Explicit PDA Stack Runs, Full-Input Acceptance and Bounded-Search Nonconclusions
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261110-r07-pushdown-automaton-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push0, pop0, accept0], [a0, b0], [Z0, A0], push0, Z0, [accept0], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
2. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push1, pop1, accept1], [a1, b1], [Z1, A1], push1, Z1, [accept1], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
3. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push2, pop2, accept2], [a2, b2], [Z2, A2], push2, Z2, [accept2], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
4. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push3, pop3, accept3], [a3, b3], [Z3, A3], push3, Z3, [accept3], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
5. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push4, pop4, accept4], [a4, b4], [Z4, A4], push4, Z4, [accept4], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
6. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push5, pop5, accept5], [a5, b5], [Z5, A5], push5, Z5, [accept5], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
7. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push6, pop6, accept6], [a6, b6], [Z6, A6], push6, Z6, [accept6], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
8. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push7, pop7, accept7], [a7, b7], [Z7, A7], push7, Z7, [accept7], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
9. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push8, pop8, accept8], [a8, b8], [Z8, A8], push8, Z8, [accept8], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
10. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push9, pop9, accept9], [a9, b9], [Z9, A9], push9, Z9, [accept9], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
11. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push10, pop10, accept10], [a10, b10], [Z10, A10], push10, Z10, [accept10], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
12. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push11, pop11, accept11], [a11, b11], [Z11, A11], push11, Z11, [accept11], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
13. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push12, pop12, accept12], [a12, b12], [Z12, A12], push12, Z12, [accept12], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
14. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push13, pop13, accept13], [a13, b13], [Z13, A13], push13, Z13, [accept13], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
15. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push14, pop14, accept14], [a14, b14], [Z14, A14], push14, Z14, [accept14], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
16. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push15, pop15, accept15], [a15, b15], [Z15, A15], push15, Z15, [accept15], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
17. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push16, pop16, accept16], [a16, b16], [Z16, A16], push16, Z16, [accept16], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
18. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push17, pop17, accept17], [a17, b17], [Z17, A17], push17, Z17, [accept17], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
19. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push18, pop18, accept18], [a18, b18], [Z18, A18], push18, Z18, [accept18], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
20. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push19, pop19, accept19], [a19, b19], [Z19, A19], push19, Z19, [accept19], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
21. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push20, pop20, accept20], [a20, b20], [Z20, A20], push20, Z20, [accept20], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
22. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push21, pop21, accept21], [a21, b21], [Z21, A21], push21, Z21, [accept21], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
23. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push22, pop22, accept22], [a22, b22], [Z22, A22], push22, Z22, [accept22], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
24. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push23, pop23, accept23], [a23, b23], [Z23, A23], push23, Z23, [accept23], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
25. A Complete PDA Definition Retains Its Final-State Acceptance Convention
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).
View answer and explanation
([push24, pop24, accept24], [a24, b24], [Z24, A24], push24, Z24, [accept24], 6, Yes, accept_only_after_full_input_in_final_state_bottom_marker_preserved)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-qualification · candidate
26. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([], [[push0, 0, [Z0]], [accept0, 0, [Z0]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
27. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
28. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
29. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
30. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([], [[push4, 0, [Z4]], [accept4, 0, [Z4]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
31. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
32. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
33. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
34. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([], [[push8, 0, [Z8]], [accept8, 0, [Z8]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
35. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
36. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
37. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
38. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([], [[push12, 0, [Z12]], [accept12, 0, [Z12]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
39. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
40. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
41. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
42. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([], [[push16, 0, [Z16]], [accept16, 0, [Z16]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
43. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
44. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
45. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
46. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([], [[push20, 0, [Z20]], [accept20, 0, [Z20]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
47. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
48. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
49. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
50. Each Explicit Stack Step Must Respect Input and Epsilon Consumption
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).
valid_run and accepted are separate: a final state before all input is consumed is not acceptance. The bottom is retained; this is not empty-stack acceptance.
View answer and explanation
([], [[push24, 0, [Z24]], [accept24, 0, [Z24]]], [4], Yes, Yes, epsilon_moves_consume_no_input_and_each_stack_update_is_explicit)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-trace-audit · candidate
51. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([], [[push0, 0, [Z0]], [accept0, 0, [Z0]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
52. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
53. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
54. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
55. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([], [[push4, 0, [Z4]], [accept4, 0, [Z4]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
56. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
57. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
58. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
59. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([], [[push8, 0, [Z8]], [accept8, 0, [Z8]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
60. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
61. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
62. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
63. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([], [[push12, 0, [Z12]], [accept12, 0, [Z12]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
64. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
65. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
66. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
67. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([], [[push16, 0, [Z16]], [accept16, 0, [Z16]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
68. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
69. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
70. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
71. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([], [[push20, 0, [Z20]], [accept20, 0, [Z20]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
72. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
73. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
74. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([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)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
75. A Finite Valid Full-Input Run Proves Acceptance, Not Search Completeness
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).
View answer and explanation
([], [[push24, 0, [Z24]], [accept24, 0, [Z24]]], [4], Yes, Yes, 1, finite_full_run_witness_proves_acceptance_under_final_state_convention)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-accepting-run-witness · candidate
76. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a0, b0], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
77. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a1, a1, b1, b1], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
78. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a2, a2, a2, b2, b2, b2], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
79. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a3, a3, a3, a3, b3, b3, b3, b3], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
80. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a4, b4], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
81. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a5, a5, b5, b5], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
82. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a6, a6, a6, b6, b6, b6], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
83. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a7, a7, a7, a7, b7, b7, b7, b7], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
84. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a8, b8], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
85. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a9, a9, b9, b9], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
86. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a10, a10, a10, b10, b10, b10], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
87. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a11, a11, a11, a11, b11, b11, b11, b11], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
88. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a12, b12], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
89. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a13, a13, b13, b13], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
90. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a14, a14, a14, b14, b14, b14], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
91. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a15, a15, a15, a15, b15, b15, b15, b15], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
92. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a16, b16], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
93. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a17, a17, b17, b17], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
94. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a18, a18, a18, b18, b18, b18], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
95. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a19, a19, a19, a19, b19, b19, b19, b19], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
96. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a20, b20], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
97. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a21, a21, b21, b21], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
98. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a22, a22, a22, b22, b22, b22], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
99. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a23, a23, a23, a23, b23, b23, b23, b23], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
100. A Bounded PDA No-Hit Cannot Certify Rejection
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).
Search bounds both steps and stack length. A no-hit is not rejection or nonmembership proof, and the fixed No does not negate the separate found flag.
View answer and explanation
([a24, b24], 1, No, No, bounded_nondeterministic_search_failure_cannot_prove_rejection_or_nonmembership)
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).
Reviewed step-by-step derivation
Pushdown automaton · qf-mod41-pda-bounded-search-boundary · candidate
Canonical question JSONL · English JSONL · Validation report