Original question and givens
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).
G001 — 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).
Step-by-step derivation
S001 · implies
Before
source operation=pda-accepting-run-witness;original snapshot and mathematics retained
After
explicit ordered source output fields=[input, trace, transition_indices, valid_run, accepted, step_count, 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]=[push19, a19, Z19, push19, [Z19, A19]];bottom=Z19
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]=[push19, a19, A19, push19, [A19, A19]];bottom=Z19
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]=[push19, b19, A19, pop19, []];bottom=Z19
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]=[pop19, b19, A19, pop19, []];bottom=Z19
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]=[push19, null, Z19, accept19, [Z19]];bottom=Z19
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]=[pop19, null, Z19, accept19, [Z19]];bottom=Z19
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=[push19, pop19, accept19],input alphabet=[a19, b19],stack alphabet=[Z19, A19];start=push19,bottom=Z19,final states=[accept19]
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=[a19, a19, a19, b19, b19, b19];ALL source configurations=[[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]]];declared start=push19,bottom=Z19
After
first configuration EXACTLY=[push19, 0, [Z19]];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=[push19, 0, [Z19]];to=[push19, 1, [Z19, A19]];actual input=[a19, a19, a19, b19, b19, b19]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, [push19, 1, [Z19, A19]]], [1, null], [2, null], [3, null], [4, [accept19, 0, [Z19]]], [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=[push19, 1, [Z19, A19]];to=[push19, 2, [Z19, A19, A19]];actual input=[a19, a19, a19, b19, b19, b19]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, [push19, 2, [Z19, A19, A19]]], [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=[push19, 2, [Z19, A19, A19]];to=[push19, 3, [Z19, A19, A19, A19]];actual input=[a19, a19, a19, b19, b19, b19]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, [push19, 3, [Z19, A19, A19, A19]]], [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, S012
S014 · implies
Before
ORIGINAL trace step=3;from=[push19, 3, [Z19, A19, A19, A19]];to=[pop19, 4, [Z19, A19, A19]];actual input=[a19, a19, a19, b19, b19, b19]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, [pop19, 4, [Z19, A19, A19]]], [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, S013
S015 · implies
Before
ORIGINAL trace step=4;from=[pop19, 4, [Z19, A19, A19]];to=[pop19, 5, [Z19, A19]];actual input=[a19, a19, a19, b19, b19, b19]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, [pop19, 5, [Z19, A19]]], [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, S014
S016 · implies
Before
ORIGINAL trace step=5;from=[pop19, 5, [Z19, A19]];to=[pop19, 6, [Z19]];actual input=[a19, a19, a19, b19, b19, b19]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, [pop19, 6, [Z19]]], [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, S015
S017 · implies
Before
ORIGINAL trace step=6;from=[pop19, 6, [Z19]];to=[accept19, 6, [Z19]];actual input=[a19, a19, a19, b19, b19, b19]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, null], [4, null], [5, [accept19, 6, [Z19]]]];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, S016
S018 · implies
Before
ORIGINAL last config=[accept19, 6, [Z19]];input length=6;matched prior indices=[0, 1, 1, 2, 3, 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, S017
S019 · calculation
Before
PDA ORIGINAL full trace length minus one := sub(8, 1)
After
PDA ORIGINAL full trace length minus one = 7
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, S018
S020 · implies
Before
ORIGINAL trace configuration count=8;actual matched transition-index count=7;accepted=Yes
After
SOURCE step_count=ORIGINAL trace.length-1=7;does NOT substitute partial matched count;acceptedTrue is a finite witness,not search completeness
Conditions: none
The source step count uses the whole supplied trace length,even on a partial failure.Only accepted=True supplies the finite accepting-run witness;it proves neither all runs,shortest length nor search completeness.
Step identifiers and dependencies
Rule: pda_original_step_count_not_partial_matched_count · Uses: G001, S019
Result
([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)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.