Original question and givens
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).
G001 — 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).
Step-by-step derivation
S001 · implies
Before
source operation=pda-trace-audit;original snapshot and mathematics retained
After
explicit ordered source output fields=[input, trace, transition_indices, valid_run, accepted, certificate]
Conditions: none
Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL transition index=0;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push14, a14, Z14, push14, [Z14, A14]];bottom=Z14
After
states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained
Conditions: none
Check each original transition's state,input and stack roles without mixing alphabets.Null is epsilon,not an input symbol.The stack is bottom-first/top-last:pop one top then append the replacement in its original order.Apply the declared bottom guard,not an invented stronger convention.
Step identifiers and dependencies
Rule: pda_EVERY_original_transition_roles_epsilon_bottom_guard · Uses: G001, S001
S003 · implies
Before
ORIGINAL transition index=1;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push14, a14, A14, push14, [A14, A14]];bottom=Z14
After
states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained
Conditions: none
Check each original transition's state,input and stack roles without mixing alphabets.Null is epsilon,not an input symbol.The stack is bottom-first/top-last:pop one top then append the replacement in its original order.Apply the declared bottom guard,not an invented stronger convention.
Step identifiers and dependencies
Rule: pda_EVERY_original_transition_roles_epsilon_bottom_guard · Uses: G001, S002
S004 · implies
Before
ORIGINAL transition index=2;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push14, b14, A14, pop14, []];bottom=Z14
After
states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained
Conditions: none
Check each original transition's state,input and stack roles without mixing alphabets.Null is epsilon,not an input symbol.The stack is bottom-first/top-last:pop one top then append the replacement in its original order.Apply the declared bottom guard,not an invented stronger convention.
Step identifiers and dependencies
Rule: pda_EVERY_original_transition_roles_epsilon_bottom_guard · Uses: G001, S003
S005 · implies
Before
ORIGINAL transition index=3;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[pop14, b14, A14, pop14, []];bottom=Z14
After
states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained
Conditions: none
Check each original transition's state,input and stack roles without mixing alphabets.Null is epsilon,not an input symbol.The stack is bottom-first/top-last:pop one top then append the replacement in its original order.Apply the declared bottom guard,not an invented stronger convention.
Step identifiers and dependencies
Rule: pda_EVERY_original_transition_roles_epsilon_bottom_guard · Uses: G001, S004
S006 · implies
Before
ORIGINAL transition index=4;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[push14, null, Z14, accept14, [Z14]];bottom=Z14
After
states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained
Conditions: none
Check each original transition's state,input and stack roles without mixing alphabets.Null is epsilon,not an input symbol.The stack is bottom-first/top-last:pop one top then append the replacement in its original order.Apply the declared bottom guard,not an invented stronger convention.
Step identifiers and dependencies
Rule: pda_EVERY_original_transition_roles_epsilon_bottom_guard · Uses: G001, S005
S007 · implies
Before
ORIGINAL transition index=5;[from,input_or_null,required_top,to,replacement_bottom_to_top]=[pop14, null, Z14, accept14, [Z14]];bottom=Z14
After
states/input/stack symbols legal;null input consumes0;top=bottom requires replacement first bottom,otherwise replacement contains NO bottom;ORIGINAL transition position retained
Conditions: none
Check each original transition's state,input and stack roles without mixing alphabets.Null is epsilon,not an input symbol.The stack is bottom-first/top-last:pop one top then append the replacement in its original order.Apply the declared bottom guard,not an invented stronger convention.
Step identifiers and dependencies
Rule: pda_EVERY_original_transition_roles_epsilon_bottom_guard · Uses: G001, S006
S008 · calculation
Before
PDA complete ORIGINAL transition count := sum(1, 1, 1, 1, 1, 1)
After
PDA complete ORIGINAL transition count = 6
Conditions: none
Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · implies
Before
ORIGINAL states=[push14, pop14, accept14],input alphabet=[a14, b14],stack alphabet=[Z14, A14];start=push14,bottom=Z14,final states=[accept14]
After
finite unique pairwise-disjoint alphabets qualified;transition count=6;acceptance convention FULL input in final state,NOT empty stack;qualification alone proves NO run,determinism or CFG equivalence
Conditions: none
A qualified complete definition gives no accepting run,determinism or CFG-equivalence proof.Acceptance may retain a nonempty stack,including its bottom,but requires full input consumption in a final state.
Step identifiers and dependencies
Rule: pda_complete_definition_not_determinism_or_accepting_run · Uses: G001, S008
S010 · implies
Before
ORIGINAL input=[a14, a14, b14, b14];ALL source configurations=[[push14, 0, [Z14]], [push14, 1, [Z14, A14]], [push14, 2, [Z14, A14, A14]], [pop14, 3, [Z14, A14]], [pop14, 4, [Z14]], [accept14, 4, [Z14]]];declared start=push14,bottom=Z14
After
first configuration EXACTLY=[push14, 0, [Z14]];all state/position/stack values legal;this is NOT yet a valid-run or acceptance certificate
Conditions: none
Require the exact initial state,position0 and one bottom.All configurations must be structurally legal before checking their transitions.A legal configuration list alone need not be a valid run.
Step identifiers and dependencies
Rule: pda_trace_ALL_configurations_qualified_actual_initial_required · Uses: G001, S009
S011 · implies
Before
ORIGINAL trace step=0;from=[push14, 0, [Z14]];to=[push14, 1, [Z14, A14]];actual input=[a14, a14, b14, b14]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, [push14, 1, [Z14, A14]]], [1, null], [2, null], [3, null], [4, [accept14, 0, [Z14]]], [5, null]];FIRST matching index=0
Conditions: none
Select the first original transition that actually yields the next configuration.Input-or-null determines consumption;replace only the top and preserve replacement order.On failure retain prior partial indices;a later final state cannot repair a bad step.
Step identifiers and dependencies
Rule: pda_each_actual_stack_update_FIRST_original_transition · Uses: G001, S010
S012 · implies
Before
ORIGINAL trace step=1;from=[push14, 1, [Z14, A14]];to=[push14, 2, [Z14, A14, A14]];actual input=[a14, a14, b14, b14]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, [push14, 2, [Z14, A14, A14]]], [2, null], [3, null], [4, null], [5, null]];FIRST matching index=1
Conditions: none
Select the first original transition that actually yields the next configuration.Input-or-null determines consumption;replace only the top and preserve replacement order.On failure retain prior partial indices;a later final state cannot repair a bad step.
Step identifiers and dependencies
Rule: pda_each_actual_stack_update_FIRST_original_transition · Uses: G001, S011
S013 · implies
Before
ORIGINAL trace step=2;from=[push14, 2, [Z14, A14, A14]];to=[pop14, 3, [Z14, A14]];actual input=[a14, a14, b14, b14]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, [pop14, 3, [Z14, A14]]], [3, null], [4, null], [5, null]];FIRST matching index=2
Conditions: none
Select the first original transition that actually yields the next configuration.Input-or-null determines consumption;replace only the top and preserve replacement order.On failure retain prior partial indices;a later final state cannot repair a bad step.
Step identifiers and dependencies
Rule: pda_each_actual_stack_update_FIRST_original_transition · Uses: G001, S012
S014 · implies
Before
ORIGINAL trace step=3;from=[pop14, 3, [Z14, A14]];to=[pop14, 4, [Z14]];actual input=[a14, a14, b14, b14]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, [pop14, 4, [Z14]]], [4, null], [5, null]];FIRST matching index=3
Conditions: none
Select the first original transition that actually yields the next configuration.Input-or-null determines consumption;replace only the top and preserve replacement order.On failure retain prior partial indices;a later final state cannot repair a bad step.
Step identifiers and dependencies
Rule: pda_each_actual_stack_update_FIRST_original_transition · Uses: G001, S013
S015 · implies
Before
ORIGINAL trace step=4;from=[pop14, 4, [Z14]];to=[accept14, 4, [Z14]];actual input=[a14, a14, b14, b14]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, null], [4, null], [5, [accept14, 4, [Z14]]]];FIRST matching index=5
Conditions: none
Select the first original transition that actually yields the next configuration.Input-or-null determines consumption;replace only the top and preserve replacement order.On failure retain prior partial indices;a later final state cannot repair a bad step.
Step identifiers and dependencies
Rule: pda_each_actual_stack_update_FIRST_original_transition · Uses: G001, S014
S016 · implies
Before
ORIGINAL last config=[accept14, 4, [Z14]];input length=4;matched prior indices=[0, 1, 2, 3, 5]
After
valid_run=Yes;FULL input consumed=Yes;last state final=Yes;accepted=Yes;residual stack may be NONEMPTY
Conditions: none
Acceptance conjoins a valid run,full input consumption and a final state.A residual bottom or other stack symbols may remain.Validity and acceptance differ;the tag cannot turn a false flag into a certificate.
Step identifiers and dependencies
Rule: pda_run_validity_full_consumption_final_state_SEPARATE · Uses: G001, S015