],[\"push12\",\"b12\",\"A12\",\"pop12\",[]],[\"push12\",null,\"Z12\",\"accept12\",[\"Z12\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push12, pop12, accept12], input_alphabet=[a12, b12], stack_alphabet=[Z12, A12], start_state=push12, bottom_symbol=Z12, final_states=[accept12], transitions=[[push12, a12, Z12, push12, [Z12, A12]], [push12, a12, A12, push12, [A12, A12]], [push12, b12, A12, pop12, []], [pop12, b12, A12, pop12, []], [push12, null, Z12, accept12, [Z12]], [pop12, null, Z12, accept12, [Z12]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[], supplied trace=[[push12, 0, [Z12]], [accept12, 0, [Z12]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([], [[push12, 0, [Z12]], [accept12, 0, [Z12]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([], [[push12, 0, [Z12]], [accept12, 0, [Z12]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0039","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d35c64cddf1e4bc125f9ff23fcc5de8968d482ad671231ed071b8884230f372e","family_id":"qf-mod41-pda-trace-audit","seed":202611107039,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z13\",\"final_states\":[\"accept13\"],\"input\":[\"a13\",\"b13\"],\"input_alphabet\":[\"a13\",\"b13\"],\"stack_alphabet\":[\"A13\",\"Z13\"],\"start_state\":\"push13\",\"states\":[\"accept13\",\"pop13\",\"push13\"],\"trace\":[[\"push13\",0,[\"Z13\"]],[\"push13\",1,[\"Z13\",\"A13\"]],[\"pop13\",2,[\"Z13\"]],[\"accept13\",2,[\"Z13\"]]],\"transitions\":[[\"pop13\",\"b13\",\"A13\",\"pop13\",[]],[\"pop13\",null,\"Z13\",\"accept13\",[\"Z13\"]],[\"push13\",\"a13\",\"A13\",\"push13\",[\"A13\",\"A13\"]],[\"push13\",\"a13\",\"Z13\",\"push13\",[\"Z13\",\"A13\"]],[\"push13\",\"b13\",\"A13\",\"pop13\",[]],[\"push13\",null,\"Z13\",\"accept13\",[\"Z13\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push13, pop13, accept13], input_alphabet=[a13, b13], stack_alphabet=[Z13, A13], start_state=push13, bottom_symbol=Z13, final_states=[accept13], transitions=[[push13, a13, Z13, push13, [Z13, A13]], [push13, a13, A13, push13, [A13, A13]], [push13, b13, A13, pop13, []], [pop13, b13, A13, pop13, []], [push13, null, Z13, accept13, [Z13]], [pop13, null, Z13, accept13, [Z13]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a13, b13], supplied trace=[[push13, 0, [Z13]], [push13, 1, [Z13, A13]], [pop13, 2, [Z13]], [accept13, 2, [Z13]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a13, b13], [[push13, 0, [Z13]], [push13, 1, [Z13, A13]], [pop13, 2, [Z13]], [accept13, 2, [Z13]]], [0, 2, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a13, b13], [[push13, 0, [Z13]], [push13, 1, [Z13, A13]], [pop13, 2, [Z13]], [accept13, 2, [Z13]]], [0, 2, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0040","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a75a48ed41d01f46c1d159453154b9c8d7f1fbee9c3b8d259a77b42848cf4da0","family_id":"qf-mod41-pda-trace-audit","seed":202611107040,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z14\",\"final_states\":[\"accept14\"],\"input\":[\"a14\",\"a14\",\"b14\",\"b14\"],\"input_alphabet\":[\"a14\",\"b14\"],\"stack_alphabet\":[\"A14\",\"Z14\"],\"start_state\":\"push14\",\"states\":[\"accept14\",\"pop14\",\"push14\"],\"trace\":[[\"push14\",0,[\"Z14\"]],[\"push14\",1,[\"Z14\",\"A14\"]],[\"push14\",2,[\"Z14\",\"A14\",\"A14\"]],[\"pop14\",3,[\"Z14\",\"A14\"]],[\"pop14\",4,[\"Z14\"]],[\"accept14\",4,[\"Z14\"]]],\"transitions\":[[\"pop14\",\"b14\",\"A14\",\"pop14\",[]],[\"pop14\",null,\"Z14\",\"accept14\",[\"Z14\"]],[\"push14\",\"a14\",\"A14\",\"push14\",[\"A14\",\"A14\"]],[\"push14\",\"a14\",\"Z14\",\"push14\",[\"Z14\",\"A14\"]],[\"push14\",\"b14\",\"A14\",\"pop14\",[]],[\"push14\",null,\"Z14\",\"accept14\",[\"Z14\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push14, pop14, accept14], input_alphabet=[a14, b14], stack_alphabet=[Z14, A14], start_state=push14, bottom_symbol=Z14, final_states=[accept14], transitions=[[push14, a14, Z14, push14, [Z14, A14]], [push14, a14, A14, push14, [A14, A14]], [push14, b14, A14, pop14, []], [pop14, b14, A14, pop14, []], [push14, null, Z14, accept14, [Z14]], [pop14, null, Z14, accept14, [Z14]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a14, a14, b14, b14], supplied trace=[[push14, 0, [Z14]], [push14, 1, [Z14, A14]], [push14, 2, [Z14, A14, A14]], [pop14, 3, [Z14, A14]], [pop14, 4, [Z14]], [accept14, 4, [Z14]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a14, a14, b14, b14], [[push14, 0, [Z14]], [push14, 1, [Z14, A14]], [push14, 2, [Z14, A14, A14]], [pop14, 3, [Z14, A14]], [pop14, 4, [Z14]], [accept14, 4, [Z14]]], [0, 1, 2, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a14, a14, b14, b14], [[push14, 0, [Z14]], [push14, 1, [Z14, A14]], [push14, 2, [Z14, A14, A14]], [pop14, 3, [Z14, A14]], [pop14, 4, [Z14]], [accept14, 4, [Z14]]], [0, 1, 2, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0041","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afd3207c9d03acc989dd7fb81763154d46bee93f8e641eedb82310684607ed49","family_id":"qf-mod41-pda-trace-audit","seed":202611107041,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z15\",\"final_states\":[\"accept15\"],\"input\":[\"a15\",\"a15\",\"a15\",\"b15\",\"b15\",\"b15\"],\"input_alphabet\":[\"a15\",\"b15\"],\"stack_alphabet\":[\"A15\",\"Z15\"],\"start_state\":\"push15\",\"states\":[\"accept15\",\"pop15\",\"push15\"],\"trace\":[[\"push15\",0,[\"Z15\"]],[\"push15\",1,[\"Z15\",\"A15\"]],[\"push15\",2,[\"Z15\",\"A15\",\"A15\"]],[\"push15\",3,[\"Z15\",\"A15\",\"A15\",\"A15\"]],[\"pop15\",4,[\"Z15\",\"A15\",\"A15\"]],[\"pop15\",5,[\"Z15\",\"A15\"]],[\"pop15\",6,[\"Z15\"]],[\"accept15\",6,[\"Z15\"]]],\"transitions\":[[\"pop15\",\"b15\",\"A15\",\"pop15\",[]],[\"pop15\",null,\"Z15\",\"accept15\",[\"Z15\"]],[\"push15\",\"a15\",\"A15\",\"push15\",[\"A15\",\"A15\"]],[\"push15\",\"a15\",\"Z15\",\"push15\",[\"Z15\",\"A15\"]],[\"push15\",\"b15\",\"A15\",\"pop15\",[]],[\"push15\",null,\"Z15\",\"accept15\",[\"Z15\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push15, pop15, accept15], input_alphabet=[a15, b15], stack_alphabet=[Z15, A15], start_state=push15, bottom_symbol=Z15, final_states=[accept15], transitions=[[push15, a15, Z15, push15, [Z15, A15]], [push15, a15, A15, push15, [A15, A15]], [push15, b15, A15, pop15, []], [pop15, b15, A15, pop15, []], [push15, null, Z15, accept15, [Z15]], [pop15, null, Z15, accept15, [Z15]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a15, a15, a15, b15, b15, b15], supplied trace=[[push15, 0, [Z15]], [push15, 1, [Z15, A15]], [push15, 2, [Z15, A15, A15]], [push15, 3, [Z15, A15, A15, A15]], [pop15, 4, [Z15, A15, A15]], [pop15, 5, [Z15, A15]], [pop15, 6, [Z15]], [accept15, 6, [Z15]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a15, a15, a15, b15, b15, b15], [[push15, 0, [Z15]], [push15, 1, [Z15, A15]], [push15, 2, [Z15, A15, A15]], [push15, 3, [Z15, A15, A15, A15]], [pop15, 4, [Z15, A15, A15]], [pop15, 5, [Z15, A15]], [pop15, 6, [Z15]], [accept15, 6, [Z15]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a15, a15, a15, b15, b15, b15], [[push15, 0, [Z15]], [push15, 1, [Z15, A15]], [push15, 2, [Z15, A15, A15]], [push15, 3, [Z15, A15, A15, A15]], [pop15, 4, [Z15, A15, A15]], [pop15, 5, [Z15, A15]], [pop15, 6, [Z15]], [accept15, 6, [Z15]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0042","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e27874ef63c15f22f435efd5ded45a7866022214234ac3784929770f1ff0be8a","family_id":"qf-mod41-pda-trace-audit","seed":202611107042,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z16\",\"final_states\":[\"accept16\"],\"input\":[],\"input_alphabet\":[\"a16\",\"b16\"],\"stack_alphabet\":[\"A16\",\"Z16\"],\"start_state\":\"push16\",\"states\":[\"accept16\",\"pop16\",\"push16\"],\"trace\":[[\"push16\",0,[\"Z16\"]],[\"accept16\",0,[\"Z16\"]]],\"transitions\":[[\"pop16\",\"b16\",\"A16\",\"pop16\",[]],[\"pop16\",null,\"Z16\",\"accept16\",[\"Z16\"]],[\"push16\",\"a16\",\"A16\",\"push16\",[\"A16\",\"A16\"]],[\"push16\",\"a16\",\"Z16\",\"push16\",[\"Z16\",\"A16\"]],[\"push16\",\"b16\",\"A16\",\"pop16\",[]],[\"push16\",null,\"Z16\",\"accept16\",[\"Z16\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push16, pop16, accept16], input_alphabet=[a16, b16], stack_alphabet=[Z16, A16], start_state=push16, bottom_symbol=Z16, final_states=[accept16], transitions=[[push16, a16, Z16, push16, [Z16, A16]], [push16, a16, A16, push16, [A16, A16]], [push16, b16, A16, pop16, []], [pop16, b16, A16, pop16, []], [push16, null, Z16, accept16, [Z16]], [pop16, null, Z16, accept16, [Z16]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[], supplied trace=[[push16, 0, [Z16]], [accept16, 0, [Z16]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([], [[push16, 0, [Z16]], [accept16, 0, [Z16]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([], [[push16, 0, [Z16]], [accept16, 0, [Z16]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0043","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e693dfa68e2b2e040f48e1ba9be7330418a70a416dba0d8040283b74fd7c709","family_id":"qf-mod41-pda-trace-audit","seed":202611107043,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z17\",\"final_states\":[\"accept17\"],\"input\":[\"a17\",\"b17\"],\"input_alphabet\":[\"a17\",\"b17\"],\"stack_alphabet\":[\"A17\",\"Z17\"],\"start_state\":\"push17\",\"states\":[\"accept17\",\"pop17\",\"push17\"],\"trace\":[[\"push17\",0,[\"Z17\"]],[\"push17\",1,[\"Z17\",\"A17\"]],[\"pop17\",2,[\"Z17\"]],[\"accept17\",2,[\"Z17\"]]],\"transitions\":[[\"pop17\",\"b17\",\"A17\",\"pop17\",[]],[\"pop17\",null,\"Z17\",\"accept17\",[\"Z17\"]],[\"push17\",\"a17\",\"A17\",\"push17\",[\"A17\",\"A17\"]],[\"push17\",\"a17\",\"Z17\",\"push17\",[\"Z17\",\"A17\"]],[\"push17\",\"b17\",\"A17\",\"pop17\",[]],[\"push17\",null,\"Z17\",\"accept17\",[\"Z17\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push17, pop17, accept17], input_alphabet=[a17, b17], stack_alphabet=[Z17, A17], start_state=push17, bottom_symbol=Z17, final_states=[accept17], transitions=[[push17, a17, Z17, push17, [Z17, A17]], [push17, a17, A17, push17, [A17, A17]], [push17, b17, A17, pop17, []], [pop17, b17, A17, pop17, []], [push17, null, Z17, accept17, [Z17]], [pop17, null, Z17, accept17, [Z17]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a17, b17], supplied trace=[[push17, 0, [Z17]], [push17, 1, [Z17, A17]], [pop17, 2, [Z17]], [accept17, 2, [Z17]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a17, b17], [[push17, 0, [Z17]], [push17, 1, [Z17, A17]], [pop17, 2, [Z17]], [accept17, 2, [Z17]]], [0, 2, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a17, b17], [[push17, 0, [Z17]], [push17, 1, [Z17, A17]], [pop17, 2, [Z17]], [accept17, 2, [Z17]]], [0, 2, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0044","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e74f6dbf7faad923ca034ca37ad0d6aa6a77dc10a735d815dfb09b7559cd949","family_id":"qf-mod41-pda-trace-audit","seed":202611107044,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z18\",\"final_states\":[\"accept18\"],\"input\":[\"a18\",\"a18\",\"b18\",\"b18\"],\"input_alphabet\":[\"a18\",\"b18\"],\"stack_alphabet\":[\"A18\",\"Z18\"],\"start_state\":\"push18\",\"states\":[\"accept18\",\"pop18\",\"push18\"],\"trace\":[[\"push18\",0,[\"Z18\"]],[\"push18\",1,[\"Z18\",\"A18\"]],[\"push18\",2,[\"Z18\",\"A18\",\"A18\"]],[\"pop18\",3,[\"Z18\",\"A18\"]],[\"pop18\",4,[\"Z18\"]],[\"accept18\",4,[\"Z18\"]]],\"transitions\":[[\"pop18\",\"b18\",\"A18\",\"pop18\",[]],[\"pop18\",null,\"Z18\",\"accept18\",[\"Z18\"]],[\"push18\",\"a18\",\"A18\",\"push18\",[\"A18\",\"A18\"]],[\"push18\",\"a18\",\"Z18\",\"push18\",[\"Z18\",\"A18\"]],[\"push18\",\"b18\",\"A18\",\"pop18\",[]],[\"push18\",null,\"Z18\",\"accept18\",[\"Z18\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push18, pop18, accept18], input_alphabet=[a18, b18], stack_alphabet=[Z18, A18], start_state=push18, bottom_symbol=Z18, final_states=[accept18], transitions=[[push18, a18, Z18, push18, [Z18, A18]], [push18, a18, A18, push18, [A18, A18]], [push18, b18, A18, pop18, []], [pop18, b18, A18, pop18, []], [push18, null, Z18, accept18, [Z18]], [pop18, null, Z18, accept18, [Z18]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a18, a18, b18, b18], supplied trace=[[push18, 0, [Z18]], [push18, 1, [Z18, A18]], [push18, 2, [Z18, A18, A18]], [pop18, 3, [Z18, A18]], [pop18, 4, [Z18]], [accept18, 4, [Z18]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a18, a18, b18, b18], [[push18, 0, [Z18]], [push18, 1, [Z18, A18]], [push18, 2, [Z18, A18, A18]], [pop18, 3, [Z18, A18]], [pop18, 4, [Z18]], [accept18, 4, [Z18]]], [0, 1, 2, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a18, a18, b18, b18], [[push18, 0, [Z18]], [push18, 1, [Z18, A18]], [push18, 2, [Z18, A18, A18]], [pop18, 3, [Z18, A18]], [pop18, 4, [Z18]], [accept18, 4, [Z18]]], [0, 1, 2, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0045","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ab3a26dbbfdf273221097b5dccc6c00e74db3786275b3a74becf4d3a455fd75","family_id":"qf-mod41-pda-trace-audit","seed":202611107045,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z19\",\"final_states\":[\"accept19\"],\"input\":[\"a19\",\"a19\",\"a19\",\"b19\",\"b19\",\"b19\"],\"input_alphabet\":[\"a19\",\"b19\"],\"stack_alphabet\":[\"A19\",\"Z19\"],\"start_state\":\"push19\",\"states\":[\"accept19\",\"pop19\",\"push19\"],\"trace\":[[\"push19\",0,[\"Z19\"]],[\"push19\",1,[\"Z19\",\"A19\"]],[\"push19\",2,[\"Z19\",\"A19\",\"A19\"]],[\"push19\",3,[\"Z19\",\"A19\",\"A19\",\"A19\"]],[\"pop19\",4,[\"Z19\",\"A19\",\"A19\"]],[\"pop19\",5,[\"Z19\",\"A19\"]],[\"pop19\",6,[\"Z19\"]],[\"accept19\",6,[\"Z19\"]]],\"transitions\":[[\"pop19\",\"b19\",\"A19\",\"pop19\",[]],[\"pop19\",null,\"Z19\",\"accept19\",[\"Z19\"]],[\"push19\",\"a19\",\"A19\",\"push19\",[\"A19\",\"A19\"]],[\"push19\",\"a19\",\"Z19\",\"push19\",[\"Z19\",\"A19\"]],[\"push19\",\"b19\",\"A19\",\"pop19\",[]],[\"push19\",null,\"Z19\",\"accept19\",[\"Z19\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push19, pop19, accept19], input_alphabet=[a19, b19], stack_alphabet=[Z19, A19], start_state=push19, bottom_symbol=Z19, final_states=[accept19], transitions=[[push19, a19, Z19, push19, [Z19, A19]], [push19, a19, A19, push19, [A19, A19]], [push19, b19, A19, pop19, []], [pop19, b19, A19, pop19, []], [push19, null, Z19, accept19, [Z19]], [pop19, null, Z19, accept19, [Z19]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a19, a19, a19, b19, b19, b19], supplied trace=[[push19, 0, [Z19]], [push19, 1, [Z19, A19]], [push19, 2, [Z19, A19, A19]], [push19, 3, [Z19, A19, A19, A19]], [pop19, 4, [Z19, A19, A19]], [pop19, 5, [Z19, A19]], [pop19, 6, [Z19]], [accept19, 6, [Z19]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a19, a19, a19, b19, b19, b19], [[push19, 0, [Z19]], [push19, 1, [Z19, A19]], [push19, 2, [Z19, A19, A19]], [push19, 3, [Z19, A19, A19, A19]], [pop19, 4, [Z19, A19, A19]], [pop19, 5, [Z19, A19]], [pop19, 6, [Z19]], [accept19, 6, [Z19]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a19, a19, a19, b19, b19, b19], [[push19, 0, [Z19]], [push19, 1, [Z19, A19]], [push19, 2, [Z19, A19, A19]], [push19, 3, [Z19, A19, A19, A19]], [pop19, 4, [Z19, A19, A19]], [pop19, 5, [Z19, A19]], [pop19, 6, [Z19]], [accept19, 6, [Z19]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0046","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ae0967335222008bcc974f580cca3d3b57ed0357330c76ae3a36b19a28e39f0","family_id":"qf-mod41-pda-trace-audit","seed":202611107046,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z20\",\"final_states\":[\"accept20\"],\"input\":[],\"input_alphabet\":[\"a20\",\"b20\"],\"stack_alphabet\":[\"A20\",\"Z20\"],\"start_state\":\"push20\",\"states\":[\"accept20\",\"pop20\",\"push20\"],\"trace\":[[\"push20\",0,[\"Z20\"]],[\"accept20\",0,[\"Z20\"]]],\"transitions\":[[\"pop20\",\"b20\",\"A20\",\"pop20\",[]],[\"pop20\",null,\"Z20\",\"accept20\",[\"Z20\"]],[\"push20\",\"a20\",\"A20\",\"push20\",[\"A20\",\"A20\"]],[\"push20\",\"a20\",\"Z20\",\"push20\",[\"Z20\",\"A20\"]],[\"push20\",\"b20\",\"A20\",\"pop20\",[]],[\"push20\",null,\"Z20\",\"accept20\",[\"Z20\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push20, pop20, accept20], input_alphabet=[a20, b20], stack_alphabet=[Z20, A20], start_state=push20, bottom_symbol=Z20, final_states=[accept20], transitions=[[push20, a20, Z20, push20, [Z20, A20]], [push20, a20, A20, push20, [A20, A20]], [push20, b20, A20, pop20, []], [pop20, b20, A20, pop20, []], [push20, null, Z20, accept20, [Z20]], [pop20, null, Z20, accept20, [Z20]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[], supplied trace=[[push20, 0, [Z20]], [accept20, 0, [Z20]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([], [[push20, 0, [Z20]], [accept20, 0, [Z20]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([], [[push20, 0, [Z20]], [accept20, 0, [Z20]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0047","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd2e842c5f0c2244caa2cd21eac3184d05265b670679772cac49798f3a58fb9a","family_id":"qf-mod41-pda-trace-audit","seed":202611107047,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z21\",\"final_states\":[\"accept21\"],\"input\":[\"a21\",\"b21\"],\"input_alphabet\":[\"a21\",\"b21\"],\"stack_alphabet\":[\"A21\",\"Z21\"],\"start_state\":\"push21\",\"states\":[\"accept21\",\"pop21\",\"push21\"],\"trace\":[[\"push21\",0,[\"Z21\"]],[\"push21\",1,[\"Z21\",\"A21\"]],[\"pop21\",2,[\"Z21\"]],[\"accept21\",2,[\"Z21\"]]],\"transitions\":[[\"pop21\",\"b21\",\"A21\",\"pop21\",[]],[\"pop21\",null,\"Z21\",\"accept21\",[\"Z21\"]],[\"push21\",\"a21\",\"A21\",\"push21\",[\"A21\",\"A21\"]],[\"push21\",\"a21\",\"Z21\",\"push21\",[\"Z21\",\"A21\"]],[\"push21\",\"b21\",\"A21\",\"pop21\",[]],[\"push21\",null,\"Z21\",\"accept21\",[\"Z21\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push21, pop21, accept21], input_alphabet=[a21, b21], stack_alphabet=[Z21, A21], start_state=push21, bottom_symbol=Z21, final_states=[accept21], transitions=[[push21, a21, Z21, push21, [Z21, A21]], [push21, a21, A21, push21, [A21, A21]], [push21, b21, A21, pop21, []], [pop21, b21, A21, pop21, []], [push21, null, Z21, accept21, [Z21]], [pop21, null, Z21, accept21, [Z21]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a21, b21], supplied trace=[[push21, 0, [Z21]], [push21, 1, [Z21, A21]], [pop21, 2, [Z21]], [accept21, 2, [Z21]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a21, b21], [[push21, 0, [Z21]], [push21, 1, [Z21, A21]], [pop21, 2, [Z21]], [accept21, 2, [Z21]]], [0, 2, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a21, b21], [[push21, 0, [Z21]], [push21, 1, [Z21, A21]], [pop21, 2, [Z21]], [accept21, 2, [Z21]]], [0, 2, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0048","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee502cc6354009c044501b96e4d94c1808d93c6c37111280ac35e1409f5b4051","family_id":"qf-mod41-pda-trace-audit","seed":202611107048,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z22\",\"final_states\":[\"accept22\"],\"input\":[\"a22\",\"a22\",\"b22\",\"b22\"],\"input_alphabet\":[\"a22\",\"b22\"],\"stack_alphabet\":[\"A22\",\"Z22\"],\"start_state\":\"push22\",\"states\":[\"accept22\",\"pop22\",\"push22\"],\"trace\":[[\"push22\",0,[\"Z22\"]],[\"push22\",1,[\"Z22\",\"A22\"]],[\"push22\",2,[\"Z22\",\"A22\",\"A22\"]],[\"pop22\",3,[\"Z22\",\"A22\"]],[\"pop22\",4,[\"Z22\"]],[\"accept22\",4,[\"Z22\"]]],\"transitions\":[[\"pop22\",\"b22\",\"A22\",\"pop22\",[]],[\"pop22\",null,\"Z22\",\"accept22\",[\"Z22\"]],[\"push22\",\"a22\",\"A22\",\"push22\",[\"A22\",\"A22\"]],[\"push22\",\"a22\",\"Z22\",\"push22\",[\"Z22\",\"A22\"]],[\"push22\",\"b22\",\"A22\",\"pop22\",[]],[\"push22\",null,\"Z22\",\"accept22\",[\"Z22\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push22, pop22, accept22], input_alphabet=[a22, b22], stack_alphabet=[Z22, A22], start_state=push22, bottom_symbol=Z22, final_states=[accept22], transitions=[[push22, a22, Z22, push22, [Z22, A22]], [push22, a22, A22, push22, [A22, A22]], [push22, b22, A22, pop22, []], [pop22, b22, A22, pop22, []], [push22, null, Z22, accept22, [Z22]], [pop22, null, Z22, accept22, [Z22]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a22, a22, b22, b22], supplied trace=[[push22, 0, [Z22]], [push22, 1, [Z22, A22]], [push22, 2, [Z22, A22, A22]], [pop22, 3, [Z22, A22]], [pop22, 4, [Z22]], [accept22, 4, [Z22]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a22, a22, b22, b22], [[push22, 0, [Z22]], [push22, 1, [Z22, A22]], [push22, 2, [Z22, A22, A22]], [pop22, 3, [Z22, A22]], [pop22, 4, [Z22]], [accept22, 4, [Z22]]], [0, 1, 2, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a22, a22, b22, b22], [[push22, 0, [Z22]], [push22, 1, [Z22, A22]], [push22, 2, [Z22, A22, A22]], [pop22, 3, [Z22, A22]], [pop22, 4, [Z22]], [accept22, 4, [Z22]]], [0, 1, 2, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0049","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd5f2eeac12d142849dadac19750cd23145e0d0eca976ece72834568018a3ada","family_id":"qf-mod41-pda-trace-audit","seed":202611107049,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z23\",\"final_states\":[\"accept23\"],\"input\":[\"a23\",\"a23\",\"a23\",\"b23\",\"b23\",\"b23\"],\"input_alphabet\":[\"a23\",\"b23\"],\"stack_alphabet\":[\"A23\",\"Z23\"],\"start_state\":\"push23\",\"states\":[\"accept23\",\"pop23\",\"push23\"],\"trace\":[[\"push23\",0,[\"Z23\"]],[\"push23\",1,[\"Z23\",\"A23\"]],[\"push23\",2,[\"Z23\",\"A23\",\"A23\"]],[\"push23\",3,[\"Z23\",\"A23\",\"A23\",\"A23\"]],[\"pop23\",4,[\"Z23\",\"A23\",\"A23\"]],[\"pop23\",5,[\"Z23\",\"A23\"]],[\"pop23\",6,[\"Z23\"]],[\"accept23\",6,[\"Z23\"]]],\"transitions\":[[\"pop23\",\"b23\",\"A23\",\"pop23\",[]],[\"pop23\",null,\"Z23\",\"accept23\",[\"Z23\"]],[\"push23\",\"a23\",\"A23\",\"push23\",[\"A23\",\"A23\"]],[\"push23\",\"a23\",\"Z23\",\"push23\",[\"Z23\",\"A23\"]],[\"push23\",\"b23\",\"A23\",\"pop23\",[]],[\"push23\",null,\"Z23\",\"accept23\",[\"Z23\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push23, pop23, accept23], input_alphabet=[a23, b23], stack_alphabet=[Z23, A23], start_state=push23, bottom_symbol=Z23, final_states=[accept23], transitions=[[push23, a23, Z23, push23, [Z23, A23]], [push23, a23, A23, push23, [A23, A23]], [push23, b23, A23, pop23, []], [pop23, b23, A23, pop23, []], [push23, null, Z23, accept23, [Z23]], [pop23, null, Z23, accept23, [Z23]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a23, a23, a23, b23, b23, b23], supplied trace=[[push23, 0, [Z23]], [push23, 1, [Z23, A23]], [push23, 2, [Z23, A23, A23]], [push23, 3, [Z23, A23, A23, A23]], [pop23, 4, [Z23, A23, A23]], [pop23, 5, [Z23, A23]], [pop23, 6, [Z23]], [accept23, 6, [Z23]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([a23, a23, a23, b23, b23, b23], [[push23, 0, [Z23]], [push23, 1, [Z23, A23]], [push23, 2, [Z23, A23, A23]], [push23, 3, [Z23, A23, A23, A23]], [pop23, 4, [Z23, A23, A23]], [pop23, 5, [Z23, A23]], [pop23, 6, [Z23]], [accept23, 6, [Z23]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([a23, a23, a23, b23, b23, b23], [[push23, 0, [Z23]], [push23, 1, [Z23, A23]], [push23, 2, [Z23, A23, A23]], [push23, 3, [Z23, A23, A23, A23]], [pop23, 4, [Z23, A23, A23]], [pop23, 5, [Z23, A23]], [pop23, 6, [Z23]], [accept23, 6, [Z23]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0050","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f10df3d92dbb9e5cc425dd7b4937d1c7b72f1c2cd0fd8c6c829e5842744fb63","family_id":"qf-mod41-pda-trace-audit","seed":202611107050,"semantic_template":"qf-mod41-pda-trace-audit/{\"bottom_symbol\":\"Z24\",\"final_states\":[\"accept24\"],\"input\":[],\"input_alphabet\":[\"a24\",\"b24\"],\"stack_alphabet\":[\"A24\",\"Z24\"],\"start_state\":\"push24\",\"states\":[\"accept24\",\"pop24\",\"push24\"],\"trace\":[[\"push24\",0,[\"Z24\"]],[\"accept24\",0,[\"Z24\"]]],\"transitions\":[[\"pop24\",\"b24\",\"A24\",\"pop24\",[]],[\"pop24\",null,\"Z24\",\"accept24\",[\"Z24\"]],[\"push24\",\"a24\",\"A24\",\"push24\",[\"A24\",\"A24\"]],[\"push24\",\"a24\",\"Z24\",\"push24\",[\"Z24\",\"A24\"]],[\"push24\",\"b24\",\"A24\",\"pop24\",[]],[\"push24\",null,\"Z24\",\"accept24\",[\"Z24\"]]]}"},"fields":{"title":"Each Explicit Stack Step Must Respect Input and Epsilon Consumption","stem":"Use the complete finite nondeterministic PDA: states=[push24, pop24, accept24], input_alphabet=[a24, b24], stack_alphabet=[Z24, A24], start_state=push24, bottom_symbol=Z24, final_states=[accept24], transitions=[[push24, a24, Z24, push24, [Z24, A24]], [push24, a24, A24, push24, [A24, A24]], [push24, b24, A24, pop24, []], [pop24, b24, A24, pop24, []], [push24, null, Z24, accept24, [Z24]], [pop24, null, Z24, accept24, [Z24]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[], supplied trace=[[push24, 0, [Z24]], [accept24, 0, [Z24]]]. Give (input, trace, transition_indices, valid_run, accepted, certificate).","answer":{"kind":"expression","value":"([], [[push24, 0, [Z24]], [accept24, 0, [Z24]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)","explanation":"Require the initial configuration [start,0,[bottom]]. Each step uses the first matching original transition index. Report run validity separately from acceptance; a valid run reaching a final state before reading all input is not accepted. The exact result, retaining the requested order, is ([], [[push24, 0, [Z24]], [accept24, 0, [Z24]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)."},"skills":["Each Explicit Stack Step Must Respect Input and Epsilon Consumption"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0051","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc8a15d54e01d27f8683ca90cd7a950fabb5cc92942dc5d2da16d0dcbad1f30a","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107051,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z0\",\"final_states\":[\"accept0\"],\"input\":[],\"input_alphabet\":[\"a0\",\"b0\"],\"stack_alphabet\":[\"A0\",\"Z0\"],\"start_state\":\"push0\",\"states\":[\"accept0\",\"pop0\",\"push0\"],\"trace\":[[\"push0\",0,[\"Z0\"]],[\"accept0\",0,[\"Z0\"]]],\"transitions\":[[\"pop0\",\"b0\",\"A0\",\"pop0\",[]],[\"pop0\",null,\"Z0\",\"accept0\",[\"Z0\"]],[\"push0\",\"a0\",\"A0\",\"push0\",[\"A0\",\"A0\"]],[\"push0\",\"a0\",\"Z0\",\"push0\",[\"Z0\",\"A0\"]],[\"push0\",\"b0\",\"A0\",\"pop0\",[]],[\"push0\",null,\"Z0\",\"accept0\",[\"Z0\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push0, pop0, accept0], input_alphabet=[a0, b0], stack_alphabet=[Z0, A0], start_state=push0, bottom_symbol=Z0, final_states=[accept0], transitions=[[push0, a0, Z0, push0, [Z0, A0]], [push0, a0, A0, push0, [A0, A0]], [push0, b0, A0, pop0, []], [pop0, b0, A0, pop0, []], [push0, null, Z0, accept0, [Z0]], [pop0, null, Z0, accept0, [Z0]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[], supplied trace=[[push0, 0, [Z0]], [accept0, 0, [Z0]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([], [[push0, 0, [Z0]], [accept0, 0, [Z0]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([], [[push0, 0, [Z0]], [accept0, 0, [Z0]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0052","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"223239e8aac23097cd5a90ec89136bfa4c74d86e640ba146d17f288992814a50","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107052,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z1\",\"final_states\":[\"accept1\"],\"input\":[\"a1\",\"b1\"],\"input_alphabet\":[\"a1\",\"b1\"],\"stack_alphabet\":[\"A1\",\"Z1\"],\"start_state\":\"push1\",\"states\":[\"accept1\",\"pop1\",\"push1\"],\"trace\":[[\"push1\",0,[\"Z1\"]],[\"push1\",1,[\"Z1\",\"A1\"]],[\"pop1\",2,[\"Z1\"]],[\"accept1\",2,[\"Z1\"]]],\"transitions\":[[\"pop1\",\"b1\",\"A1\",\"pop1\",[]],[\"pop1\",null,\"Z1\",\"accept1\",[\"Z1\"]],[\"push1\",\"a1\",\"A1\",\"push1\",[\"A1\",\"A1\"]],[\"push1\",\"a1\",\"Z1\",\"push1\",[\"Z1\",\"A1\"]],[\"push1\",\"b1\",\"A1\",\"pop1\",[]],[\"push1\",null,\"Z1\",\"accept1\",[\"Z1\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push1, pop1, accept1], input_alphabet=[a1, b1], stack_alphabet=[Z1, A1], start_state=push1, bottom_symbol=Z1, final_states=[accept1], transitions=[[push1, a1, Z1, push1, [Z1, A1]], [push1, a1, A1, push1, [A1, A1]], [push1, b1, A1, pop1, []], [pop1, b1, A1, pop1, []], [push1, null, Z1, accept1, [Z1]], [pop1, null, Z1, accept1, [Z1]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a1, b1], supplied trace=[[push1, 0, [Z1]], [push1, 1, [Z1, A1]], [pop1, 2, [Z1]], [accept1, 2, [Z1]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a1, b1], [[push1, 0, [Z1]], [push1, 1, [Z1, A1]], [pop1, 2, [Z1]], [accept1, 2, [Z1]]], [0, 2, 5], Yes, Yes, 3, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a1, b1], [[push1, 0, [Z1]], [push1, 1, [Z1, A1]], [pop1, 2, [Z1]], [accept1, 2, [Z1]]], [0, 2, 5], Yes, Yes, 3, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0053","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85c588f85157fd624fc4dc35c06455288efb1d965208d78fcf8b9d9fb89aa2a1","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107053,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z2\",\"final_states\":[\"accept2\"],\"input\":[\"a2\",\"a2\",\"b2\",\"b2\"],\"input_alphabet\":[\"a2\",\"b2\"],\"stack_alphabet\":[\"A2\",\"Z2\"],\"start_state\":\"push2\",\"states\":[\"accept2\",\"pop2\",\"push2\"],\"trace\":[[\"push2\",0,[\"Z2\"]],[\"push2\",1,[\"Z2\",\"A2\"]],[\"push2\",2,[\"Z2\",\"A2\",\"A2\"]],[\"pop2\",3,[\"Z2\",\"A2\"]],[\"pop2\",4,[\"Z2\"]],[\"accept2\",4,[\"Z2\"]]],\"transitions\":[[\"pop2\",\"b2\",\"A2\",\"pop2\",[]],[\"pop2\",null,\"Z2\",\"accept2\",[\"Z2\"]],[\"push2\",\"a2\",\"A2\",\"push2\",[\"A2\",\"A2\"]],[\"push2\",\"a2\",\"Z2\",\"push2\",[\"Z2\",\"A2\"]],[\"push2\",\"b2\",\"A2\",\"pop2\",[]],[\"push2\",null,\"Z2\",\"accept2\",[\"Z2\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push2, pop2, accept2], input_alphabet=[a2, b2], stack_alphabet=[Z2, A2], start_state=push2, bottom_symbol=Z2, final_states=[accept2], transitions=[[push2, a2, Z2, push2, [Z2, A2]], [push2, a2, A2, push2, [A2, A2]], [push2, b2, A2, pop2, []], [pop2, b2, A2, pop2, []], [push2, null, Z2, accept2, [Z2]], [pop2, null, Z2, accept2, [Z2]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a2, a2, b2, b2], supplied trace=[[push2, 0, [Z2]], [push2, 1, [Z2, A2]], [push2, 2, [Z2, A2, A2]], [pop2, 3, [Z2, A2]], [pop2, 4, [Z2]], [accept2, 4, [Z2]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a2, a2, b2, b2], [[push2, 0, [Z2]], [push2, 1, [Z2, A2]], [push2, 2, [Z2, A2, A2]], [pop2, 3, [Z2, A2]], [pop2, 4, [Z2]], [accept2, 4, [Z2]]], [0, 1, 2, 3, 5], Yes, Yes, 5, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a2, a2, b2, b2], [[push2, 0, [Z2]], [push2, 1, [Z2, A2]], [push2, 2, [Z2, A2, A2]], [pop2, 3, [Z2, A2]], [pop2, 4, [Z2]], [accept2, 4, [Z2]]], [0, 1, 2, 3, 5], Yes, Yes, 5, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0054","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a76d445b049ee2bb208b286cb3eb82ac2f96acf90c8f3eb4642154b4d71bb16b","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107054,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z3\",\"final_states\":[\"accept3\"],\"input\":[\"a3\",\"a3\",\"a3\",\"b3\",\"b3\",\"b3\"],\"input_alphabet\":[\"a3\",\"b3\"],\"stack_alphabet\":[\"A3\",\"Z3\"],\"start_state\":\"push3\",\"states\":[\"accept3\",\"pop3\",\"push3\"],\"trace\":[[\"push3\",0,[\"Z3\"]],[\"push3\",1,[\"Z3\",\"A3\"]],[\"push3\",2,[\"Z3\",\"A3\",\"A3\"]],[\"push3\",3,[\"Z3\",\"A3\",\"A3\",\"A3\"]],[\"pop3\",4,[\"Z3\",\"A3\",\"A3\"]],[\"pop3\",5,[\"Z3\",\"A3\"]],[\"pop3\",6,[\"Z3\"]],[\"accept3\",6,[\"Z3\"]]],\"transitions\":[[\"pop3\",\"b3\",\"A3\",\"pop3\",[]],[\"pop3\",null,\"Z3\",\"accept3\",[\"Z3\"]],[\"push3\",\"a3\",\"A3\",\"push3\",[\"A3\",\"A3\"]],[\"push3\",\"a3\",\"Z3\",\"push3\",[\"Z3\",\"A3\"]],[\"push3\",\"b3\",\"A3\",\"pop3\",[]],[\"push3\",null,\"Z3\",\"accept3\",[\"Z3\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push3, pop3, accept3], input_alphabet=[a3, b3], stack_alphabet=[Z3, A3], start_state=push3, bottom_symbol=Z3, final_states=[accept3], transitions=[[push3, a3, Z3, push3, [Z3, A3]], [push3, a3, A3, push3, [A3, A3]], [push3, b3, A3, pop3, []], [pop3, b3, A3, pop3, []], [push3, null, Z3, accept3, [Z3]], [pop3, null, Z3, accept3, [Z3]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a3, a3, a3, b3, b3, b3], supplied trace=[[push3, 0, [Z3]], [push3, 1, [Z3, A3]], [push3, 2, [Z3, A3, A3]], [push3, 3, [Z3, A3, A3, A3]], [pop3, 4, [Z3, A3, A3]], [pop3, 5, [Z3, A3]], [pop3, 6, [Z3]], [accept3, 6, [Z3]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a3, a3, a3, b3, b3, b3], [[push3, 0, [Z3]], [push3, 1, [Z3, A3]], [push3, 2, [Z3, A3, A3]], [push3, 3, [Z3, A3, A3, A3]], [pop3, 4, [Z3, A3, A3]], [pop3, 5, [Z3, A3]], [pop3, 6, [Z3]], [accept3, 6, [Z3]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, 7, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a3, a3, a3, b3, b3, b3], [[push3, 0, [Z3]], [push3, 1, [Z3, A3]], [push3, 2, [Z3, A3, A3]], [push3, 3, [Z3, A3, A3, A3]], [pop3, 4, [Z3, A3, A3]], [pop3, 5, [Z3, A3]], [pop3, 6, [Z3]], [accept3, 6, [Z3]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, 7, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0055","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e70d48f147747688363d142f7ee391475cbafd3ab6bf045e60f51c57258d51c7","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107055,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z4\",\"final_states\":[\"accept4\"],\"input\":[],\"input_alphabet\":[\"a4\",\"b4\"],\"stack_alphabet\":[\"A4\",\"Z4\"],\"start_state\":\"push4\",\"states\":[\"accept4\",\"pop4\",\"push4\"],\"trace\":[[\"push4\",0,[\"Z4\"]],[\"accept4\",0,[\"Z4\"]]],\"transitions\":[[\"pop4\",\"b4\",\"A4\",\"pop4\",[]],[\"pop4\",null,\"Z4\",\"accept4\",[\"Z4\"]],[\"push4\",\"a4\",\"A4\",\"push4\",[\"A4\",\"A4\"]],[\"push4\",\"a4\",\"Z4\",\"push4\",[\"Z4\",\"A4\"]],[\"push4\",\"b4\",\"A4\",\"pop4\",[]],[\"push4\",null,\"Z4\",\"accept4\",[\"Z4\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push4, pop4, accept4], input_alphabet=[a4, b4], stack_alphabet=[Z4, A4], start_state=push4, bottom_symbol=Z4, final_states=[accept4], transitions=[[push4, a4, Z4, push4, [Z4, A4]], [push4, a4, A4, push4, [A4, A4]], [push4, b4, A4, pop4, []], [pop4, b4, A4, pop4, []], [push4, null, Z4, accept4, [Z4]], [pop4, null, Z4, accept4, [Z4]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[], supplied trace=[[push4, 0, [Z4]], [accept4, 0, [Z4]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([], [[push4, 0, [Z4]], [accept4, 0, [Z4]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([], [[push4, 0, [Z4]], [accept4, 0, [Z4]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0056","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa9ed4480f8a5d2bd559e334914ed24fd523f683d7d6363877b04e106098d6b1","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107056,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z5\",\"final_states\":[\"accept5\"],\"input\":[\"a5\",\"b5\"],\"input_alphabet\":[\"a5\",\"b5\"],\"stack_alphabet\":[\"A5\",\"Z5\"],\"start_state\":\"push5\",\"states\":[\"accept5\",\"pop5\",\"push5\"],\"trace\":[[\"push5\",0,[\"Z5\"]],[\"push5\",1,[\"Z5\",\"A5\"]],[\"pop5\",2,[\"Z5\"]],[\"accept5\",2,[\"Z5\"]]],\"transitions\":[[\"pop5\",\"b5\",\"A5\",\"pop5\",[]],[\"pop5\",null,\"Z5\",\"accept5\",[\"Z5\"]],[\"push5\",\"a5\",\"A5\",\"push5\",[\"A5\",\"A5\"]],[\"push5\",\"a5\",\"Z5\",\"push5\",[\"Z5\",\"A5\"]],[\"push5\",\"b5\",\"A5\",\"pop5\",[]],[\"push5\",null,\"Z5\",\"accept5\",[\"Z5\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push5, pop5, accept5], input_alphabet=[a5, b5], stack_alphabet=[Z5, A5], start_state=push5, bottom_symbol=Z5, final_states=[accept5], transitions=[[push5, a5, Z5, push5, [Z5, A5]], [push5, a5, A5, push5, [A5, A5]], [push5, b5, A5, pop5, []], [pop5, b5, A5, pop5, []], [push5, null, Z5, accept5, [Z5]], [pop5, null, Z5, accept5, [Z5]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a5, b5], supplied trace=[[push5, 0, [Z5]], [push5, 1, [Z5, A5]], [pop5, 2, [Z5]], [accept5, 2, [Z5]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a5, b5], [[push5, 0, [Z5]], [push5, 1, [Z5, A5]], [pop5, 2, [Z5]], [accept5, 2, [Z5]]], [0, 2, 5], Yes, Yes, 3, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a5, b5], [[push5, 0, [Z5]], [push5, 1, [Z5, A5]], [pop5, 2, [Z5]], [accept5, 2, [Z5]]], [0, 2, 5], Yes, Yes, 3, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0057","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6932bb338a154bd5e25b2fb468f9851be71eaf644bd40c7f0ac6dd40991bc71","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107057,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z6\",\"final_states\":[\"accept6\"],\"input\":[\"a6\",\"a6\",\"b6\",\"b6\"],\"input_alphabet\":[\"a6\",\"b6\"],\"stack_alphabet\":[\"A6\",\"Z6\"],\"start_state\":\"push6\",\"states\":[\"accept6\",\"pop6\",\"push6\"],\"trace\":[[\"push6\",0,[\"Z6\"]],[\"push6\",1,[\"Z6\",\"A6\"]],[\"push6\",2,[\"Z6\",\"A6\",\"A6\"]],[\"pop6\",3,[\"Z6\",\"A6\"]],[\"pop6\",4,[\"Z6\"]],[\"accept6\",4,[\"Z6\"]]],\"transitions\":[[\"pop6\",\"b6\",\"A6\",\"pop6\",[]],[\"pop6\",null,\"Z6\",\"accept6\",[\"Z6\"]],[\"push6\",\"a6\",\"A6\",\"push6\",[\"A6\",\"A6\"]],[\"push6\",\"a6\",\"Z6\",\"push6\",[\"Z6\",\"A6\"]],[\"push6\",\"b6\",\"A6\",\"pop6\",[]],[\"push6\",null,\"Z6\",\"accept6\",[\"Z6\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push6, pop6, accept6], input_alphabet=[a6, b6], stack_alphabet=[Z6, A6], start_state=push6, bottom_symbol=Z6, final_states=[accept6], transitions=[[push6, a6, Z6, push6, [Z6, A6]], [push6, a6, A6, push6, [A6, A6]], [push6, b6, A6, pop6, []], [pop6, b6, A6, pop6, []], [push6, null, Z6, accept6, [Z6]], [pop6, null, Z6, accept6, [Z6]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a6, a6, b6, b6], supplied trace=[[push6, 0, [Z6]], [push6, 1, [Z6, A6]], [push6, 2, [Z6, A6, A6]], [pop6, 3, [Z6, A6]], [pop6, 4, [Z6]], [accept6, 4, [Z6]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a6, a6, b6, b6], [[push6, 0, [Z6]], [push6, 1, [Z6, A6]], [push6, 2, [Z6, A6, A6]], [pop6, 3, [Z6, A6]], [pop6, 4, [Z6]], [accept6, 4, [Z6]]], [0, 1, 2, 3, 5], Yes, Yes, 5, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a6, a6, b6, b6], [[push6, 0, [Z6]], [push6, 1, [Z6, A6]], [push6, 2, [Z6, A6, A6]], [pop6, 3, [Z6, A6]], [pop6, 4, [Z6]], [accept6, 4, [Z6]]], [0, 1, 2, 3, 5], Yes, Yes, 5, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0058","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42f426d1a3db9d338a864344a91eea878d4c8a2169fb58f906778ec7ef89af91","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107058,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z7\",\"final_states\":[\"accept7\"],\"input\":[\"a7\",\"a7\",\"a7\",\"b7\",\"b7\",\"b7\"],\"input_alphabet\":[\"a7\",\"b7\"],\"stack_alphabet\":[\"A7\",\"Z7\"],\"start_state\":\"push7\",\"states\":[\"accept7\",\"pop7\",\"push7\"],\"trace\":[[\"push7\",0,[\"Z7\"]],[\"push7\",1,[\"Z7\",\"A7\"]],[\"push7\",2,[\"Z7\",\"A7\",\"A7\"]],[\"push7\",3,[\"Z7\",\"A7\",\"A7\",\"A7\"]],[\"pop7\",4,[\"Z7\",\"A7\",\"A7\"]],[\"pop7\",5,[\"Z7\",\"A7\"]],[\"pop7\",6,[\"Z7\"]],[\"accept7\",6,[\"Z7\"]]],\"transitions\":[[\"pop7\",\"b7\",\"A7\",\"pop7\",[]],[\"pop7\",null,\"Z7\",\"accept7\",[\"Z7\"]],[\"push7\",\"a7\",\"A7\",\"push7\",[\"A7\",\"A7\"]],[\"push7\",\"a7\",\"Z7\",\"push7\",[\"Z7\",\"A7\"]],[\"push7\",\"b7\",\"A7\",\"pop7\",[]],[\"push7\",null,\"Z7\",\"accept7\",[\"Z7\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push7, pop7, accept7], input_alphabet=[a7, b7], stack_alphabet=[Z7, A7], start_state=push7, bottom_symbol=Z7, final_states=[accept7], transitions=[[push7, a7, Z7, push7, [Z7, A7]], [push7, a7, A7, push7, [A7, A7]], [push7, b7, A7, pop7, []], [pop7, b7, A7, pop7, []], [push7, null, Z7, accept7, [Z7]], [pop7, null, Z7, accept7, [Z7]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a7, a7, a7, b7, b7, b7], supplied trace=[[push7, 0, [Z7]], [push7, 1, [Z7, A7]], [push7, 2, [Z7, A7, A7]], [push7, 3, [Z7, A7, A7, A7]], [pop7, 4, [Z7, A7, A7]], [pop7, 5, [Z7, A7]], [pop7, 6, [Z7]], [accept7, 6, [Z7]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a7, a7, a7, b7, b7, b7], [[push7, 0, [Z7]], [push7, 1, [Z7, A7]], [push7, 2, [Z7, A7, A7]], [push7, 3, [Z7, A7, A7, A7]], [pop7, 4, [Z7, A7, A7]], [pop7, 5, [Z7, A7]], [pop7, 6, [Z7]], [accept7, 6, [Z7]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, 7, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a7, a7, a7, b7, b7, b7], [[push7, 0, [Z7]], [push7, 1, [Z7, A7]], [push7, 2, [Z7, A7, A7]], [push7, 3, [Z7, A7, A7, A7]], [pop7, 4, [Z7, A7, A7]], [pop7, 5, [Z7, A7]], [pop7, 6, [Z7]], [accept7, 6, [Z7]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, 7, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0059","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d1b1e59c6c0642bc1fbe4fa829d4705d6d65639d1b9a4afc1eb103a54a32098","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107059,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z8\",\"final_states\":[\"accept8\"],\"input\":[],\"input_alphabet\":[\"a8\",\"b8\"],\"stack_alphabet\":[\"A8\",\"Z8\"],\"start_state\":\"push8\",\"states\":[\"accept8\",\"pop8\",\"push8\"],\"trace\":[[\"push8\",0,[\"Z8\"]],[\"accept8\",0,[\"Z8\"]]],\"transitions\":[[\"pop8\",\"b8\",\"A8\",\"pop8\",[]],[\"pop8\",null,\"Z8\",\"accept8\",[\"Z8\"]],[\"push8\",\"a8\",\"A8\",\"push8\",[\"A8\",\"A8\"]],[\"push8\",\"a8\",\"Z8\",\"push8\",[\"Z8\",\"A8\"]],[\"push8\",\"b8\",\"A8\",\"pop8\",[]],[\"push8\",null,\"Z8\",\"accept8\",[\"Z8\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push8, pop8, accept8], input_alphabet=[a8, b8], stack_alphabet=[Z8, A8], start_state=push8, bottom_symbol=Z8, final_states=[accept8], transitions=[[push8, a8, Z8, push8, [Z8, A8]], [push8, a8, A8, push8, [A8, A8]], [push8, b8, A8, pop8, []], [pop8, b8, A8, pop8, []], [push8, null, Z8, accept8, [Z8]], [pop8, null, Z8, accept8, [Z8]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[], supplied trace=[[push8, 0, [Z8]], [accept8, 0, [Z8]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([], [[push8, 0, [Z8]], [accept8, 0, [Z8]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([], [[push8, 0, [Z8]], [accept8, 0, [Z8]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0060","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5ad7a39b12b503887352900971dd11e6b521b20d06c31e7566244d197507be8","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107060,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z9\",\"final_states\":[\"accept9\"],\"input\":[\"a9\",\"b9\"],\"input_alphabet\":[\"a9\",\"b9\"],\"stack_alphabet\":[\"A9\",\"Z9\"],\"start_state\":\"push9\",\"states\":[\"accept9\",\"pop9\",\"push9\"],\"trace\":[[\"push9\",0,[\"Z9\"]],[\"push9\",1,[\"Z9\",\"A9\"]],[\"pop9\",2,[\"Z9\"]],[\"accept9\",2,[\"Z9\"]]],\"transitions\":[[\"pop9\",\"b9\",\"A9\",\"pop9\",[]],[\"pop9\",null,\"Z9\",\"accept9\",[\"Z9\"]],[\"push9\",\"a9\",\"A9\",\"push9\",[\"A9\",\"A9\"]],[\"push9\",\"a9\",\"Z9\",\"push9\",[\"Z9\",\"A9\"]],[\"push9\",\"b9\",\"A9\",\"pop9\",[]],[\"push9\",null,\"Z9\",\"accept9\",[\"Z9\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push9, pop9, accept9], input_alphabet=[a9, b9], stack_alphabet=[Z9, A9], start_state=push9, bottom_symbol=Z9, final_states=[accept9], transitions=[[push9, a9, Z9, push9, [Z9, A9]], [push9, a9, A9, push9, [A9, A9]], [push9, b9, A9, pop9, []], [pop9, b9, A9, pop9, []], [push9, null, Z9, accept9, [Z9]], [pop9, null, Z9, accept9, [Z9]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a9, b9], supplied trace=[[push9, 0, [Z9]], [push9, 1, [Z9, A9]], [pop9, 2, [Z9]], [accept9, 2, [Z9]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a9, b9], [[push9, 0, [Z9]], [push9, 1, [Z9, A9]], [pop9, 2, [Z9]], [accept9, 2, [Z9]]], [0, 2, 5], Yes, Yes, 3, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a9, b9], [[push9, 0, [Z9]], [push9, 1, [Z9, A9]], [pop9, 2, [Z9]], [accept9, 2, [Z9]]], [0, 2, 5], Yes, Yes, 3, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0061","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"267eed6dad15a9fb60bac514d97d441f52e605a009c4ca43576cdd0fa02e38b6","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107061,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z10\",\"final_states\":[\"accept10\"],\"input\":[\"a10\",\"a10\",\"b10\",\"b10\"],\"input_alphabet\":[\"a10\",\"b10\"],\"stack_alphabet\":[\"A10\",\"Z10\"],\"start_state\":\"push10\",\"states\":[\"accept10\",\"pop10\",\"push10\"],\"trace\":[[\"push10\",0,[\"Z10\"]],[\"push10\",1,[\"Z10\",\"A10\"]],[\"push10\",2,[\"Z10\",\"A10\",\"A10\"]],[\"pop10\",3,[\"Z10\",\"A10\"]],[\"pop10\",4,[\"Z10\"]],[\"accept10\",4,[\"Z10\"]]],\"transitions\":[[\"pop10\",\"b10\",\"A10\",\"pop10\",[]],[\"pop10\",null,\"Z10\",\"accept10\",[\"Z10\"]],[\"push10\",\"a10\",\"A10\",\"push10\",[\"A10\",\"A10\"]],[\"push10\",\"a10\",\"Z10\",\"push10\",[\"Z10\",\"A10\"]],[\"push10\",\"b10\",\"A10\",\"pop10\",[]],[\"push10\",null,\"Z10\",\"accept10\",[\"Z10\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push10, pop10, accept10], input_alphabet=[a10, b10], stack_alphabet=[Z10, A10], start_state=push10, bottom_symbol=Z10, final_states=[accept10], transitions=[[push10, a10, Z10, push10, [Z10, A10]], [push10, a10, A10, push10, [A10, A10]], [push10, b10, A10, pop10, []], [pop10, b10, A10, pop10, []], [push10, null, Z10, accept10, [Z10]], [pop10, null, Z10, accept10, [Z10]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a10, a10, b10, b10], supplied trace=[[push10, 0, [Z10]], [push10, 1, [Z10, A10]], [push10, 2, [Z10, A10, A10]], [pop10, 3, [Z10, A10]], [pop10, 4, [Z10]], [accept10, 4, [Z10]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a10, a10, b10, b10], [[push10, 0, [Z10]], [push10, 1, [Z10, A10]], [push10, 2, [Z10, A10, A10]], [pop10, 3, [Z10, A10]], [pop10, 4, [Z10]], [accept10, 4, [Z10]]], [0, 1, 2, 3, 5], Yes, Yes, 5, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a10, a10, b10, b10], [[push10, 0, [Z10]], [push10, 1, [Z10, A10]], [push10, 2, [Z10, A10, A10]], [pop10, 3, [Z10, A10]], [pop10, 4, [Z10]], [accept10, 4, [Z10]]], [0, 1, 2, 3, 5], Yes, Yes, 5, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0062","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73b2d73ca3f4ecea4eced23ec24ecbb1adb1929539172b47cad1730de0d9a879","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107062,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z11\",\"final_states\":[\"accept11\"],\"input\":[\"a11\",\"a11\",\"a11\",\"b11\",\"b11\",\"b11\"],\"input_alphabet\":[\"a11\",\"b11\"],\"stack_alphabet\":[\"A11\",\"Z11\"],\"start_state\":\"push11\",\"states\":[\"accept11\",\"pop11\",\"push11\"],\"trace\":[[\"push11\",0,[\"Z11\"]],[\"push11\",1,[\"Z11\",\"A11\"]],[\"push11\",2,[\"Z11\",\"A11\",\"A11\"]],[\"push11\",3,[\"Z11\",\"A11\",\"A11\",\"A11\"]],[\"pop11\",4,[\"Z11\",\"A11\",\"A11\"]],[\"pop11\",5,[\"Z11\",\"A11\"]],[\"pop11\",6,[\"Z11\"]],[\"accept11\",6,[\"Z11\"]]],\"transitions\":[[\"pop11\",\"b11\",\"A11\",\"pop11\",[]],[\"pop11\",null,\"Z11\",\"accept11\",[\"Z11\"]],[\"push11\",\"a11\",\"A11\",\"push11\",[\"A11\",\"A11\"]],[\"push11\",\"a11\",\"Z11\",\"push11\",[\"Z11\",\"A11\"]],[\"push11\",\"b11\",\"A11\",\"pop11\",[]],[\"push11\",null,\"Z11\",\"accept11\",[\"Z11\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push11, pop11, accept11], input_alphabet=[a11, b11], stack_alphabet=[Z11, A11], start_state=push11, bottom_symbol=Z11, final_states=[accept11], transitions=[[push11, a11, Z11, push11, [Z11, A11]], [push11, a11, A11, push11, [A11, A11]], [push11, b11, A11, pop11, []], [pop11, b11, A11, pop11, []], [push11, null, Z11, accept11, [Z11]], [pop11, null, Z11, accept11, [Z11]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a11, a11, a11, b11, b11, b11], supplied trace=[[push11, 0, [Z11]], [push11, 1, [Z11, A11]], [push11, 2, [Z11, A11, A11]], [push11, 3, [Z11, A11, A11, A11]], [pop11, 4, [Z11, A11, A11]], [pop11, 5, [Z11, A11]], [pop11, 6, [Z11]], [accept11, 6, [Z11]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a11, a11, a11, b11, b11, b11], [[push11, 0, [Z11]], [push11, 1, [Z11, A11]], [push11, 2, [Z11, A11, A11]], [push11, 3, [Z11, A11, A11, A11]], [pop11, 4, [Z11, A11, A11]], [pop11, 5, [Z11, A11]], [pop11, 6, [Z11]], [accept11, 6, [Z11]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, 7, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a11, a11, a11, b11, b11, b11], [[push11, 0, [Z11]], [push11, 1, [Z11, A11]], [push11, 2, [Z11, A11, A11]], [push11, 3, [Z11, A11, A11, A11]], [pop11, 4, [Z11, A11, A11]], [pop11, 5, [Z11, A11]], [pop11, 6, [Z11]], [accept11, 6, [Z11]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, 7, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0063","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"868f6a23d663fea859767c6569bf0605193f240551b9f7f1a878aa31b9cfa956","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107063,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z12\",\"final_states\":[\"accept12\"],\"input\":[],\"input_alphabet\":[\"a12\",\"b12\"],\"stack_alphabet\":[\"A12\",\"Z12\"],\"start_state\":\"push12\",\"states\":[\"accept12\",\"pop12\",\"push12\"],\"trace\":[[\"push12\",0,[\"Z12\"]],[\"accept12\",0,[\"Z12\"]]],\"transitions\":[[\"pop12\",\"b12\",\"A12\",\"pop12\",[]],[\"pop12\",null,\"Z12\",\"accept12\",[\"Z12\"]],[\"push12\",\"a12\",\"A12\",\"push12\",[\"A12\",\"A12\"]],[\"push12\",\"a12\",\"Z12\",\"push12\",[\"Z12\",\"A12\"]],[\"push12\",\"b12\",\"A12\",\"pop12\",[]],[\"push12\",null,\"Z12\",\"accept12\",[\"Z12\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push12, pop12, accept12], input_alphabet=[a12, b12], stack_alphabet=[Z12, A12], start_state=push12, bottom_symbol=Z12, final_states=[accept12], transitions=[[push12, a12, Z12, push12, [Z12, A12]], [push12, a12, A12, push12, [A12, A12]], [push12, b12, A12, pop12, []], [pop12, b12, A12, pop12, []], [push12, null, Z12, accept12, [Z12]], [pop12, null, Z12, accept12, [Z12]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[], supplied trace=[[push12, 0, [Z12]], [accept12, 0, [Z12]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([], [[push12, 0, [Z12]], [accept12, 0, [Z12]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([], [[push12, 0, [Z12]], [accept12, 0, [Z12]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0064","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bf9bd9146b70f6cfc673bdcdbc59f3204c57f445934ffc400ef72c797e3bb70","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107064,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z13\",\"final_states\":[\"accept13\"],\"input\":[\"a13\",\"b13\"],\"input_alphabet\":[\"a13\",\"b13\"],\"stack_alphabet\":[\"A13\",\"Z13\"],\"start_state\":\"push13\",\"states\":[\"accept13\",\"pop13\",\"push13\"],\"trace\":[[\"push13\",0,[\"Z13\"]],[\"push13\",1,[\"Z13\",\"A13\"]],[\"pop13\",2,[\"Z13\"]],[\"accept13\",2,[\"Z13\"]]],\"transitions\":[[\"pop13\",\"b13\",\"A13\",\"pop13\",[]],[\"pop13\",null,\"Z13\",\"accept13\",[\"Z13\"]],[\"push13\",\"a13\",\"A13\",\"push13\",[\"A13\",\"A13\"]],[\"push13\",\"a13\",\"Z13\",\"push13\",[\"Z13\",\"A13\"]],[\"push13\",\"b13\",\"A13\",\"pop13\",[]],[\"push13\",null,\"Z13\",\"accept13\",[\"Z13\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push13, pop13, accept13], input_alphabet=[a13, b13], stack_alphabet=[Z13, A13], start_state=push13, bottom_symbol=Z13, final_states=[accept13], transitions=[[push13, a13, Z13, push13, [Z13, A13]], [push13, a13, A13, push13, [A13, A13]], [push13, b13, A13, pop13, []], [pop13, b13, A13, pop13, []], [push13, null, Z13, accept13, [Z13]], [pop13, null, Z13, accept13, [Z13]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a13, b13], supplied trace=[[push13, 0, [Z13]], [push13, 1, [Z13, A13]], [pop13, 2, [Z13]], [accept13, 2, [Z13]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a13, b13], [[push13, 0, [Z13]], [push13, 1, [Z13, A13]], [pop13, 2, [Z13]], [accept13, 2, [Z13]]], [0, 2, 5], Yes, Yes, 3, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a13, b13], [[push13, 0, [Z13]], [push13, 1, [Z13, A13]], [pop13, 2, [Z13]], [accept13, 2, [Z13]]], [0, 2, 5], Yes, Yes, 3, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0065","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"845b33a2ad8cfdc78b7aeb18a3088492454825395c6ec9b64604523abe99d063","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107065,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z14\",\"final_states\":[\"accept14\"],\"input\":[\"a14\",\"a14\",\"b14\",\"b14\"],\"input_alphabet\":[\"a14\",\"b14\"],\"stack_alphabet\":[\"A14\",\"Z14\"],\"start_state\":\"push14\",\"states\":[\"accept14\",\"pop14\",\"push14\"],\"trace\":[[\"push14\",0,[\"Z14\"]],[\"push14\",1,[\"Z14\",\"A14\"]],[\"push14\",2,[\"Z14\",\"A14\",\"A14\"]],[\"pop14\",3,[\"Z14\",\"A14\"]],[\"pop14\",4,[\"Z14\"]],[\"accept14\",4,[\"Z14\"]]],\"transitions\":[[\"pop14\",\"b14\",\"A14\",\"pop14\",[]],[\"pop14\",null,\"Z14\",\"accept14\",[\"Z14\"]],[\"push14\",\"a14\",\"A14\",\"push14\",[\"A14\",\"A14\"]],[\"push14\",\"a14\",\"Z14\",\"push14\",[\"Z14\",\"A14\"]],[\"push14\",\"b14\",\"A14\",\"pop14\",[]],[\"push14\",null,\"Z14\",\"accept14\",[\"Z14\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push14, pop14, accept14], input_alphabet=[a14, b14], stack_alphabet=[Z14, A14], start_state=push14, bottom_symbol=Z14, final_states=[accept14], transitions=[[push14, a14, Z14, push14, [Z14, A14]], [push14, a14, A14, push14, [A14, A14]], [push14, b14, A14, pop14, []], [pop14, b14, A14, pop14, []], [push14, null, Z14, accept14, [Z14]], [pop14, null, Z14, accept14, [Z14]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a14, a14, b14, b14], supplied trace=[[push14, 0, [Z14]], [push14, 1, [Z14, A14]], [push14, 2, [Z14, A14, A14]], [pop14, 3, [Z14, A14]], [pop14, 4, [Z14]], [accept14, 4, [Z14]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a14, a14, b14, b14], [[push14, 0, [Z14]], [push14, 1, [Z14, A14]], [push14, 2, [Z14, A14, A14]], [pop14, 3, [Z14, A14]], [pop14, 4, [Z14]], [accept14, 4, [Z14]]], [0, 1, 2, 3, 5], Yes, Yes, 5, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a14, a14, b14, b14], [[push14, 0, [Z14]], [push14, 1, [Z14, A14]], [push14, 2, [Z14, A14, A14]], [pop14, 3, [Z14, A14]], [pop14, 4, [Z14]], [accept14, 4, [Z14]]], [0, 1, 2, 3, 5], Yes, Yes, 5, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0066","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8074cea164e157dcd7f2b7641ef2eb42efbeca0484638706b1dbf62fc8aa41b","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107066,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z15\",\"final_states\":[\"accept15\"],\"input\":[\"a15\",\"a15\",\"a15\",\"b15\",\"b15\",\"b15\"],\"input_alphabet\":[\"a15\",\"b15\"],\"stack_alphabet\":[\"A15\",\"Z15\"],\"start_state\":\"push15\",\"states\":[\"accept15\",\"pop15\",\"push15\"],\"trace\":[[\"push15\",0,[\"Z15\"]],[\"push15\",1,[\"Z15\",\"A15\"]],[\"push15\",2,[\"Z15\",\"A15\",\"A15\"]],[\"push15\",3,[\"Z15\",\"A15\",\"A15\",\"A15\"]],[\"pop15\",4,[\"Z15\",\"A15\",\"A15\"]],[\"pop15\",5,[\"Z15\",\"A15\"]],[\"pop15\",6,[\"Z15\"]],[\"accept15\",6,[\"Z15\"]]],\"transitions\":[[\"pop15\",\"b15\",\"A15\",\"pop15\",[]],[\"pop15\",null,\"Z15\",\"accept15\",[\"Z15\"]],[\"push15\",\"a15\",\"A15\",\"push15\",[\"A15\",\"A15\"]],[\"push15\",\"a15\",\"Z15\",\"push15\",[\"Z15\",\"A15\"]],[\"push15\",\"b15\",\"A15\",\"pop15\",[]],[\"push15\",null,\"Z15\",\"accept15\",[\"Z15\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push15, pop15, accept15], input_alphabet=[a15, b15], stack_alphabet=[Z15, A15], start_state=push15, bottom_symbol=Z15, final_states=[accept15], transitions=[[push15, a15, Z15, push15, [Z15, A15]], [push15, a15, A15, push15, [A15, A15]], [push15, b15, A15, pop15, []], [pop15, b15, A15, pop15, []], [push15, null, Z15, accept15, [Z15]], [pop15, null, Z15, accept15, [Z15]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a15, a15, a15, b15, b15, b15], supplied trace=[[push15, 0, [Z15]], [push15, 1, [Z15, A15]], [push15, 2, [Z15, A15, A15]], [push15, 3, [Z15, A15, A15, A15]], [pop15, 4, [Z15, A15, A15]], [pop15, 5, [Z15, A15]], [pop15, 6, [Z15]], [accept15, 6, [Z15]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a15, a15, a15, b15, b15, b15], [[push15, 0, [Z15]], [push15, 1, [Z15, A15]], [push15, 2, [Z15, A15, A15]], [push15, 3, [Z15, A15, A15, A15]], [pop15, 4, [Z15, A15, A15]], [pop15, 5, [Z15, A15]], [pop15, 6, [Z15]], [accept15, 6, [Z15]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, 7, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a15, a15, a15, b15, b15, b15], [[push15, 0, [Z15]], [push15, 1, [Z15, A15]], [push15, 2, [Z15, A15, A15]], [push15, 3, [Z15, A15, A15, A15]], [pop15, 4, [Z15, A15, A15]], [pop15, 5, [Z15, A15]], [pop15, 6, [Z15]], [accept15, 6, [Z15]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, 7, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0067","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e9b045183e29c8f44ee14f069c3201372ae0786980a6fc8d1e62453e874bb1b","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107067,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z16\",\"final_states\":[\"accept16\"],\"input\":[],\"input_alphabet\":[\"a16\",\"b16\"],\"stack_alphabet\":[\"A16\",\"Z16\"],\"start_state\":\"push16\",\"states\":[\"accept16\",\"pop16\",\"push16\"],\"trace\":[[\"push16\",0,[\"Z16\"]],[\"accept16\",0,[\"Z16\"]]],\"transitions\":[[\"pop16\",\"b16\",\"A16\",\"pop16\",[]],[\"pop16\",null,\"Z16\",\"accept16\",[\"Z16\"]],[\"push16\",\"a16\",\"A16\",\"push16\",[\"A16\",\"A16\"]],[\"push16\",\"a16\",\"Z16\",\"push16\",[\"Z16\",\"A16\"]],[\"push16\",\"b16\",\"A16\",\"pop16\",[]],[\"push16\",null,\"Z16\",\"accept16\",[\"Z16\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push16, pop16, accept16], input_alphabet=[a16, b16], stack_alphabet=[Z16, A16], start_state=push16, bottom_symbol=Z16, final_states=[accept16], transitions=[[push16, a16, Z16, push16, [Z16, A16]], [push16, a16, A16, push16, [A16, A16]], [push16, b16, A16, pop16, []], [pop16, b16, A16, pop16, []], [push16, null, Z16, accept16, [Z16]], [pop16, null, Z16, accept16, [Z16]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[], supplied trace=[[push16, 0, [Z16]], [accept16, 0, [Z16]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([], [[push16, 0, [Z16]], [accept16, 0, [Z16]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([], [[push16, 0, [Z16]], [accept16, 0, [Z16]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0068","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c7d6b66fc6161398f0cd8d59b58bab65e69ab4e40247db8425aa2245d1e96d0","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107068,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z17\",\"final_states\":[\"accept17\"],\"input\":[\"a17\",\"b17\"],\"input_alphabet\":[\"a17\",\"b17\"],\"stack_alphabet\":[\"A17\",\"Z17\"],\"start_state\":\"push17\",\"states\":[\"accept17\",\"pop17\",\"push17\"],\"trace\":[[\"push17\",0,[\"Z17\"]],[\"push17\",1,[\"Z17\",\"A17\"]],[\"pop17\",2,[\"Z17\"]],[\"accept17\",2,[\"Z17\"]]],\"transitions\":[[\"pop17\",\"b17\",\"A17\",\"pop17\",[]],[\"pop17\",null,\"Z17\",\"accept17\",[\"Z17\"]],[\"push17\",\"a17\",\"A17\",\"push17\",[\"A17\",\"A17\"]],[\"push17\",\"a17\",\"Z17\",\"push17\",[\"Z17\",\"A17\"]],[\"push17\",\"b17\",\"A17\",\"pop17\",[]],[\"push17\",null,\"Z17\",\"accept17\",[\"Z17\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push17, pop17, accept17], input_alphabet=[a17, b17], stack_alphabet=[Z17, A17], start_state=push17, bottom_symbol=Z17, final_states=[accept17], transitions=[[push17, a17, Z17, push17, [Z17, A17]], [push17, a17, A17, push17, [A17, A17]], [push17, b17, A17, pop17, []], [pop17, b17, A17, pop17, []], [push17, null, Z17, accept17, [Z17]], [pop17, null, Z17, accept17, [Z17]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a17, b17], supplied trace=[[push17, 0, [Z17]], [push17, 1, [Z17, A17]], [pop17, 2, [Z17]], [accept17, 2, [Z17]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a17, b17], [[push17, 0, [Z17]], [push17, 1, [Z17, A17]], [pop17, 2, [Z17]], [accept17, 2, [Z17]]], [0, 2, 5], Yes, Yes, 3, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a17, b17], [[push17, 0, [Z17]], [push17, 1, [Z17, A17]], [pop17, 2, [Z17]], [accept17, 2, [Z17]]], [0, 2, 5], Yes, Yes, 3, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0069","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cdfeb3922f3312a26a45bfe6e7febc9b8ce0f956dcc94547e63f897acc5f3558","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107069,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z18\",\"final_states\":[\"accept18\"],\"input\":[\"a18\",\"a18\",\"b18\",\"b18\"],\"input_alphabet\":[\"a18\",\"b18\"],\"stack_alphabet\":[\"A18\",\"Z18\"],\"start_state\":\"push18\",\"states\":[\"accept18\",\"pop18\",\"push18\"],\"trace\":[[\"push18\",0,[\"Z18\"]],[\"push18\",1,[\"Z18\",\"A18\"]],[\"push18\",2,[\"Z18\",\"A18\",\"A18\"]],[\"pop18\",3,[\"Z18\",\"A18\"]],[\"pop18\",4,[\"Z18\"]],[\"accept18\",4,[\"Z18\"]]],\"transitions\":[[\"pop18\",\"b18\",\"A18\",\"pop18\",[]],[\"pop18\",null,\"Z18\",\"accept18\",[\"Z18\"]],[\"push18\",\"a18\",\"A18\",\"push18\",[\"A18\",\"A18\"]],[\"push18\",\"a18\",\"Z18\",\"push18\",[\"Z18\",\"A18\"]],[\"push18\",\"b18\",\"A18\",\"pop18\",[]],[\"push18\",null,\"Z18\",\"accept18\",[\"Z18\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push18, pop18, accept18], input_alphabet=[a18, b18], stack_alphabet=[Z18, A18], start_state=push18, bottom_symbol=Z18, final_states=[accept18], transitions=[[push18, a18, Z18, push18, [Z18, A18]], [push18, a18, A18, push18, [A18, A18]], [push18, b18, A18, pop18, []], [pop18, b18, A18, pop18, []], [push18, null, Z18, accept18, [Z18]], [pop18, null, Z18, accept18, [Z18]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a18, a18, b18, b18], supplied trace=[[push18, 0, [Z18]], [push18, 1, [Z18, A18]], [push18, 2, [Z18, A18, A18]], [pop18, 3, [Z18, A18]], [pop18, 4, [Z18]], [accept18, 4, [Z18]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a18, a18, b18, b18], [[push18, 0, [Z18]], [push18, 1, [Z18, A18]], [push18, 2, [Z18, A18, A18]], [pop18, 3, [Z18, A18]], [pop18, 4, [Z18]], [accept18, 4, [Z18]]], [0, 1, 2, 3, 5], Yes, Yes, 5, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a18, a18, b18, b18], [[push18, 0, [Z18]], [push18, 1, [Z18, A18]], [push18, 2, [Z18, A18, A18]], [pop18, 3, [Z18, A18]], [pop18, 4, [Z18]], [accept18, 4, [Z18]]], [0, 1, 2, 3, 5], Yes, Yes, 5, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0070","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a62ac9284a93446ae8a5f2b64eb06bfe3e5ead74e81e1aafbeaee6094847c7b","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107070,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z19\",\"final_states\":[\"accept19\"],\"input\":[\"a19\",\"a19\",\"a19\",\"b19\",\"b19\",\"b19\"],\"input_alphabet\":[\"a19\",\"b19\"],\"stack_alphabet\":[\"A19\",\"Z19\"],\"start_state\":\"push19\",\"states\":[\"accept19\",\"pop19\",\"push19\"],\"trace\":[[\"push19\",0,[\"Z19\"]],[\"push19\",1,[\"Z19\",\"A19\"]],[\"push19\",2,[\"Z19\",\"A19\",\"A19\"]],[\"push19\",3,[\"Z19\",\"A19\",\"A19\",\"A19\"]],[\"pop19\",4,[\"Z19\",\"A19\",\"A19\"]],[\"pop19\",5,[\"Z19\",\"A19\"]],[\"pop19\",6,[\"Z19\"]],[\"accept19\",6,[\"Z19\"]]],\"transitions\":[[\"pop19\",\"b19\",\"A19\",\"pop19\",[]],[\"pop19\",null,\"Z19\",\"accept19\",[\"Z19\"]],[\"push19\",\"a19\",\"A19\",\"push19\",[\"A19\",\"A19\"]],[\"push19\",\"a19\",\"Z19\",\"push19\",[\"Z19\",\"A19\"]],[\"push19\",\"b19\",\"A19\",\"pop19\",[]],[\"push19\",null,\"Z19\",\"accept19\",[\"Z19\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push19, pop19, accept19], input_alphabet=[a19, b19], stack_alphabet=[Z19, A19], start_state=push19, bottom_symbol=Z19, final_states=[accept19], transitions=[[push19, a19, Z19, push19, [Z19, A19]], [push19, a19, A19, push19, [A19, A19]], [push19, b19, A19, pop19, []], [pop19, b19, A19, pop19, []], [push19, null, Z19, accept19, [Z19]], [pop19, null, Z19, accept19, [Z19]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a19, a19, a19, b19, b19, b19], supplied trace=[[push19, 0, [Z19]], [push19, 1, [Z19, A19]], [push19, 2, [Z19, A19, A19]], [push19, 3, [Z19, A19, A19, A19]], [pop19, 4, [Z19, A19, A19]], [pop19, 5, [Z19, A19]], [pop19, 6, [Z19]], [accept19, 6, [Z19]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a19, a19, a19, b19, b19, b19], [[push19, 0, [Z19]], [push19, 1, [Z19, A19]], [push19, 2, [Z19, A19, A19]], [push19, 3, [Z19, A19, A19, A19]], [pop19, 4, [Z19, A19, A19]], [pop19, 5, [Z19, A19]], [pop19, 6, [Z19]], [accept19, 6, [Z19]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, 7, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a19, a19, a19, b19, b19, b19], [[push19, 0, [Z19]], [push19, 1, [Z19, A19]], [push19, 2, [Z19, A19, A19]], [push19, 3, [Z19, A19, A19, A19]], [pop19, 4, [Z19, A19, A19]], [pop19, 5, [Z19, A19]], [pop19, 6, [Z19]], [accept19, 6, [Z19]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, 7, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0071","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e99800a9c4eb0ccb5f16a89132e38cfeba3b10de81bb20d1963bd77bf14357d","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107071,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z20\",\"final_states\":[\"accept20\"],\"input\":[],\"input_alphabet\":[\"a20\",\"b20\"],\"stack_alphabet\":[\"A20\",\"Z20\"],\"start_state\":\"push20\",\"states\":[\"accept20\",\"pop20\",\"push20\"],\"trace\":[[\"push20\",0,[\"Z20\"]],[\"accept20\",0,[\"Z20\"]]],\"transitions\":[[\"pop20\",\"b20\",\"A20\",\"pop20\",[]],[\"pop20\",null,\"Z20\",\"accept20\",[\"Z20\"]],[\"push20\",\"a20\",\"A20\",\"push20\",[\"A20\",\"A20\"]],[\"push20\",\"a20\",\"Z20\",\"push20\",[\"Z20\",\"A20\"]],[\"push20\",\"b20\",\"A20\",\"pop20\",[]],[\"push20\",null,\"Z20\",\"accept20\",[\"Z20\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push20, pop20, accept20], input_alphabet=[a20, b20], stack_alphabet=[Z20, A20], start_state=push20, bottom_symbol=Z20, final_states=[accept20], transitions=[[push20, a20, Z20, push20, [Z20, A20]], [push20, a20, A20, push20, [A20, A20]], [push20, b20, A20, pop20, []], [pop20, b20, A20, pop20, []], [push20, null, Z20, accept20, [Z20]], [pop20, null, Z20, accept20, [Z20]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[], supplied trace=[[push20, 0, [Z20]], [accept20, 0, [Z20]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([], [[push20, 0, [Z20]], [accept20, 0, [Z20]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([], [[push20, 0, [Z20]], [accept20, 0, [Z20]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0072","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a98d5370fa018149db58dc7d2caa348826ca20b14cc54222100ff8c9df5e3ea","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107072,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z21\",\"final_states\":[\"accept21\"],\"input\":[\"a21\",\"b21\"],\"input_alphabet\":[\"a21\",\"b21\"],\"stack_alphabet\":[\"A21\",\"Z21\"],\"start_state\":\"push21\",\"states\":[\"accept21\",\"pop21\",\"push21\"],\"trace\":[[\"push21\",0,[\"Z21\"]],[\"push21\",1,[\"Z21\",\"A21\"]],[\"pop21\",2,[\"Z21\"]],[\"accept21\",2,[\"Z21\"]]],\"transitions\":[[\"pop21\",\"b21\",\"A21\",\"pop21\",[]],[\"pop21\",null,\"Z21\",\"accept21\",[\"Z21\"]],[\"push21\",\"a21\",\"A21\",\"push21\",[\"A21\",\"A21\"]],[\"push21\",\"a21\",\"Z21\",\"push21\",[\"Z21\",\"A21\"]],[\"push21\",\"b21\",\"A21\",\"pop21\",[]],[\"push21\",null,\"Z21\",\"accept21\",[\"Z21\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push21, pop21, accept21], input_alphabet=[a21, b21], stack_alphabet=[Z21, A21], start_state=push21, bottom_symbol=Z21, final_states=[accept21], transitions=[[push21, a21, Z21, push21, [Z21, A21]], [push21, a21, A21, push21, [A21, A21]], [push21, b21, A21, pop21, []], [pop21, b21, A21, pop21, []], [push21, null, Z21, accept21, [Z21]], [pop21, null, Z21, accept21, [Z21]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a21, b21], supplied trace=[[push21, 0, [Z21]], [push21, 1, [Z21, A21]], [pop21, 2, [Z21]], [accept21, 2, [Z21]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a21, b21], [[push21, 0, [Z21]], [push21, 1, [Z21, A21]], [pop21, 2, [Z21]], [accept21, 2, [Z21]]], [0, 2, 5], Yes, Yes, 3, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a21, b21], [[push21, 0, [Z21]], [push21, 1, [Z21, A21]], [pop21, 2, [Z21]], [accept21, 2, [Z21]]], [0, 2, 5], Yes, Yes, 3, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0073","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2eab32edb27fd71fd258d1bd924fa42b1bf14969b0fbba97d858ba3217ef17e2","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107073,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z22\",\"final_states\":[\"accept22\"],\"input\":[\"a22\",\"a22\",\"b22\",\"b22\"],\"input_alphabet\":[\"a22\",\"b22\"],\"stack_alphabet\":[\"A22\",\"Z22\"],\"start_state\":\"push22\",\"states\":[\"accept22\",\"pop22\",\"push22\"],\"trace\":[[\"push22\",0,[\"Z22\"]],[\"push22\",1,[\"Z22\",\"A22\"]],[\"push22\",2,[\"Z22\",\"A22\",\"A22\"]],[\"pop22\",3,[\"Z22\",\"A22\"]],[\"pop22\",4,[\"Z22\"]],[\"accept22\",4,[\"Z22\"]]],\"transitions\":[[\"pop22\",\"b22\",\"A22\",\"pop22\",[]],[\"pop22\",null,\"Z22\",\"accept22\",[\"Z22\"]],[\"push22\",\"a22\",\"A22\",\"push22\",[\"A22\",\"A22\"]],[\"push22\",\"a22\",\"Z22\",\"push22\",[\"Z22\",\"A22\"]],[\"push22\",\"b22\",\"A22\",\"pop22\",[]],[\"push22\",null,\"Z22\",\"accept22\",[\"Z22\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push22, pop22, accept22], input_alphabet=[a22, b22], stack_alphabet=[Z22, A22], start_state=push22, bottom_symbol=Z22, final_states=[accept22], transitions=[[push22, a22, Z22, push22, [Z22, A22]], [push22, a22, A22, push22, [A22, A22]], [push22, b22, A22, pop22, []], [pop22, b22, A22, pop22, []], [push22, null, Z22, accept22, [Z22]], [pop22, null, Z22, accept22, [Z22]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a22, a22, b22, b22], supplied trace=[[push22, 0, [Z22]], [push22, 1, [Z22, A22]], [push22, 2, [Z22, A22, A22]], [pop22, 3, [Z22, A22]], [pop22, 4, [Z22]], [accept22, 4, [Z22]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a22, a22, b22, b22], [[push22, 0, [Z22]], [push22, 1, [Z22, A22]], [push22, 2, [Z22, A22, A22]], [pop22, 3, [Z22, A22]], [pop22, 4, [Z22]], [accept22, 4, [Z22]]], [0, 1, 2, 3, 5], Yes, Yes, 5, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a22, a22, b22, b22], [[push22, 0, [Z22]], [push22, 1, [Z22, A22]], [push22, 2, [Z22, A22, A22]], [pop22, 3, [Z22, A22]], [pop22, 4, [Z22]], [accept22, 4, [Z22]]], [0, 1, 2, 3, 5], Yes, Yes, 5, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0074","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b8c5aeaf4ececa97063d2f5acd2a8c410972fb98b1654aea05578355e2c7123","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107074,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z23\",\"final_states\":[\"accept23\"],\"input\":[\"a23\",\"a23\",\"a23\",\"b23\",\"b23\",\"b23\"],\"input_alphabet\":[\"a23\",\"b23\"],\"stack_alphabet\":[\"A23\",\"Z23\"],\"start_state\":\"push23\",\"states\":[\"accept23\",\"pop23\",\"push23\"],\"trace\":[[\"push23\",0,[\"Z23\"]],[\"push23\",1,[\"Z23\",\"A23\"]],[\"push23\",2,[\"Z23\",\"A23\",\"A23\"]],[\"push23\",3,[\"Z23\",\"A23\",\"A23\",\"A23\"]],[\"pop23\",4,[\"Z23\",\"A23\",\"A23\"]],[\"pop23\",5,[\"Z23\",\"A23\"]],[\"pop23\",6,[\"Z23\"]],[\"accept23\",6,[\"Z23\"]]],\"transitions\":[[\"pop23\",\"b23\",\"A23\",\"pop23\",[]],[\"pop23\",null,\"Z23\",\"accept23\",[\"Z23\"]],[\"push23\",\"a23\",\"A23\",\"push23\",[\"A23\",\"A23\"]],[\"push23\",\"a23\",\"Z23\",\"push23\",[\"Z23\",\"A23\"]],[\"push23\",\"b23\",\"A23\",\"pop23\",[]],[\"push23\",null,\"Z23\",\"accept23\",[\"Z23\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push23, pop23, accept23], input_alphabet=[a23, b23], stack_alphabet=[Z23, A23], start_state=push23, bottom_symbol=Z23, final_states=[accept23], transitions=[[push23, a23, Z23, push23, [Z23, A23]], [push23, a23, A23, push23, [A23, A23]], [push23, b23, A23, pop23, []], [pop23, b23, A23, pop23, []], [push23, null, Z23, accept23, [Z23]], [pop23, null, Z23, accept23, [Z23]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a23, a23, a23, b23, b23, b23], supplied trace=[[push23, 0, [Z23]], [push23, 1, [Z23, A23]], [push23, 2, [Z23, A23, A23]], [push23, 3, [Z23, A23, A23, A23]], [pop23, 4, [Z23, A23, A23]], [pop23, 5, [Z23, A23]], [pop23, 6, [Z23]], [accept23, 6, [Z23]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([a23, a23, a23, b23, b23, b23], [[push23, 0, [Z23]], [push23, 1, [Z23, A23]], [push23, 2, [Z23, A23, A23]], [push23, 3, [Z23, A23, A23, A23]], [pop23, 4, [Z23, A23, A23]], [pop23, 5, [Z23, A23]], [pop23, 6, [Z23]], [accept23, 6, [Z23]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, 7, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([a23, a23, a23, b23, b23, b23], [[push23, 0, [Z23]], [push23, 1, [Z23, A23]], [push23, 2, [Z23, A23, A23]], [push23, 3, [Z23, A23, A23, A23]], [pop23, 4, [Z23, A23, A23]], [pop23, 5, [Z23, A23]], [pop23, 6, [Z23]], [accept23, 6, [Z23]]], [0, 1, 1, 2, 3, 3, 5], Yes, Yes, 7, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0075","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c46aca70220ec167dd2d3b0c2e4776b586f0da725a4dd88c5378476c27de708","family_id":"qf-mod41-pda-accepting-run-witness","seed":202611107075,"semantic_template":"qf-mod41-pda-accepting-run-witness/{\"bottom_symbol\":\"Z24\",\"final_states\":[\"accept24\"],\"input\":[],\"input_alphabet\":[\"a24\",\"b24\"],\"stack_alphabet\":[\"A24\",\"Z24\"],\"start_state\":\"push24\",\"states\":[\"accept24\",\"pop24\",\"push24\"],\"trace\":[[\"push24\",0,[\"Z24\"]],[\"accept24\",0,[\"Z24\"]]],\"transitions\":[[\"pop24\",\"b24\",\"A24\",\"pop24\",[]],[\"pop24\",null,\"Z24\",\"accept24\",[\"Z24\"]],[\"push24\",\"a24\",\"A24\",\"push24\",[\"A24\",\"A24\"]],[\"push24\",\"a24\",\"Z24\",\"push24\",[\"Z24\",\"A24\"]],[\"push24\",\"b24\",\"A24\",\"pop24\",[]],[\"push24\",null,\"Z24\",\"accept24\",[\"Z24\"]]]}"},"fields":{"title":"A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness","stem":"Use the complete finite nondeterministic PDA: states=[push24, pop24, accept24], input_alphabet=[a24, b24], stack_alphabet=[Z24, A24], start_state=push24, bottom_symbol=Z24, final_states=[accept24], transitions=[[push24, a24, Z24, push24, [Z24, A24]], [push24, a24, A24, push24, [A24, A24]], [push24, b24, A24, pop24, []], [pop24, b24, A24, pop24, []], [push24, null, Z24, accept24, [Z24]], [pop24, null, Z24, accept24, [Z24]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[], supplied trace=[[push24, 0, [Z24]], [accept24, 0, [Z24]]]. Give (input, trace, transition_indices, valid_run, accepted, step_count, certificate).","answer":{"kind":"expression","value":"([], [[push24, 0, [Z24]], [accept24, 0, [Z24]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)","explanation":"Audit every state/input/stack update and the initial configuration, then require full consumption and final state. Report the step count. A nonempty residual stack is compatible with this convention, including the retained bottom. The exact result, retaining the requested order, is ([], [[push24, 0, [Z24]], [accept24, 0, [Z24]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)."},"skills":["A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0076","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6af943c0f6919ea7c7118833954e7e77d284017efc6a56fed4af0fc13c72002c","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107076,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z0\",\"final_states\":[\"accept0\"],\"input\":[\"a0\",\"b0\"],\"input_alphabet\":[\"a0\",\"b0\"],\"max_steps\":1,\"stack_alphabet\":[\"A0\",\"Z0\"],\"start_state\":\"push0\",\"states\":[\"accept0\",\"pop0\",\"push0\"],\"transitions\":[[\"pop0\",\"b0\",\"A0\",\"pop0\",[]],[\"pop0\",null,\"Z0\",\"accept0\",[\"Z0\"]],[\"push0\",\"a0\",\"A0\",\"push0\",[\"A0\",\"A0\"]],[\"push0\",\"a0\",\"Z0\",\"push0\",[\"Z0\",\"A0\"]],[\"push0\",\"b0\",\"A0\",\"pop0\",[]],[\"push0\",null,\"Z0\",\"accept0\",[\"Z0\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push0, pop0, accept0], input_alphabet=[a0, b0], stack_alphabet=[Z0, A0], start_state=push0, bottom_symbol=Z0, final_states=[accept0], transitions=[[push0, a0, Z0, push0, [Z0, A0]], [push0, a0, A0, push0, [A0, A0]], [push0, b0, A0, pop0, []], [pop0, b0, A0, pop0, []], [push0, null, Z0, accept0, [Z0]], [pop0, null, Z0, accept0, [Z0]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a0, b0], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a0, b0], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a0, b0], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0077","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f64319b91a2c1f03a32bcdfbcd6a0aa8b9b2a92d454624009e35a0ab0adb309","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107077,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z1\",\"final_states\":[\"accept1\"],\"input\":[\"a1\",\"a1\",\"b1\",\"b1\"],\"input_alphabet\":[\"a1\",\"b1\"],\"max_steps\":1,\"stack_alphabet\":[\"A1\",\"Z1\"],\"start_state\":\"push1\",\"states\":[\"accept1\",\"pop1\",\"push1\"],\"transitions\":[[\"pop1\",\"b1\",\"A1\",\"pop1\",[]],[\"pop1\",null,\"Z1\",\"accept1\",[\"Z1\"]],[\"push1\",\"a1\",\"A1\",\"push1\",[\"A1\",\"A1\"]],[\"push1\",\"a1\",\"Z1\",\"push1\",[\"Z1\",\"A1\"]],[\"push1\",\"b1\",\"A1\",\"pop1\",[]],[\"push1\",null,\"Z1\",\"accept1\",[\"Z1\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push1, pop1, accept1], input_alphabet=[a1, b1], stack_alphabet=[Z1, A1], start_state=push1, bottom_symbol=Z1, final_states=[accept1], transitions=[[push1, a1, Z1, push1, [Z1, A1]], [push1, a1, A1, push1, [A1, A1]], [push1, b1, A1, pop1, []], [pop1, b1, A1, pop1, []], [push1, null, Z1, accept1, [Z1]], [pop1, null, Z1, accept1, [Z1]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a1, a1, b1, b1], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a1, a1, b1, b1], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a1, a1, b1, b1], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0078","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"437d4938550108519b538ca9eecc1b2a91a51f4e077e32cf7094917d87d19b5a","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107078,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z2\",\"final_states\":[\"accept2\"],\"input\":[\"a2\",\"a2\",\"a2\",\"b2\",\"b2\",\"b2\"],\"input_alphabet\":[\"a2\",\"b2\"],\"max_steps\":1,\"stack_alphabet\":[\"A2\",\"Z2\"],\"start_state\":\"push2\",\"states\":[\"accept2\",\"pop2\",\"push2\"],\"transitions\":[[\"pop2\",\"b2\",\"A2\",\"pop2\",[]],[\"pop2\",null,\"Z2\",\"accept2\",[\"Z2\"]],[\"push2\",\"a2\",\"A2\",\"push2\",[\"A2\",\"A2\"]],[\"push2\",\"a2\",\"Z2\",\"push2\",[\"Z2\",\"A2\"]],[\"push2\",\"b2\",\"A2\",\"pop2\",[]],[\"push2\",null,\"Z2\",\"accept2\",[\"Z2\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push2, pop2, accept2], input_alphabet=[a2, b2], stack_alphabet=[Z2, A2], start_state=push2, bottom_symbol=Z2, final_states=[accept2], transitions=[[push2, a2, Z2, push2, [Z2, A2]], [push2, a2, A2, push2, [A2, A2]], [push2, b2, A2, pop2, []], [pop2, b2, A2, pop2, []], [push2, null, Z2, accept2, [Z2]], [pop2, null, Z2, accept2, [Z2]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a2, a2, a2, b2, b2, b2], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a2, a2, a2, b2, b2, b2], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a2, a2, a2, b2, b2, b2], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0079","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b1acf901d494af3a393d40913215ec21b74ca64f04e9c1f26092517495e0a5e","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107079,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z3\",\"final_states\":[\"accept3\"],\"input\":[\"a3\",\"a3\",\"a3\",\"a3\",\"b3\",\"b3\",\"b3\",\"b3\"],\"input_alphabet\":[\"a3\",\"b3\"],\"max_steps\":1,\"stack_alphabet\":[\"A3\",\"Z3\"],\"start_state\":\"push3\",\"states\":[\"accept3\",\"pop3\",\"push3\"],\"transitions\":[[\"pop3\",\"b3\",\"A3\",\"pop3\",[]],[\"pop3\",null,\"Z3\",\"accept3\",[\"Z3\"]],[\"push3\",\"a3\",\"A3\",\"push3\",[\"A3\",\"A3\"]],[\"push3\",\"a3\",\"Z3\",\"push3\",[\"Z3\",\"A3\"]],[\"push3\",\"b3\",\"A3\",\"pop3\",[]],[\"push3\",null,\"Z3\",\"accept3\",[\"Z3\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push3, pop3, accept3], input_alphabet=[a3, b3], stack_alphabet=[Z3, A3], start_state=push3, bottom_symbol=Z3, final_states=[accept3], transitions=[[push3, a3, Z3, push3, [Z3, A3]], [push3, a3, A3, push3, [A3, A3]], [push3, b3, A3, pop3, []], [pop3, b3, A3, pop3, []], [push3, null, Z3, accept3, [Z3]], [pop3, null, Z3, accept3, [Z3]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a3, a3, a3, a3, b3, b3, b3, b3], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a3, a3, a3, a3, b3, b3, b3, b3], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a3, a3, a3, a3, b3, b3, b3, b3], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0080","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9d49b266932696197ba5433971e29d9fe184694e703ee0263a8461333c38207","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107080,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z4\",\"final_states\":[\"accept4\"],\"input\":[\"a4\",\"b4\"],\"input_alphabet\":[\"a4\",\"b4\"],\"max_steps\":1,\"stack_alphabet\":[\"A4\",\"Z4\"],\"start_state\":\"push4\",\"states\":[\"accept4\",\"pop4\",\"push4\"],\"transitions\":[[\"pop4\",\"b4\",\"A4\",\"pop4\",[]],[\"pop4\",null,\"Z4\",\"accept4\",[\"Z4\"]],[\"push4\",\"a4\",\"A4\",\"push4\",[\"A4\",\"A4\"]],[\"push4\",\"a4\",\"Z4\",\"push4\",[\"Z4\",\"A4\"]],[\"push4\",\"b4\",\"A4\",\"pop4\",[]],[\"push4\",null,\"Z4\",\"accept4\",[\"Z4\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push4, pop4, accept4], input_alphabet=[a4, b4], stack_alphabet=[Z4, A4], start_state=push4, bottom_symbol=Z4, final_states=[accept4], transitions=[[push4, a4, Z4, push4, [Z4, A4]], [push4, a4, A4, push4, [A4, A4]], [push4, b4, A4, pop4, []], [pop4, b4, A4, pop4, []], [push4, null, Z4, accept4, [Z4]], [pop4, null, Z4, accept4, [Z4]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a4, b4], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a4, b4], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a4, b4], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0081","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07e3a03dc20735307558c8a99f6dda19b2e33ba159a1c4a0624ebcad173bea75","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107081,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z5\",\"final_states\":[\"accept5\"],\"input\":[\"a5\",\"a5\",\"b5\",\"b5\"],\"input_alphabet\":[\"a5\",\"b5\"],\"max_steps\":1,\"stack_alphabet\":[\"A5\",\"Z5\"],\"start_state\":\"push5\",\"states\":[\"accept5\",\"pop5\",\"push5\"],\"transitions\":[[\"pop5\",\"b5\",\"A5\",\"pop5\",[]],[\"pop5\",null,\"Z5\",\"accept5\",[\"Z5\"]],[\"push5\",\"a5\",\"A5\",\"push5\",[\"A5\",\"A5\"]],[\"push5\",\"a5\",\"Z5\",\"push5\",[\"Z5\",\"A5\"]],[\"push5\",\"b5\",\"A5\",\"pop5\",[]],[\"push5\",null,\"Z5\",\"accept5\",[\"Z5\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push5, pop5, accept5], input_alphabet=[a5, b5], stack_alphabet=[Z5, A5], start_state=push5, bottom_symbol=Z5, final_states=[accept5], transitions=[[push5, a5, Z5, push5, [Z5, A5]], [push5, a5, A5, push5, [A5, A5]], [push5, b5, A5, pop5, []], [pop5, b5, A5, pop5, []], [push5, null, Z5, accept5, [Z5]], [pop5, null, Z5, accept5, [Z5]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a5, a5, b5, b5], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a5, a5, b5, b5], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a5, a5, b5, b5], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0082","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"505b7c13b73ba5f1f0f63856c64e5e7a96bd78358d7c05c9422a2a3f8c8c97bf","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107082,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z6\",\"final_states\":[\"accept6\"],\"input\":[\"a6\",\"a6\",\"a6\",\"b6\",\"b6\",\"b6\"],\"input_alphabet\":[\"a6\",\"b6\"],\"max_steps\":1,\"stack_alphabet\":[\"A6\",\"Z6\"],\"start_state\":\"push6\",\"states\":[\"accept6\",\"pop6\",\"push6\"],\"transitions\":[[\"pop6\",\"b6\",\"A6\",\"pop6\",[]],[\"pop6\",null,\"Z6\",\"accept6\",[\"Z6\"]],[\"push6\",\"a6\",\"A6\",\"push6\",[\"A6\",\"A6\"]],[\"push6\",\"a6\",\"Z6\",\"push6\",[\"Z6\",\"A6\"]],[\"push6\",\"b6\",\"A6\",\"pop6\",[]],[\"push6\",null,\"Z6\",\"accept6\",[\"Z6\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push6, pop6, accept6], input_alphabet=[a6, b6], stack_alphabet=[Z6, A6], start_state=push6, bottom_symbol=Z6, final_states=[accept6], transitions=[[push6, a6, Z6, push6, [Z6, A6]], [push6, a6, A6, push6, [A6, A6]], [push6, b6, A6, pop6, []], [pop6, b6, A6, pop6, []], [push6, null, Z6, accept6, [Z6]], [pop6, null, Z6, accept6, [Z6]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a6, a6, a6, b6, b6, b6], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a6, a6, a6, b6, b6, b6], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a6, a6, a6, b6, b6, b6], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0083","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6dc2548e749dc09f7ad423aa71bd2abf4de27996435f650c8d8d85df151e4d9c","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107083,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z7\",\"final_states\":[\"accept7\"],\"input\":[\"a7\",\"a7\",\"a7\",\"a7\",\"b7\",\"b7\",\"b7\",\"b7\"],\"input_alphabet\":[\"a7\",\"b7\"],\"max_steps\":1,\"stack_alphabet\":[\"A7\",\"Z7\"],\"start_state\":\"push7\",\"states\":[\"accept7\",\"pop7\",\"push7\"],\"transitions\":[[\"pop7\",\"b7\",\"A7\",\"pop7\",[]],[\"pop7\",null,\"Z7\",\"accept7\",[\"Z7\"]],[\"push7\",\"a7\",\"A7\",\"push7\",[\"A7\",\"A7\"]],[\"push7\",\"a7\",\"Z7\",\"push7\",[\"Z7\",\"A7\"]],[\"push7\",\"b7\",\"A7\",\"pop7\",[]],[\"push7\",null,\"Z7\",\"accept7\",[\"Z7\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push7, pop7, accept7], input_alphabet=[a7, b7], stack_alphabet=[Z7, A7], start_state=push7, bottom_symbol=Z7, final_states=[accept7], transitions=[[push7, a7, Z7, push7, [Z7, A7]], [push7, a7, A7, push7, [A7, A7]], [push7, b7, A7, pop7, []], [pop7, b7, A7, pop7, []], [push7, null, Z7, accept7, [Z7]], [pop7, null, Z7, accept7, [Z7]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a7, a7, a7, a7, b7, b7, b7, b7], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a7, a7, a7, a7, b7, b7, b7, b7], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a7, a7, a7, a7, b7, b7, b7, b7], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0084","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2db6aa6d120eb2bf4b04045b3a130efd57da7e024b4b8d71848385a6f61159e7","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107084,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z8\",\"final_states\":[\"accept8\"],\"input\":[\"a8\",\"b8\"],\"input_alphabet\":[\"a8\",\"b8\"],\"max_steps\":1,\"stack_alphabet\":[\"A8\",\"Z8\"],\"start_state\":\"push8\",\"states\":[\"accept8\",\"pop8\",\"push8\"],\"transitions\":[[\"pop8\",\"b8\",\"A8\",\"pop8\",[]],[\"pop8\",null,\"Z8\",\"accept8\",[\"Z8\"]],[\"push8\",\"a8\",\"A8\",\"push8\",[\"A8\",\"A8\"]],[\"push8\",\"a8\",\"Z8\",\"push8\",[\"Z8\",\"A8\"]],[\"push8\",\"b8\",\"A8\",\"pop8\",[]],[\"push8\",null,\"Z8\",\"accept8\",[\"Z8\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push8, pop8, accept8], input_alphabet=[a8, b8], stack_alphabet=[Z8, A8], start_state=push8, bottom_symbol=Z8, final_states=[accept8], transitions=[[push8, a8, Z8, push8, [Z8, A8]], [push8, a8, A8, push8, [A8, A8]], [push8, b8, A8, pop8, []], [pop8, b8, A8, pop8, []], [push8, null, Z8, accept8, [Z8]], [pop8, null, Z8, accept8, [Z8]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a8, b8], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a8, b8], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a8, b8], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0085","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cc4b207a326b14e6537bf6bd15c074fc826476c7bac9a772a3baf942235b3f4","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107085,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z9\",\"final_states\":[\"accept9\"],\"input\":[\"a9\",\"a9\",\"b9\",\"b9\"],\"input_alphabet\":[\"a9\",\"b9\"],\"max_steps\":1,\"stack_alphabet\":[\"A9\",\"Z9\"],\"start_state\":\"push9\",\"states\":[\"accept9\",\"pop9\",\"push9\"],\"transitions\":[[\"pop9\",\"b9\",\"A9\",\"pop9\",[]],[\"pop9\",null,\"Z9\",\"accept9\",[\"Z9\"]],[\"push9\",\"a9\",\"A9\",\"push9\",[\"A9\",\"A9\"]],[\"push9\",\"a9\",\"Z9\",\"push9\",[\"Z9\",\"A9\"]],[\"push9\",\"b9\",\"A9\",\"pop9\",[]],[\"push9\",null,\"Z9\",\"accept9\",[\"Z9\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push9, pop9, accept9], input_alphabet=[a9, b9], stack_alphabet=[Z9, A9], start_state=push9, bottom_symbol=Z9, final_states=[accept9], transitions=[[push9, a9, Z9, push9, [Z9, A9]], [push9, a9, A9, push9, [A9, A9]], [push9, b9, A9, pop9, []], [pop9, b9, A9, pop9, []], [push9, null, Z9, accept9, [Z9]], [pop9, null, Z9, accept9, [Z9]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a9, a9, b9, b9], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a9, a9, b9, b9], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a9, a9, b9, b9], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0086","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18338d0e202a048e56a5b9681e380359f35dfad5802e1997b90383ec108ac47c","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107086,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z10\",\"final_states\":[\"accept10\"],\"input\":[\"a10\",\"a10\",\"a10\",\"b10\",\"b10\",\"b10\"],\"input_alphabet\":[\"a10\",\"b10\"],\"max_steps\":1,\"stack_alphabet\":[\"A10\",\"Z10\"],\"start_state\":\"push10\",\"states\":[\"accept10\",\"pop10\",\"push10\"],\"transitions\":[[\"pop10\",\"b10\",\"A10\",\"pop10\",[]],[\"pop10\",null,\"Z10\",\"accept10\",[\"Z10\"]],[\"push10\",\"a10\",\"A10\",\"push10\",[\"A10\",\"A10\"]],[\"push10\",\"a10\",\"Z10\",\"push10\",[\"Z10\",\"A10\"]],[\"push10\",\"b10\",\"A10\",\"pop10\",[]],[\"push10\",null,\"Z10\",\"accept10\",[\"Z10\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push10, pop10, accept10], input_alphabet=[a10, b10], stack_alphabet=[Z10, A10], start_state=push10, bottom_symbol=Z10, final_states=[accept10], transitions=[[push10, a10, Z10, push10, [Z10, A10]], [push10, a10, A10, push10, [A10, A10]], [push10, b10, A10, pop10, []], [pop10, b10, A10, pop10, []], [push10, null, Z10, accept10, [Z10]], [pop10, null, Z10, accept10, [Z10]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a10, a10, a10, b10, b10, b10], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a10, a10, a10, b10, b10, b10], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a10, a10, a10, b10, b10, b10], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0087","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c82206911e3542fc1ca56206377925cf32725b23090ba8322f10c2101ec5c0a","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107087,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z11\",\"final_states\":[\"accept11\"],\"input\":[\"a11\",\"a11\",\"a11\",\"a11\",\"b11\",\"b11\",\"b11\",\"b11\"],\"input_alphabet\":[\"a11\",\"b11\"],\"max_steps\":1,\"stack_alphabet\":[\"A11\",\"Z11\"],\"start_state\":\"push11\",\"states\":[\"accept11\",\"pop11\",\"push11\"],\"transitions\":[[\"pop11\",\"b11\",\"A11\",\"pop11\",[]],[\"pop11\",null,\"Z11\",\"accept11\",[\"Z11\"]],[\"push11\",\"a11\",\"A11\",\"push11\",[\"A11\",\"A11\"]],[\"push11\",\"a11\",\"Z11\",\"push11\",[\"Z11\",\"A11\"]],[\"push11\",\"b11\",\"A11\",\"pop11\",[]],[\"push11\",null,\"Z11\",\"accept11\",[\"Z11\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push11, pop11, accept11], input_alphabet=[a11, b11], stack_alphabet=[Z11, A11], start_state=push11, bottom_symbol=Z11, final_states=[accept11], transitions=[[push11, a11, Z11, push11, [Z11, A11]], [push11, a11, A11, push11, [A11, A11]], [push11, b11, A11, pop11, []], [pop11, b11, A11, pop11, []], [push11, null, Z11, accept11, [Z11]], [pop11, null, Z11, accept11, [Z11]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a11, a11, a11, a11, b11, b11, b11, b11], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a11, a11, a11, a11, b11, b11, b11, b11], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a11, a11, a11, a11, b11, b11, b11, b11], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0088","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6440166261d92700ba760a5c1776cd30763c459e30f3707e5460be76b44b1e2b","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107088,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z12\",\"final_states\":[\"accept12\"],\"input\":[\"a12\",\"b12\"],\"input_alphabet\":[\"a12\",\"b12\"],\"max_steps\":1,\"stack_alphabet\":[\"A12\",\"Z12\"],\"start_state\":\"push12\",\"states\":[\"accept12\",\"pop12\",\"push12\"],\"transitions\":[[\"pop12\",\"b12\",\"A12\",\"pop12\",[]],[\"pop12\",null,\"Z12\",\"accept12\",[\"Z12\"]],[\"push12\",\"a12\",\"A12\",\"push12\",[\"A12\",\"A12\"]],[\"push12\",\"a12\",\"Z12\",\"push12\",[\"Z12\",\"A12\"]],[\"push12\",\"b12\",\"A12\",\"pop12\",[]],[\"push12\",null,\"Z12\",\"accept12\",[\"Z12\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push12, pop12, accept12], input_alphabet=[a12, b12], stack_alphabet=[Z12, A12], start_state=push12, bottom_symbol=Z12, final_states=[accept12], transitions=[[push12, a12, Z12, push12, [Z12, A12]], [push12, a12, A12, push12, [A12, A12]], [push12, b12, A12, pop12, []], [pop12, b12, A12, pop12, []], [push12, null, Z12, accept12, [Z12]], [pop12, null, Z12, accept12, [Z12]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a12, b12], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a12, b12], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a12, b12], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0089","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25d90e8026cda548792b30ba450f32267899c620c070e5367b23c3b2d5909c60","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107089,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z13\",\"final_states\":[\"accept13\"],\"input\":[\"a13\",\"a13\",\"b13\",\"b13\"],\"input_alphabet\":[\"a13\",\"b13\"],\"max_steps\":1,\"stack_alphabet\":[\"A13\",\"Z13\"],\"start_state\":\"push13\",\"states\":[\"accept13\",\"pop13\",\"push13\"],\"transitions\":[[\"pop13\",\"b13\",\"A13\",\"pop13\",[]],[\"pop13\",null,\"Z13\",\"accept13\",[\"Z13\"]],[\"push13\",\"a13\",\"A13\",\"push13\",[\"A13\",\"A13\"]],[\"push13\",\"a13\",\"Z13\",\"push13\",[\"Z13\",\"A13\"]],[\"push13\",\"b13\",\"A13\",\"pop13\",[]],[\"push13\",null,\"Z13\",\"accept13\",[\"Z13\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push13, pop13, accept13], input_alphabet=[a13, b13], stack_alphabet=[Z13, A13], start_state=push13, bottom_symbol=Z13, final_states=[accept13], transitions=[[push13, a13, Z13, push13, [Z13, A13]], [push13, a13, A13, push13, [A13, A13]], [push13, b13, A13, pop13, []], [pop13, b13, A13, pop13, []], [push13, null, Z13, accept13, [Z13]], [pop13, null, Z13, accept13, [Z13]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a13, a13, b13, b13], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a13, a13, b13, b13], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a13, a13, b13, b13], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0090","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27af4ea372db36d90f66487909b82765af1f74547e64a5ed66cb4416bdcadc29","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107090,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z14\",\"final_states\":[\"accept14\"],\"input\":[\"a14\",\"a14\",\"a14\",\"b14\",\"b14\",\"b14\"],\"input_alphabet\":[\"a14\",\"b14\"],\"max_steps\":1,\"stack_alphabet\":[\"A14\",\"Z14\"],\"start_state\":\"push14\",\"states\":[\"accept14\",\"pop14\",\"push14\"],\"transitions\":[[\"pop14\",\"b14\",\"A14\",\"pop14\",[]],[\"pop14\",null,\"Z14\",\"accept14\",[\"Z14\"]],[\"push14\",\"a14\",\"A14\",\"push14\",[\"A14\",\"A14\"]],[\"push14\",\"a14\",\"Z14\",\"push14\",[\"Z14\",\"A14\"]],[\"push14\",\"b14\",\"A14\",\"pop14\",[]],[\"push14\",null,\"Z14\",\"accept14\",[\"Z14\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push14, pop14, accept14], input_alphabet=[a14, b14], stack_alphabet=[Z14, A14], start_state=push14, bottom_symbol=Z14, final_states=[accept14], transitions=[[push14, a14, Z14, push14, [Z14, A14]], [push14, a14, A14, push14, [A14, A14]], [push14, b14, A14, pop14, []], [pop14, b14, A14, pop14, []], [push14, null, Z14, accept14, [Z14]], [pop14, null, Z14, accept14, [Z14]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a14, a14, a14, b14, b14, b14], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a14, a14, a14, b14, b14, b14], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a14, a14, a14, b14, b14, b14], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0091","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab5c761347c3158c513b0abbfe23cc24ad6c123caf6f841bf65b918a41336f77","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107091,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z15\",\"final_states\":[\"accept15\"],\"input\":[\"a15\",\"a15\",\"a15\",\"a15\",\"b15\",\"b15\",\"b15\",\"b15\"],\"input_alphabet\":[\"a15\",\"b15\"],\"max_steps\":1,\"stack_alphabet\":[\"A15\",\"Z15\"],\"start_state\":\"push15\",\"states\":[\"accept15\",\"pop15\",\"push15\"],\"transitions\":[[\"pop15\",\"b15\",\"A15\",\"pop15\",[]],[\"pop15\",null,\"Z15\",\"accept15\",[\"Z15\"]],[\"push15\",\"a15\",\"A15\",\"push15\",[\"A15\",\"A15\"]],[\"push15\",\"a15\",\"Z15\",\"push15\",[\"Z15\",\"A15\"]],[\"push15\",\"b15\",\"A15\",\"pop15\",[]],[\"push15\",null,\"Z15\",\"accept15\",[\"Z15\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push15, pop15, accept15], input_alphabet=[a15, b15], stack_alphabet=[Z15, A15], start_state=push15, bottom_symbol=Z15, final_states=[accept15], transitions=[[push15, a15, Z15, push15, [Z15, A15]], [push15, a15, A15, push15, [A15, A15]], [push15, b15, A15, pop15, []], [pop15, b15, A15, pop15, []], [push15, null, Z15, accept15, [Z15]], [pop15, null, Z15, accept15, [Z15]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a15, a15, a15, a15, b15, b15, b15, b15], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a15, a15, a15, a15, b15, b15, b15, b15], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a15, a15, a15, a15, b15, b15, b15, b15], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0092","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4825b0dc775fbc7076d44fa466a5ff326ce029a081cc30bddc3cdfea4102b4b8","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107092,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z16\",\"final_states\":[\"accept16\"],\"input\":[\"a16\",\"b16\"],\"input_alphabet\":[\"a16\",\"b16\"],\"max_steps\":1,\"stack_alphabet\":[\"A16\",\"Z16\"],\"start_state\":\"push16\",\"states\":[\"accept16\",\"pop16\",\"push16\"],\"transitions\":[[\"pop16\",\"b16\",\"A16\",\"pop16\",[]],[\"pop16\",null,\"Z16\",\"accept16\",[\"Z16\"]],[\"push16\",\"a16\",\"A16\",\"push16\",[\"A16\",\"A16\"]],[\"push16\",\"a16\",\"Z16\",\"push16\",[\"Z16\",\"A16\"]],[\"push16\",\"b16\",\"A16\",\"pop16\",[]],[\"push16\",null,\"Z16\",\"accept16\",[\"Z16\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push16, pop16, accept16], input_alphabet=[a16, b16], stack_alphabet=[Z16, A16], start_state=push16, bottom_symbol=Z16, final_states=[accept16], transitions=[[push16, a16, Z16, push16, [Z16, A16]], [push16, a16, A16, push16, [A16, A16]], [push16, b16, A16, pop16, []], [pop16, b16, A16, pop16, []], [push16, null, Z16, accept16, [Z16]], [pop16, null, Z16, accept16, [Z16]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a16, b16], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a16, b16], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a16, b16], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0093","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68ee6b4b009972c673c39c666a8cf6fb86fce54ca9e0273518f893cbf75abca6","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107093,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z17\",\"final_states\":[\"accept17\"],\"input\":[\"a17\",\"a17\",\"b17\",\"b17\"],\"input_alphabet\":[\"a17\",\"b17\"],\"max_steps\":1,\"stack_alphabet\":[\"A17\",\"Z17\"],\"start_state\":\"push17\",\"states\":[\"accept17\",\"pop17\",\"push17\"],\"transitions\":[[\"pop17\",\"b17\",\"A17\",\"pop17\",[]],[\"pop17\",null,\"Z17\",\"accept17\",[\"Z17\"]],[\"push17\",\"a17\",\"A17\",\"push17\",[\"A17\",\"A17\"]],[\"push17\",\"a17\",\"Z17\",\"push17\",[\"Z17\",\"A17\"]],[\"push17\",\"b17\",\"A17\",\"pop17\",[]],[\"push17\",null,\"Z17\",\"accept17\",[\"Z17\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push17, pop17, accept17], input_alphabet=[a17, b17], stack_alphabet=[Z17, A17], start_state=push17, bottom_symbol=Z17, final_states=[accept17], transitions=[[push17, a17, Z17, push17, [Z17, A17]], [push17, a17, A17, push17, [A17, A17]], [push17, b17, A17, pop17, []], [pop17, b17, A17, pop17, []], [push17, null, Z17, accept17, [Z17]], [pop17, null, Z17, accept17, [Z17]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a17, a17, b17, b17], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a17, a17, b17, b17], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a17, a17, b17, b17], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0094","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21027e28063552f3c2123244225a63b6e5a6a6a29c7284bf3e8a020cd791062e","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107094,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z18\",\"final_states\":[\"accept18\"],\"input\":[\"a18\",\"a18\",\"a18\",\"b18\",\"b18\",\"b18\"],\"input_alphabet\":[\"a18\",\"b18\"],\"max_steps\":1,\"stack_alphabet\":[\"A18\",\"Z18\"],\"start_state\":\"push18\",\"states\":[\"accept18\",\"pop18\",\"push18\"],\"transitions\":[[\"pop18\",\"b18\",\"A18\",\"pop18\",[]],[\"pop18\",null,\"Z18\",\"accept18\",[\"Z18\"]],[\"push18\",\"a18\",\"A18\",\"push18\",[\"A18\",\"A18\"]],[\"push18\",\"a18\",\"Z18\",\"push18\",[\"Z18\",\"A18\"]],[\"push18\",\"b18\",\"A18\",\"pop18\",[]],[\"push18\",null,\"Z18\",\"accept18\",[\"Z18\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push18, pop18, accept18], input_alphabet=[a18, b18], stack_alphabet=[Z18, A18], start_state=push18, bottom_symbol=Z18, final_states=[accept18], transitions=[[push18, a18, Z18, push18, [Z18, A18]], [push18, a18, A18, push18, [A18, A18]], [push18, b18, A18, pop18, []], [pop18, b18, A18, pop18, []], [push18, null, Z18, accept18, [Z18]], [pop18, null, Z18, accept18, [Z18]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a18, a18, a18, b18, b18, b18], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a18, a18, a18, b18, b18, b18], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a18, a18, a18, b18, b18, b18], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0095","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69e8ea005befa6831266556ff9b6efe28ad079845a1ebb09fa067651691b6ffe","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107095,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z19\",\"final_states\":[\"accept19\"],\"input\":[\"a19\",\"a19\",\"a19\",\"a19\",\"b19\",\"b19\",\"b19\",\"b19\"],\"input_alphabet\":[\"a19\",\"b19\"],\"max_steps\":1,\"stack_alphabet\":[\"A19\",\"Z19\"],\"start_state\":\"push19\",\"states\":[\"accept19\",\"pop19\",\"push19\"],\"transitions\":[[\"pop19\",\"b19\",\"A19\",\"pop19\",[]],[\"pop19\",null,\"Z19\",\"accept19\",[\"Z19\"]],[\"push19\",\"a19\",\"A19\",\"push19\",[\"A19\",\"A19\"]],[\"push19\",\"a19\",\"Z19\",\"push19\",[\"Z19\",\"A19\"]],[\"push19\",\"b19\",\"A19\",\"pop19\",[]],[\"push19\",null,\"Z19\",\"accept19\",[\"Z19\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push19, pop19, accept19], input_alphabet=[a19, b19], stack_alphabet=[Z19, A19], start_state=push19, bottom_symbol=Z19, final_states=[accept19], transitions=[[push19, a19, Z19, push19, [Z19, A19]], [push19, a19, A19, push19, [A19, A19]], [push19, b19, A19, pop19, []], [pop19, b19, A19, pop19, []], [push19, null, Z19, accept19, [Z19]], [pop19, null, Z19, accept19, [Z19]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a19, a19, a19, a19, b19, b19, b19, b19], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a19, a19, a19, a19, b19, b19, b19, b19], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a19, a19, a19, a19, b19, b19, b19, b19], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0096","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ce5c224a281e8f888478adf0d0ffb563f071507373558bc6ce95147f71d426e","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107096,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z20\",\"final_states\":[\"accept20\"],\"input\":[\"a20\",\"b20\"],\"input_alphabet\":[\"a20\",\"b20\"],\"max_steps\":1,\"stack_alphabet\":[\"A20\",\"Z20\"],\"start_state\":\"push20\",\"states\":[\"accept20\",\"pop20\",\"push20\"],\"transitions\":[[\"pop20\",\"b20\",\"A20\",\"pop20\",[]],[\"pop20\",null,\"Z20\",\"accept20\",[\"Z20\"]],[\"push20\",\"a20\",\"A20\",\"push20\",[\"A20\",\"A20\"]],[\"push20\",\"a20\",\"Z20\",\"push20\",[\"Z20\",\"A20\"]],[\"push20\",\"b20\",\"A20\",\"pop20\",[]],[\"push20\",null,\"Z20\",\"accept20\",[\"Z20\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push20, pop20, accept20], input_alphabet=[a20, b20], stack_alphabet=[Z20, A20], start_state=push20, bottom_symbol=Z20, final_states=[accept20], transitions=[[push20, a20, Z20, push20, [Z20, A20]], [push20, a20, A20, push20, [A20, A20]], [push20, b20, A20, pop20, []], [pop20, b20, A20, pop20, []], [push20, null, Z20, accept20, [Z20]], [pop20, null, Z20, accept20, [Z20]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a20, b20], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a20, b20], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a20, b20], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0097","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74560ca40d5951886fa5cc3ae65d602ac25d1adaa2cab9317051207a2ac1bcd8","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107097,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z21\",\"final_states\":[\"accept21\"],\"input\":[\"a21\",\"a21\",\"b21\",\"b21\"],\"input_alphabet\":[\"a21\",\"b21\"],\"max_steps\":1,\"stack_alphabet\":[\"A21\",\"Z21\"],\"start_state\":\"push21\",\"states\":[\"accept21\",\"pop21\",\"push21\"],\"transitions\":[[\"pop21\",\"b21\",\"A21\",\"pop21\",[]],[\"pop21\",null,\"Z21\",\"accept21\",[\"Z21\"]],[\"push21\",\"a21\",\"A21\",\"push21\",[\"A21\",\"A21\"]],[\"push21\",\"a21\",\"Z21\",\"push21\",[\"Z21\",\"A21\"]],[\"push21\",\"b21\",\"A21\",\"pop21\",[]],[\"push21\",null,\"Z21\",\"accept21\",[\"Z21\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push21, pop21, accept21], input_alphabet=[a21, b21], stack_alphabet=[Z21, A21], start_state=push21, bottom_symbol=Z21, final_states=[accept21], transitions=[[push21, a21, Z21, push21, [Z21, A21]], [push21, a21, A21, push21, [A21, A21]], [push21, b21, A21, pop21, []], [pop21, b21, A21, pop21, []], [push21, null, Z21, accept21, [Z21]], [pop21, null, Z21, accept21, [Z21]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a21, a21, b21, b21], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a21, a21, b21, b21], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a21, a21, b21, b21], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0098","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd1e9fbad95dad2b7a091f14831cd14aa0f8ce7367aa7a411e802ea75751fd60","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107098,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z22\",\"final_states\":[\"accept22\"],\"input\":[\"a22\",\"a22\",\"a22\",\"b22\",\"b22\",\"b22\"],\"input_alphabet\":[\"a22\",\"b22\"],\"max_steps\":1,\"stack_alphabet\":[\"A22\",\"Z22\"],\"start_state\":\"push22\",\"states\":[\"accept22\",\"pop22\",\"push22\"],\"transitions\":[[\"pop22\",\"b22\",\"A22\",\"pop22\",[]],[\"pop22\",null,\"Z22\",\"accept22\",[\"Z22\"]],[\"push22\",\"a22\",\"A22\",\"push22\",[\"A22\",\"A22\"]],[\"push22\",\"a22\",\"Z22\",\"push22\",[\"Z22\",\"A22\"]],[\"push22\",\"b22\",\"A22\",\"pop22\",[]],[\"push22\",null,\"Z22\",\"accept22\",[\"Z22\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push22, pop22, accept22], input_alphabet=[a22, b22], stack_alphabet=[Z22, A22], start_state=push22, bottom_symbol=Z22, final_states=[accept22], transitions=[[push22, a22, Z22, push22, [Z22, A22]], [push22, a22, A22, push22, [A22, A22]], [push22, b22, A22, pop22, []], [pop22, b22, A22, pop22, []], [push22, null, Z22, accept22, [Z22]], [pop22, null, Z22, accept22, [Z22]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a22, a22, a22, b22, b22, b22], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a22, a22, a22, b22, b22, b22], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a22, a22, a22, b22, b22, b22], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0099","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"381769b3ed6e28ddeae0f29f46c25d0b517c6b069ddd1aeb0bc65f2115094530","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107099,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z23\",\"final_states\":[\"accept23\"],\"input\":[\"a23\",\"a23\",\"a23\",\"a23\",\"b23\",\"b23\",\"b23\",\"b23\"],\"input_alphabet\":[\"a23\",\"b23\"],\"max_steps\":1,\"stack_alphabet\":[\"A23\",\"Z23\"],\"start_state\":\"push23\",\"states\":[\"accept23\",\"pop23\",\"push23\"],\"transitions\":[[\"pop23\",\"b23\",\"A23\",\"pop23\",[]],[\"pop23\",null,\"Z23\",\"accept23\",[\"Z23\"]],[\"push23\",\"a23\",\"A23\",\"push23\",[\"A23\",\"A23\"]],[\"push23\",\"a23\",\"Z23\",\"push23\",[\"Z23\",\"A23\"]],[\"push23\",\"b23\",\"A23\",\"pop23\",[]],[\"push23\",null,\"Z23\",\"accept23\",[\"Z23\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push23, pop23, accept23], input_alphabet=[a23, b23], stack_alphabet=[Z23, A23], start_state=push23, bottom_symbol=Z23, final_states=[accept23], transitions=[[push23, a23, Z23, push23, [Z23, A23]], [push23, a23, A23, push23, [A23, A23]], [push23, b23, A23, pop23, []], [pop23, b23, A23, pop23, []], [push23, null, Z23, accept23, [Z23]], [pop23, null, Z23, accept23, [Z23]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a23, a23, a23, a23, b23, b23, b23, b23], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a23, a23, a23, a23, b23, b23, b23, b23], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a23, a23, a23, a23, b23, b23, b23, b23], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261110-r07-pushdown-automaton-0100","batch_id":"batch-20261014-advance-20261110-r07-pushdown-automaton-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"683999dea9361ef8fde1f33171180787dfb856d293fe69025d923f8fa6f97b8c","family_id":"qf-mod41-pda-bounded-search-boundary","seed":202611107100,"semantic_template":"qf-mod41-pda-bounded-search-boundary/{\"bottom_symbol\":\"Z24\",\"final_states\":[\"accept24\"],\"input\":[\"a24\",\"b24\"],\"input_alphabet\":[\"a24\",\"b24\"],\"max_steps\":1,\"stack_alphabet\":[\"A24\",\"Z24\"],\"start_state\":\"push24\",\"states\":[\"accept24\",\"pop24\",\"push24\"],\"transitions\":[[\"pop24\",\"b24\",\"A24\",\"pop24\",[]],[\"pop24\",null,\"Z24\",\"accept24\",[\"Z24\"]],[\"push24\",\"a24\",\"A24\",\"push24\",[\"A24\",\"A24\"]],[\"push24\",\"a24\",\"Z24\",\"push24\",[\"Z24\",\"A24\"]],[\"push24\",\"b24\",\"A24\",\"pop24\",[]],[\"push24\",null,\"Z24\",\"accept24\",[\"Z24\"]]]}"},"fields":{"title":"A Bounded PDA No-Hit Cannot Certify Rejection","stem":"Use the complete finite nondeterministic PDA: states=[push24, pop24, accept24], input_alphabet=[a24, b24], stack_alphabet=[Z24, A24], start_state=push24, bottom_symbol=Z24, final_states=[accept24], transitions=[[push24, a24, Z24, push24, [Z24, A24]], [push24, a24, A24, push24, [A24, A24]], [push24, b24, A24, pop24, []], [pop24, b24, A24, pop24, []], [push24, null, Z24, accept24, [Z24]], [pop24, null, Z24, accept24, [Z24]]]. Alphabets are unique and pairwise disjoint. Each transition is [from,input symbol or null,required top,to,replacement bottom-to-top]. null is epsilon and consumes no input. Configurations are [state,input position,stack] with bottom first and top last. Preserve the bottom marker and accept only after consuming the entire input in a final state, not by empty stack. Input=[a24, b24], max_steps=1. Give (input, max_steps, found_accepting_run, rejection_or_nonmembership_claim, boundary).","answer":{"kind":"expression","value":"([a24, b24], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)","explanation":"Search from the initial configuration for a fully consumed final-state run, bounding steps and stack length at input.length+max_steps+2. The found flag concerns this search only. The fixed No withholds rejection/nonmembership, even on failure; a longer accepting run may exist. No empty-stack convention, determinism or CFG equivalence is inferred. The exact result, retaining the requested order, is ([a24, b24], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)."},"skills":["A Bounded PDA No-Hit Cannot Certify Rejection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}