Original question and givens
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).
G001 — 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).
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]=[push1, a1, Z1, push1, [Z1, A1]];bottom=Z1
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]=[push1, a1, A1, push1, [A1, A1]];bottom=Z1
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]=[push1, b1, A1, pop1, []];bottom=Z1
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]=[pop1, b1, A1, pop1, []];bottom=Z1
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]=[push1, null, Z1, accept1, [Z1]];bottom=Z1
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]=[pop1, null, Z1, accept1, [Z1]];bottom=Z1
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=[push1, pop1, accept1],input alphabet=[a1, b1],stack alphabet=[Z1, A1];start=push1,bottom=Z1,final states=[accept1]
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=[a1, b1];ALL source configurations=[[push1, 0, [Z1]], [push1, 1, [Z1, A1]], [pop1, 2, [Z1]], [accept1, 2, [Z1]]];declared start=push1,bottom=Z1
After
first configuration EXACTLY=[push1, 0, [Z1]];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=[push1, 0, [Z1]];to=[push1, 1, [Z1, A1]];actual input=[a1, b1]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, [push1, 1, [Z1, A1]]], [1, null], [2, null], [3, null], [4, [accept1, 0, [Z1]]], [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=[push1, 1, [Z1, A1]];to=[pop1, 2, [Z1]];actual input=[a1, b1]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, [pop1, 2, [Z1]]], [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, S011
S013 · implies
Before
ORIGINAL trace step=2;from=[pop1, 2, [Z1]];to=[accept1, 2, [Z1]];actual input=[a1, b1]
After
ALL [ORIGINAL transition index,applied_result_or_null]=[[0, null], [1, null], [2, null], [3, null], [4, null], [5, [accept1, 2, [Z1]]]];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, S012
S014 · implies
Before
ORIGINAL last config=[accept1, 2, [Z1]];input length=2;matched prior indices=[0, 2, 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, S013
S015 · calculation
Before
PDA ORIGINAL full trace length minus one := sub(4, 1)
After
PDA ORIGINAL full trace length minus one = 3
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, S014
S016 · implies
Before
ORIGINAL trace configuration count=4;actual matched transition-index count=3;accepted=Yes
After
SOURCE step_count=ORIGINAL trace.length-1=3;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, S015